Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 194704 5 pageshttpdxdoiorg1011552013194704

Research ArticleSelf-Similar Solutions of the CompressibleFlow in One-Space Dimension

Tailong Li1 Ping Chen2 and Jian Xie3

1 School of Economics amp Management Zhejiang Sci-Tech University Hangzhou 310018 China2 Shanghai Baosteel Industry Technological Service Co Ltd Tongji Road 3521 Baoshan Area Shanghai 201900 China3Department of Mathematics Hangzhou Normal University Hangzhou 310016 China

Correspondence should be addressed to Tailong Li m9845163com

Received 29 July 2013 Accepted 17 September 2013

Academic Editor Hui-Shen Shen

Copyright copy 2013 Tailong Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For the isentropic compressible fluids in one-space dimension we prove that the Navier-Stokes equations with density-dependentviscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy Moreover we obtain thesame result for the nonisentropic compressible gas flow that is for the fluid dynamics of the Navier-Stokes equations coupledwith a transport equation of entropy These results generalize those in Guo and Jiangrsquos work (2006) where the one-dimensionalcompressible fluids with constant viscosity are considered

1 Introduction

Self-similar solutions have attracted much attention in math-ematical physics because understanding them is fundamentaland important for investigating the well-posedness regu-larity and asymptotic behavior of differential equations inphysics Since the pioneering work of Leray [1] self-similarsolutions of the Navier-Stokes equations for incompressiblefluids have been widely studied in different settings (eg [2page 207] [3 page 120] [4ndash10] [11 Chapter 23] [12ndash20]) Onthe contrary studies on the self-similar solutions of the com-pressible Navier-Stokes equations have been limited partiallydue to the complicated nonlinearities in the equations (see[21ndash24])

In one-space dimension the isentropic compressible fluidflow is governed by the Navier-Stokes equations

120588119905+ (120588119906)

119909= 0

(120588119906)119905+ (120588119906

2)119909+ 119875(120588)

119909= (120583 (120588) 119906

119909)119909 (119909 119905) isin R timesR

+

(1)

where 120588 = 120588(119909 119905) and 119906 = 119906(119909 119905) are the density and velocityof the fluid 120583(120588) and 119875(120588) denote the density-dependentviscosity and pressure respectively and the subscripts mean

partial derivations Guo and Jiang [21] considered (1) withconstant viscosity 120583(120588) equiv 120583 gt 0 and linear density-dependent pressure119875(120588) = 119886120588 where 119886 gt 0 is a constant andproved that there exist neither forward nor backward self-similar solutions with finite total energy Their investigationgeneralized the results for 3D incompressible fluids in Necaset alrsquos work [6] to the 1D compressible case with 119875(120588) =

120588120574 where 120574 = 1 The problem with 120574 gt 1 however is

open From a physical point of view one can derive thecompressible Navier-Stokes equations from the Boltzmannequations by exploiting the Chapman-Enskog expansion upto the second order and then find that the viscosity dependson the temperature If considering an isentropic process thisdependence can be translated into that on the density such as120583(120588) = 120588

120579 where 120579 gt 0 is a constant (see [25]) Okada et al[26] pointed out that because of the hard sphere interactionthe relation between indices 120579 and 120574 is 120579 = (120574 minus 1)2 In thefirst part of this paper we are concerned with (1) where

120583 (120588) = 120588120579 119875 (120588) = 120588

120574

120574 minus 1

2le 120579 lt

120574 + 1

2

120574 ge 1

(2)

When considering an ideal compressible gas flow partic-ularly in the thermodynamic analysis with exergy loss and

2 Journal of Applied Mathematics

entropy generation both the viscosity and pressure rely onthe entropy so it is necessary to extend the nonisentropicfluid dynamics to include the transport of entropy (see [1327ndash35]) We consider the following coupled system of theNavier-Stokes equations with an entropy transport equationin a pure form

120588119904119905+ 120588119906119904

119909= 0 (3)

120588119905+ (120588119906)

119909= 0 (4)

(120588119906)119905+ (120588119906

2)119909+ 119875(120588 119904)

119909= (120583 (120588 119904) 119906

119909)119909

(119909 119905) isin R timesR+

(5)

where 119904 = 119904(119909 119905) is the entropy of the fluid and 120583(120588 119904) and119875(120588 119904) denote the density-entropy-dependent viscosity andpressure respectively In this system we assume that

120583 (120588 119904) = 120588120579119890119904 119875 (120588 119904) = 120588

120574119890119904 0 le 120579 lt

120574 + 1

2

120574 ge 1

(6)

Navier-Stokes equations enjoy a scaling property if (120588 119906)solves (1)-(2) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)) (7)

does so for any 120582 gt 0 by setting 119886 = 1(120574 minus 120579) 119887 = (120574 minus

1)2(120574 minus 120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) and 119889 = 1 Note thatfrom (2) 119886 ge 119888 gt 0 and 119887 ge 0 Solution (120588 119906) is called forwardself-similar if

(120588 119906) = (120588(120582) 119906(120582)) for every 120582 gt 0 (8)

In that case 120588(119909 119905) and 119906(119909 119905) are decided by their values atthe instant of 119905 = 1120582

120588 (119909 119905) =1

119905119886119876(

119909

119905119888) 119906 (119909 119905) =

1

119905119887119880(

119909

119905119888) 119905 gt 0 (9)

where 119876(119910) = 120588(119910 1) and 119880(119910) = 119906(119910 1) are defined on RIn the same manner the backward self-similar solutions areof the form

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119880(

119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(10)

where 119876(119910) = 120588(119910 119879 minus 1) and 119880(119910) = 119906(119910 119879 minus 1) for 119879 gt 1Substitution of (9) or (10) into (1) gives

(119876119880)1015840minus 119888(119910119876)

1015840

+ (119888 minus 119886)119876 = 0 (11)

(1198761205791198801015840)1015840

minus (1198761198802)1015840

+ 119888(119910119876119880)1015840

minus (119876120574)1015840

+ (119886 + 119887 minus 119888)119876119880 = 0

(12)

for forward self-similar solutions or

(119876119880)1015840+ 119888(119910119876)

1015840

minus (119888 minus 119886)119876 = 0 (13)

(1198761205791198801015840)1015840

minus (1198761198802)1015840

minus 119888(119910119876119880)1015840

minus (119876120574)1015840

minus (119886 + 119887 minus 119888)119876119880 = 0

(14)

for backward self-similar solutions respectively In compar-ison with those in Guo and Jiang [21] forward (backward)self-similar equations above process necessary modificationsand additional difficulties For instance (11) and (13) havesolutions with an additional integral term and thus themodified blow-up analysis needs an 119871

infin estimate on thedensity and a new large-scale argument on the energy Inaddition conditions on 120579 and 120574 proposed in (2) are directlyrelated to the energy estimate

Mellet and Vasseur [25] obtained the global existence ofstrong solutions for the Cauchy problem of (1) with positiveinitial density having (possibly different) positive limits at119909 =plusmninfin Precisely fix constant positive density120588

+gt 0 and120588

minusgt 0

and let 120588(119909) be a smooth monotone function satisfying

120588 (119909) = 120588plusmn

when plusmn 119909 ge 1 120588 (119909) gt 0 forall119909 isin R (15)

Assume that the initial data 120588(119909 119905)|119905=0

= 1205880(119909) and

119906(119909 119905)|119905=0

= 1199060(119909) satisfy

0 lt 1205810le 1205880(119909) le 120581

0lt infin

1205880minus 120588 isin 119867

1(R) 119906

0isin 1198671(R)

(16)

for some constants 1205810and 1205810 Assume also that 120583(120588) and 119875(120588)

verify (2) Mellet and Vasseur [25] proved that there exists aglobal strong solution (120588 119906) of (1) on R times R

+such that for

every 119879 gt 0

120588 minus 120588 isin 119871infin(0 119879119867

1(R))

119906 isin 119871infin(0 119879119867

1(R)) cap 119871

2(0 119879119867

2(R))

(17)

Moreover for every 119879 gt 0 there exist uniform bounds awayfrom zero with respect to all strong solutions having the sameinitial data Precisely there exist some constants 119862(119879) 120581(119879)and 120581(119879) depending only on 119879 120588

0(119909) and 119906

0(119909) such that

the following bounds hold uniformly for any strong solution(120588 119906)

1003817100381710038171003817120588 minus 1205881003817100381710038171003817119871infin(01198791198671(R)) le 119862 (119879)

119906119871infin(0119879119867

1(R)) le 119862 (119879)

(18)

0 lt 120581 (119879) le 120588 (119909 119905) le 120581 (119879) forall (119909 119905) isin R times [0 119879] (19)

Define

119901 (120588 | 120588) =1

120574 minus 1120588120574minus

1

120574 minus 1120588120574minus

120574

120574 minus 1120588120574minus1

(120588 minus 120588) (20)

as the relative potential energy density of (1) and

119864 (119905) = intR

1

2120588 (119909 119905) 119906

2(119909 119905) 119889119909 (21)

Journal of Applied Mathematics 3

as the kinetic energy Note that since 119901 is strictly convex119901(120588 | 120588) is nonnegative for every 120588 and 119901(120588 | 120588) = 0 if andonly if 120588 = 120588 Mellet and Vasseur [25] also showed that if theinitial total energy is finite that is the sum of the kinetic andpotential energy at time 0 satisfies

intR

[1

212058801199062

0+ 119901 (120588

0| 120588)] 119889119909 lt +infin (22)

then the following global-energy estimate on (minusinfininfin) times

[0 119879] holds uniformly with respect to all strong solutionsthat is for every 119879 gt 0 there exists a positive constant 119862(119879)depending only on 119879 120588

0(119909) and 119906

0(119909) such that

sup[0119879]

intR

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

0

intR

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119879)

(23)

holds for any strong solution (120588 119906) Correspondingly for 119877 gt0 0 lt 119905

1lt 119879 and some constant 119862(119877 119905

1 119879) we call

sup1199051le119905le119879

int

119877

minus119877

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

1199051

int

119877

minus119877

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119877 1199051 119879)

(24)

the local-energy estimate on [minus119877 119877] times [1199051 119879] Note that the

global-energy estimate implies the local-energy estimateThe main result for the self-similar solutions of the

isentropic compressible Navier-Stokes equations is as follows

Theorem 1 Assume that 120583(120588) and 119875(120588) in (1) verify (2) Thenthe following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

For the self-similar solutions of the coupled system of thenonisentropic compressible Navier-Stokes equations with anentropy transport equation the main result is as follows

Theorem 2 Assume that 120583(120588 119904) and 119875(120588 119904) in (3)ndash(5) verify(6) Then the following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

Theorem 1 is proved in Section 2 andTheorem 2 in Section 3

2 Proof of Theorem 1

Any self-similar solution of (1) is either forward or backwardsowefirst proveTheorem 1 for forward and then for backwardself-similar solutions

21 Forward Self-Similar Solutions

Lemma 3 If (119876 119880) solves (11)-(12) then the correspondingstrong solution (120588 119906) defined by (9) of (1) does not satisfy theglobal-energy estimate (23)

Proof From (11)

119876 (119910)119880 (119910) = 119888119910119876 (119910) + (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (25)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (1) le 120588 (119910 1) = 119876 (119910) le 120581 (1) forall119910 isin R (26)

Since 119886 ge 119888 gt 0 (25) and (26) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) =119888

2119910119876 (119910) + 119888

119910

2119876 (119910) + (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

ge119888

2119910119876 (119910) +

119888

2119910119876 (119910) + 119862

0

ge119888

2119910119876 (119910) +

119888

21198840120581 (1) + 119862

0

ge119888

2119910119876 (119910)

(27)

Thus from (9) and (21) for any 119905 gt 0

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot

1

11990521198871198802(119909

119905119888)119889119909

=1

2119905119886+2119887minus119888intR

1

119876 (119910)[119876 (119910)119880 (119910)]

2

119889119910

ge1

2119905119886+2119887minus119888int

+infin

1198840

1

119876 (119910)[119888

2119910119876 (119910)]

2

119889119910

ge1198882120581 (1)

8119905119886+2119887minus119888int

+infin

1198840

1199102119889119910

= +infin

(28)

This proves the lemma

Lemma 4 If (119876 119880) solves (11)-(12) and the correspondingstrong solution (120588 119906) defined by (9) of (1) satisfies the local-energy estimate (24) then as 119905 darr 0 the kinetic energy (21)mustblow up

Proof Similar to the proof of Lemma 3 for any 119905 gt 0 and119877 gt 119884

0119905119888gt 0

119864 (119905) ge1198882120581 (1)

8119905119886+2119887minus119888int

119877119905119888

1198840

1199102119889119910

=1198882120581 (1)

24119905119886+2119887minus119888(1198773

1199053119888minus 1198843

0)

=1198882120581 (1)

24119905119886+2119887+2119888[1198773minus (1198840119905119888)3

] 997888rarr +infin as 119905 darr 0

(29)

This proves the lemma

4 Journal of Applied Mathematics

22 Backward Self-Similar Solutions

Lemma 5 If (119876 119880) solves (13)-(14) then the correspondingstrong solution (120588 119906) defined by (10) of (1) does not satisfy theglobal-energy estimate (23)

Proof Fix 119879 gt 1 From (13)

119876 (119910)119880 (119910) = minus119888119910119876 (119910) minus (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (30)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (119879) le 120588 (119910 119879 minus 1) = 119876 (119910) le 120581 (119879) forall119910 isin R (31)

Since 119886 ge 119888 gt 0 (30) and (31) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) = minus119888

2119910119876 (119910) minus 119888

119910

2119876 (119910) minus (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

le minus119888

2119910119876 (119910)

(32)

Thus from (10) and (21) for any 119905 lt 119879

119864 (119905) = intR

1

2sdot

1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888) sdot

1

(119879 minus 119905)21198871198802

times (119909

(119879 minus 119905)119888)119889119909

ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

+infin

1198840

1199102119889119910

= +infin

(33)

This proves the lemma

Lemma 6 If (119876 119880) solves (13)-(14) and the correspondingstrong solution (120588 119906) defined by (10) of (1) satisfies the local-energy estimate (24) then as 119905 uarr 119879 the kinetic energy (21)mustblow up

Proof Recalling the proofs of Lemmas 4 and 5 for119877 gt 1198840(119879minus

119905)119888gt 0

119864 (119905) ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

119877(119879minus119905)119888

1198840

1199102119889119910

=1198882120581 (119879)

24(119879 minus 119905)119886+2119887+2119888

times [1198773minus 1198843

0(119879 minus 119905)

3119888] 997888rarr +infin as 119905 uarr 119879

(34)

This proves the lemma

Now Theorem 1 follows from the four lemmas above

3 Proof of Theorem 2

If (120588 119906 119904) solves (3)ndash(6) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)

120582119897119904 (120582119888119909 120582119889119905))

(35)

does so for any120582 gt 0 by setting 119886 = 1(120574minus120579) 119887 = (120574minus1)2(120574minus120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) 119889 = 1 and 119897 = 0 Note that from(6) 119886 gt 0 119887 ge 0 and 119888 gt 0The forward self-similar solutionshave the following form

120588 (119909 119905) =1

119905119886119876(

119909

119905119888)

119906 (119909 119905) =1

119905119887119880(

119909

119905119888)

119904 (119909 119905) = 119878 (119909

119905119888)

119905 gt 0

(36)

where119876(119910) = 120588(119910 1)119880(119910) = 119906(119910 1) and 119878(119910) = 119904(119910 1)Thebackward self-similar solutions are

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119876(

119909

(119879 minus 119905)119888)

119904 (119909 119905) = 119878 (119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(37)

where 119876(119910) = 120588(119910 119879 minus 1) 119880(119910) = 119906(119910 119879 minus 1) and 119878(119910) =119904(119910 119879 minus 1) for 119879 gt 1

Lions [29] investigated the coupled system of the Navier-Stokes equations with an entropy transport equation in a pureform and obtained the existence of weak solutions satisfying(19) and (23)

Proof of Theorem 2 Suppose that (120588 119906 119904) is a forward self-similar solution Inserting (36) into (3) one gets 119880(119910) = 119888119910and thus 119906(119909 119905) = 119888119909119905 Therefore for any 119905 gt 0 (36) (19)and (21) yield

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot (

119888119909

119905)

2

ge1198882120581 (1)

2119905119886+2intR

1199092119889119909 = +infin

(38)

Thismeans that the global-energy estimate (23) does not holdand that the kinetic energy (21) blows up as 119905 darr 0

The case of backward self-similar solutions can be provedsimilarly so Theorem 2 is proved

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 5

Acknowledgments

This work is partially supported by Zhejiang ProvincialNatural Science Foundation of China (no LQ13G030018) andNational Natural Science Foundation of China (nos 11001049and 11226184)

References

[1] J Leray ldquoSur le mouvement drsquoun liquide visqueux emplissantlrsquoespacerdquo Acta Mathematica vol 63 no 1 pp 193ndash248 1934

[2] G K Batchelor An Introduction to Fluid Dynamics CambridgeUniversity Press Cambridge UK 1999

[3] L I Sedov Similarity and Dimensional Methods in MechanicsCRC Press Boca Raton Fla USA 10th edition 1993

[4] O A Barraza ldquoSelf-similar solutions in weak 119871119901-spaces of theNavier-Stokes equationsrdquo Revista Matematica Iberoamericanavol 12 no 2 pp 411ndash439 1996

[5] M Cannone and F Planchon ldquoSelf-similar solutions forNavier-Stokes equations inR3rdquo Communications in Partial DifferentialEquations vol 21 no 1-2 pp 179ndash193 1996

[6] J Necas M Ruzicka and V Sverak ldquoOn Lerayrsquos self-similarsolutions of the Navier-Stokes equationsrdquo Acta Mathematicavol 176 no 2 pp 283ndash294 1996

[7] F Planchon ldquoAsymptotic behavior of global solutions to theNavier-Stokes equations inR3rdquo Revista Matematica Iberoamer-icana vol 14 no 1 pp 71ndash93 1998

[8] T-P Tsai ldquoOn Lerayrsquos self-similar solutions of theNavier-Stokesequations satisfying local energy estimatesrdquoArchive for RationalMechanics and Analysis vol 143 no 1 pp 29ndash51 1998

[9] T-P Tasi ldquoForward discrete self-similarsolutions of the Navier-Stokes equationsrdquo submitted httparxivorgabs12102783

[10] J R Miller M OrsquoLeary and M Schonbek ldquoNonexistenceof singular pseudo-self-similar solutions of the Navier-Stokessystemrdquo Mathematische Annalen vol 319 no 4 pp 809ndash8152001

[11] P G Lemarie-Rieusset Recent Developments in the Navier-Stokes Problem vol 431 ofChapmanampHallCRCResearchNotesin Mathematics Chapman amp HallCRC Boca Raton Fla USA2002

[12] D Chae ldquoNonexistence of asymptotically self-similar singular-ities in the Euler and the Navier-Stokes equationsrdquoMathematis-che Annalen vol 338 no 2 pp 435ndash449 2007

[13] D Chae ldquoNotes on the incompressible Euler and relatedequations on 119877119873rdquo Chinese Annals of Mathematics B vol 30 no5 pp 513ndash526 2009

[14] D Chae ldquoNonexistence of self-similar singularities in the idealmagnetohydrodynamicsrdquo Archive for Rational Mechanics andAnalysis vol 194 no 3 pp 1011ndash1027 2009

[15] L Brandolese ldquoFine properties of self-similar solutions of theNavier-Stokes equationsrdquo Archive for Rational Mechanics andAnalysis vol 192 no 3 pp 375ndash401 2009

[16] X-Y Jiao ldquoSome similarity reduction solutions to two-dimensional incompressible Navier-Stokes equationrdquo Commu-nications inTheoretical Physics vol 52 no 3 pp 389ndash394 2009

[17] I F Barna ldquoSelf-similar solutions of three-dimensionalNaviermdashstokes equationrdquo Communications in TheoreticalPhysics vol 56 no 4 pp 745ndash750 2011

[18] G Karch andD Pilarczyk ldquoAsymptotic stability of Landau solu-tions to Navier-Stokes systemrdquo Archive for Rational Mechanicsand Analysis vol 202 no 1 pp 115ndash131 2011

[19] V Sverak ldquoOn Landaursquos solutions of the Navier-Stokes equa-tionsrdquo Journal of Mathematical Sciences vol 179 no 1 pp 208ndash228 2011

[20] H Jia and V Sverak ldquoLocal-in-space estimates near initial timefor weak solutions of the Navier-Stokes equations and forwardself-similar solutionsrdquo Inventiones Mathematicae 2013

[21] Z Guo and S Jiang ldquoSelf-similar solutions to the isothermalcompressible Navier-Stokes equationsrdquo IMA Journal of AppliedMathematics vol 71 no 5 pp 658ndash669 2006

[22] Z Guo and Z Xin ldquoAnalytical solutions to the compressibleNavier-Stokes equations with density-dependent viscosity coef-ficients and free boundariesrdquo Journal of Differential Equationsvol 253 no 1 pp 1ndash19 2012

[23] M Yuen ldquoSelf-similar solutions with elliptic symmetry for thecompressible Euler and Navier-Stokes equations in R119873rdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 12 pp 4524ndash4528 2012

[24] H An andM Yuen ldquoSupplement to lsquoSelf-similar solutions withelliptic symmetry for the compressible Euler and Navier-Stokesequations in R119873rsquordquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1558ndash1561 2013

[25] A Mellet and A Vasseur ldquoExistence and uniqueness of globalstrong solutions for one-dimensional compressible Navier-Stokes equationsrdquo SIAM Journal on Mathematical Analysis vol39 no 4 pp 1344ndash1365 2008

[26] M Okada S Matusocircu-Necasova and T Makino ldquoFreeboundary problem for the equation of one-dimensional motionof compressible gas with density-dependent viscosityrdquo AnnalidellrsquoUniversita di Ferrara Nuova Serie Sezione VII ScienzeMatematiche vol 48 pp 1ndash20 2002

[27] H Teng C M Kinoshita S M Masutani and J ZhouldquoEntropy generation in multicomponent reacting flowsrdquo Jour-nal of Energy Resources Technology Transactions of the ASMEvol 120 no 3 pp 226ndash232 1998

[28] J O Hirschfelder C F Curtiss and R B BirdMolecularTheoryof Gases and Liquids John Wiley amp Sons New York NY USA1954

[29] P-L Lions Mathematical Topics in Fluid Mechanics vol 2of Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press New York NY USA 1996

[30] K Nishida T Takagi and S Kinoshita ldquoAnalysis of entropygeneration and exergy loss during combustionrdquo Proceedings ofthe Combustion Institute vol 29 pp 869ndash874 2002

[31] Z W Li S K Chou C Shu and W M Yang ldquoEntropy gener-ation during microcombustionrdquo Journal of Applied Physics vol97 no 8 Article ID 084914 2005

[32] H Yapıcı N Kayatas B Albayrak and G Basturk ldquoNumericalcalculation of local entropy generation in a methaneair burnerrdquoEnergy Conversion and Management vol 46 pp 1885ndash19192005

[33] N Lior W Sarmiento-Darkin and H S Al-Sharqawi ldquoTheexergy fields in transport processes their calculation and userdquoEnergy vol 31 no 5 pp 553ndash578 2006

[34] D Stanciu MMarinescu and A Dobrovicescu ldquoThe influenceof swirl angle on the irreversibilities in turbulent diffusionflamesrdquo International Journal of Thermodynamics vol 10 no 4pp 143ndash153 2007

[35] M Safari M R H Sheikhi M Janbozorgi and H MetghalchildquoEntropy transport equation in large eddy simulation for exergyanalysis of turbulent combustion systemsrdquo Entropy vol 12 no3 pp 434ndash444 2010

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

entropy generation both the viscosity and pressure rely onthe entropy so it is necessary to extend the nonisentropicfluid dynamics to include the transport of entropy (see [1327ndash35]) We consider the following coupled system of theNavier-Stokes equations with an entropy transport equationin a pure form

120588119904119905+ 120588119906119904

119909= 0 (3)

120588119905+ (120588119906)

119909= 0 (4)

(120588119906)119905+ (120588119906

2)119909+ 119875(120588 119904)

119909= (120583 (120588 119904) 119906

119909)119909

(119909 119905) isin R timesR+

(5)

where 119904 = 119904(119909 119905) is the entropy of the fluid and 120583(120588 119904) and119875(120588 119904) denote the density-entropy-dependent viscosity andpressure respectively In this system we assume that

120583 (120588 119904) = 120588120579119890119904 119875 (120588 119904) = 120588

120574119890119904 0 le 120579 lt

120574 + 1

2

120574 ge 1

(6)

Navier-Stokes equations enjoy a scaling property if (120588 119906)solves (1)-(2) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)) (7)

does so for any 120582 gt 0 by setting 119886 = 1(120574 minus 120579) 119887 = (120574 minus

1)2(120574 minus 120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) and 119889 = 1 Note thatfrom (2) 119886 ge 119888 gt 0 and 119887 ge 0 Solution (120588 119906) is called forwardself-similar if

(120588 119906) = (120588(120582) 119906(120582)) for every 120582 gt 0 (8)

In that case 120588(119909 119905) and 119906(119909 119905) are decided by their values atthe instant of 119905 = 1120582

120588 (119909 119905) =1

119905119886119876(

119909

119905119888) 119906 (119909 119905) =

1

119905119887119880(

119909

119905119888) 119905 gt 0 (9)

where 119876(119910) = 120588(119910 1) and 119880(119910) = 119906(119910 1) are defined on RIn the same manner the backward self-similar solutions areof the form

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119880(

119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(10)

where 119876(119910) = 120588(119910 119879 minus 1) and 119880(119910) = 119906(119910 119879 minus 1) for 119879 gt 1Substitution of (9) or (10) into (1) gives

(119876119880)1015840minus 119888(119910119876)

1015840

+ (119888 minus 119886)119876 = 0 (11)

(1198761205791198801015840)1015840

minus (1198761198802)1015840

+ 119888(119910119876119880)1015840

minus (119876120574)1015840

+ (119886 + 119887 minus 119888)119876119880 = 0

(12)

for forward self-similar solutions or

(119876119880)1015840+ 119888(119910119876)

1015840

minus (119888 minus 119886)119876 = 0 (13)

(1198761205791198801015840)1015840

minus (1198761198802)1015840

minus 119888(119910119876119880)1015840

minus (119876120574)1015840

minus (119886 + 119887 minus 119888)119876119880 = 0

(14)

for backward self-similar solutions respectively In compar-ison with those in Guo and Jiang [21] forward (backward)self-similar equations above process necessary modificationsand additional difficulties For instance (11) and (13) havesolutions with an additional integral term and thus themodified blow-up analysis needs an 119871

infin estimate on thedensity and a new large-scale argument on the energy Inaddition conditions on 120579 and 120574 proposed in (2) are directlyrelated to the energy estimate

Mellet and Vasseur [25] obtained the global existence ofstrong solutions for the Cauchy problem of (1) with positiveinitial density having (possibly different) positive limits at119909 =plusmninfin Precisely fix constant positive density120588

+gt 0 and120588

minusgt 0

and let 120588(119909) be a smooth monotone function satisfying

120588 (119909) = 120588plusmn

when plusmn 119909 ge 1 120588 (119909) gt 0 forall119909 isin R (15)

Assume that the initial data 120588(119909 119905)|119905=0

= 1205880(119909) and

119906(119909 119905)|119905=0

= 1199060(119909) satisfy

0 lt 1205810le 1205880(119909) le 120581

0lt infin

1205880minus 120588 isin 119867

1(R) 119906

0isin 1198671(R)

(16)

for some constants 1205810and 1205810 Assume also that 120583(120588) and 119875(120588)

verify (2) Mellet and Vasseur [25] proved that there exists aglobal strong solution (120588 119906) of (1) on R times R

+such that for

every 119879 gt 0

120588 minus 120588 isin 119871infin(0 119879119867

1(R))

119906 isin 119871infin(0 119879119867

1(R)) cap 119871

2(0 119879119867

2(R))

(17)

Moreover for every 119879 gt 0 there exist uniform bounds awayfrom zero with respect to all strong solutions having the sameinitial data Precisely there exist some constants 119862(119879) 120581(119879)and 120581(119879) depending only on 119879 120588

0(119909) and 119906

0(119909) such that

the following bounds hold uniformly for any strong solution(120588 119906)

1003817100381710038171003817120588 minus 1205881003817100381710038171003817119871infin(01198791198671(R)) le 119862 (119879)

119906119871infin(0119879119867

1(R)) le 119862 (119879)

(18)

0 lt 120581 (119879) le 120588 (119909 119905) le 120581 (119879) forall (119909 119905) isin R times [0 119879] (19)

Define

119901 (120588 | 120588) =1

120574 minus 1120588120574minus

1

120574 minus 1120588120574minus

120574

120574 minus 1120588120574minus1

(120588 minus 120588) (20)

as the relative potential energy density of (1) and

119864 (119905) = intR

1

2120588 (119909 119905) 119906

2(119909 119905) 119889119909 (21)

Journal of Applied Mathematics 3

as the kinetic energy Note that since 119901 is strictly convex119901(120588 | 120588) is nonnegative for every 120588 and 119901(120588 | 120588) = 0 if andonly if 120588 = 120588 Mellet and Vasseur [25] also showed that if theinitial total energy is finite that is the sum of the kinetic andpotential energy at time 0 satisfies

intR

[1

212058801199062

0+ 119901 (120588

0| 120588)] 119889119909 lt +infin (22)

then the following global-energy estimate on (minusinfininfin) times

[0 119879] holds uniformly with respect to all strong solutionsthat is for every 119879 gt 0 there exists a positive constant 119862(119879)depending only on 119879 120588

0(119909) and 119906

0(119909) such that

sup[0119879]

intR

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

0

intR

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119879)

(23)

holds for any strong solution (120588 119906) Correspondingly for 119877 gt0 0 lt 119905

1lt 119879 and some constant 119862(119877 119905

1 119879) we call

sup1199051le119905le119879

int

119877

minus119877

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

1199051

int

119877

minus119877

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119877 1199051 119879)

(24)

the local-energy estimate on [minus119877 119877] times [1199051 119879] Note that the

global-energy estimate implies the local-energy estimateThe main result for the self-similar solutions of the

isentropic compressible Navier-Stokes equations is as follows

Theorem 1 Assume that 120583(120588) and 119875(120588) in (1) verify (2) Thenthe following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

For the self-similar solutions of the coupled system of thenonisentropic compressible Navier-Stokes equations with anentropy transport equation the main result is as follows

Theorem 2 Assume that 120583(120588 119904) and 119875(120588 119904) in (3)ndash(5) verify(6) Then the following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

Theorem 1 is proved in Section 2 andTheorem 2 in Section 3

2 Proof of Theorem 1

Any self-similar solution of (1) is either forward or backwardsowefirst proveTheorem 1 for forward and then for backwardself-similar solutions

21 Forward Self-Similar Solutions

Lemma 3 If (119876 119880) solves (11)-(12) then the correspondingstrong solution (120588 119906) defined by (9) of (1) does not satisfy theglobal-energy estimate (23)

Proof From (11)

119876 (119910)119880 (119910) = 119888119910119876 (119910) + (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (25)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (1) le 120588 (119910 1) = 119876 (119910) le 120581 (1) forall119910 isin R (26)

Since 119886 ge 119888 gt 0 (25) and (26) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) =119888

2119910119876 (119910) + 119888

119910

2119876 (119910) + (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

ge119888

2119910119876 (119910) +

119888

2119910119876 (119910) + 119862

0

ge119888

2119910119876 (119910) +

119888

21198840120581 (1) + 119862

0

ge119888

2119910119876 (119910)

(27)

Thus from (9) and (21) for any 119905 gt 0

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot

1

11990521198871198802(119909

119905119888)119889119909

=1

2119905119886+2119887minus119888intR

1

119876 (119910)[119876 (119910)119880 (119910)]

2

119889119910

ge1

2119905119886+2119887minus119888int

+infin

1198840

1

119876 (119910)[119888

2119910119876 (119910)]

2

119889119910

ge1198882120581 (1)

8119905119886+2119887minus119888int

+infin

1198840

1199102119889119910

= +infin

(28)

This proves the lemma

Lemma 4 If (119876 119880) solves (11)-(12) and the correspondingstrong solution (120588 119906) defined by (9) of (1) satisfies the local-energy estimate (24) then as 119905 darr 0 the kinetic energy (21)mustblow up

Proof Similar to the proof of Lemma 3 for any 119905 gt 0 and119877 gt 119884

0119905119888gt 0

119864 (119905) ge1198882120581 (1)

8119905119886+2119887minus119888int

119877119905119888

1198840

1199102119889119910

=1198882120581 (1)

24119905119886+2119887minus119888(1198773

1199053119888minus 1198843

0)

=1198882120581 (1)

24119905119886+2119887+2119888[1198773minus (1198840119905119888)3

] 997888rarr +infin as 119905 darr 0

(29)

This proves the lemma

4 Journal of Applied Mathematics

22 Backward Self-Similar Solutions

Lemma 5 If (119876 119880) solves (13)-(14) then the correspondingstrong solution (120588 119906) defined by (10) of (1) does not satisfy theglobal-energy estimate (23)

Proof Fix 119879 gt 1 From (13)

119876 (119910)119880 (119910) = minus119888119910119876 (119910) minus (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (30)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (119879) le 120588 (119910 119879 minus 1) = 119876 (119910) le 120581 (119879) forall119910 isin R (31)

Since 119886 ge 119888 gt 0 (30) and (31) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) = minus119888

2119910119876 (119910) minus 119888

119910

2119876 (119910) minus (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

le minus119888

2119910119876 (119910)

(32)

Thus from (10) and (21) for any 119905 lt 119879

119864 (119905) = intR

1

2sdot

1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888) sdot

1

(119879 minus 119905)21198871198802

times (119909

(119879 minus 119905)119888)119889119909

ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

+infin

1198840

1199102119889119910

= +infin

(33)

This proves the lemma

Lemma 6 If (119876 119880) solves (13)-(14) and the correspondingstrong solution (120588 119906) defined by (10) of (1) satisfies the local-energy estimate (24) then as 119905 uarr 119879 the kinetic energy (21)mustblow up

Proof Recalling the proofs of Lemmas 4 and 5 for119877 gt 1198840(119879minus

119905)119888gt 0

119864 (119905) ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

119877(119879minus119905)119888

1198840

1199102119889119910

=1198882120581 (119879)

24(119879 minus 119905)119886+2119887+2119888

times [1198773minus 1198843

0(119879 minus 119905)

3119888] 997888rarr +infin as 119905 uarr 119879

(34)

This proves the lemma

Now Theorem 1 follows from the four lemmas above

3 Proof of Theorem 2

If (120588 119906 119904) solves (3)ndash(6) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)

120582119897119904 (120582119888119909 120582119889119905))

(35)

does so for any120582 gt 0 by setting 119886 = 1(120574minus120579) 119887 = (120574minus1)2(120574minus120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) 119889 = 1 and 119897 = 0 Note that from(6) 119886 gt 0 119887 ge 0 and 119888 gt 0The forward self-similar solutionshave the following form

120588 (119909 119905) =1

119905119886119876(

119909

119905119888)

119906 (119909 119905) =1

119905119887119880(

119909

119905119888)

119904 (119909 119905) = 119878 (119909

119905119888)

119905 gt 0

(36)

where119876(119910) = 120588(119910 1)119880(119910) = 119906(119910 1) and 119878(119910) = 119904(119910 1)Thebackward self-similar solutions are

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119876(

119909

(119879 minus 119905)119888)

119904 (119909 119905) = 119878 (119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(37)

where 119876(119910) = 120588(119910 119879 minus 1) 119880(119910) = 119906(119910 119879 minus 1) and 119878(119910) =119904(119910 119879 minus 1) for 119879 gt 1

Lions [29] investigated the coupled system of the Navier-Stokes equations with an entropy transport equation in a pureform and obtained the existence of weak solutions satisfying(19) and (23)

Proof of Theorem 2 Suppose that (120588 119906 119904) is a forward self-similar solution Inserting (36) into (3) one gets 119880(119910) = 119888119910and thus 119906(119909 119905) = 119888119909119905 Therefore for any 119905 gt 0 (36) (19)and (21) yield

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot (

119888119909

119905)

2

ge1198882120581 (1)

2119905119886+2intR

1199092119889119909 = +infin

(38)

Thismeans that the global-energy estimate (23) does not holdand that the kinetic energy (21) blows up as 119905 darr 0

The case of backward self-similar solutions can be provedsimilarly so Theorem 2 is proved

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 5

Acknowledgments

This work is partially supported by Zhejiang ProvincialNatural Science Foundation of China (no LQ13G030018) andNational Natural Science Foundation of China (nos 11001049and 11226184)

References

[1] J Leray ldquoSur le mouvement drsquoun liquide visqueux emplissantlrsquoespacerdquo Acta Mathematica vol 63 no 1 pp 193ndash248 1934

[2] G K Batchelor An Introduction to Fluid Dynamics CambridgeUniversity Press Cambridge UK 1999

[3] L I Sedov Similarity and Dimensional Methods in MechanicsCRC Press Boca Raton Fla USA 10th edition 1993

[4] O A Barraza ldquoSelf-similar solutions in weak 119871119901-spaces of theNavier-Stokes equationsrdquo Revista Matematica Iberoamericanavol 12 no 2 pp 411ndash439 1996

[5] M Cannone and F Planchon ldquoSelf-similar solutions forNavier-Stokes equations inR3rdquo Communications in Partial DifferentialEquations vol 21 no 1-2 pp 179ndash193 1996

[6] J Necas M Ruzicka and V Sverak ldquoOn Lerayrsquos self-similarsolutions of the Navier-Stokes equationsrdquo Acta Mathematicavol 176 no 2 pp 283ndash294 1996

[7] F Planchon ldquoAsymptotic behavior of global solutions to theNavier-Stokes equations inR3rdquo Revista Matematica Iberoamer-icana vol 14 no 1 pp 71ndash93 1998

[8] T-P Tsai ldquoOn Lerayrsquos self-similar solutions of theNavier-Stokesequations satisfying local energy estimatesrdquoArchive for RationalMechanics and Analysis vol 143 no 1 pp 29ndash51 1998

[9] T-P Tasi ldquoForward discrete self-similarsolutions of the Navier-Stokes equationsrdquo submitted httparxivorgabs12102783

[10] J R Miller M OrsquoLeary and M Schonbek ldquoNonexistenceof singular pseudo-self-similar solutions of the Navier-Stokessystemrdquo Mathematische Annalen vol 319 no 4 pp 809ndash8152001

[11] P G Lemarie-Rieusset Recent Developments in the Navier-Stokes Problem vol 431 ofChapmanampHallCRCResearchNotesin Mathematics Chapman amp HallCRC Boca Raton Fla USA2002

[12] D Chae ldquoNonexistence of asymptotically self-similar singular-ities in the Euler and the Navier-Stokes equationsrdquoMathematis-che Annalen vol 338 no 2 pp 435ndash449 2007

[13] D Chae ldquoNotes on the incompressible Euler and relatedequations on 119877119873rdquo Chinese Annals of Mathematics B vol 30 no5 pp 513ndash526 2009

[14] D Chae ldquoNonexistence of self-similar singularities in the idealmagnetohydrodynamicsrdquo Archive for Rational Mechanics andAnalysis vol 194 no 3 pp 1011ndash1027 2009

[15] L Brandolese ldquoFine properties of self-similar solutions of theNavier-Stokes equationsrdquo Archive for Rational Mechanics andAnalysis vol 192 no 3 pp 375ndash401 2009

[16] X-Y Jiao ldquoSome similarity reduction solutions to two-dimensional incompressible Navier-Stokes equationrdquo Commu-nications inTheoretical Physics vol 52 no 3 pp 389ndash394 2009

[17] I F Barna ldquoSelf-similar solutions of three-dimensionalNaviermdashstokes equationrdquo Communications in TheoreticalPhysics vol 56 no 4 pp 745ndash750 2011

[18] G Karch andD Pilarczyk ldquoAsymptotic stability of Landau solu-tions to Navier-Stokes systemrdquo Archive for Rational Mechanicsand Analysis vol 202 no 1 pp 115ndash131 2011

[19] V Sverak ldquoOn Landaursquos solutions of the Navier-Stokes equa-tionsrdquo Journal of Mathematical Sciences vol 179 no 1 pp 208ndash228 2011

[20] H Jia and V Sverak ldquoLocal-in-space estimates near initial timefor weak solutions of the Navier-Stokes equations and forwardself-similar solutionsrdquo Inventiones Mathematicae 2013

[21] Z Guo and S Jiang ldquoSelf-similar solutions to the isothermalcompressible Navier-Stokes equationsrdquo IMA Journal of AppliedMathematics vol 71 no 5 pp 658ndash669 2006

[22] Z Guo and Z Xin ldquoAnalytical solutions to the compressibleNavier-Stokes equations with density-dependent viscosity coef-ficients and free boundariesrdquo Journal of Differential Equationsvol 253 no 1 pp 1ndash19 2012

[23] M Yuen ldquoSelf-similar solutions with elliptic symmetry for thecompressible Euler and Navier-Stokes equations in R119873rdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 12 pp 4524ndash4528 2012

[24] H An andM Yuen ldquoSupplement to lsquoSelf-similar solutions withelliptic symmetry for the compressible Euler and Navier-Stokesequations in R119873rsquordquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1558ndash1561 2013

[25] A Mellet and A Vasseur ldquoExistence and uniqueness of globalstrong solutions for one-dimensional compressible Navier-Stokes equationsrdquo SIAM Journal on Mathematical Analysis vol39 no 4 pp 1344ndash1365 2008

[26] M Okada S Matusocircu-Necasova and T Makino ldquoFreeboundary problem for the equation of one-dimensional motionof compressible gas with density-dependent viscosityrdquo AnnalidellrsquoUniversita di Ferrara Nuova Serie Sezione VII ScienzeMatematiche vol 48 pp 1ndash20 2002

[27] H Teng C M Kinoshita S M Masutani and J ZhouldquoEntropy generation in multicomponent reacting flowsrdquo Jour-nal of Energy Resources Technology Transactions of the ASMEvol 120 no 3 pp 226ndash232 1998

[28] J O Hirschfelder C F Curtiss and R B BirdMolecularTheoryof Gases and Liquids John Wiley amp Sons New York NY USA1954

[29] P-L Lions Mathematical Topics in Fluid Mechanics vol 2of Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press New York NY USA 1996

[30] K Nishida T Takagi and S Kinoshita ldquoAnalysis of entropygeneration and exergy loss during combustionrdquo Proceedings ofthe Combustion Institute vol 29 pp 869ndash874 2002

[31] Z W Li S K Chou C Shu and W M Yang ldquoEntropy gener-ation during microcombustionrdquo Journal of Applied Physics vol97 no 8 Article ID 084914 2005

[32] H Yapıcı N Kayatas B Albayrak and G Basturk ldquoNumericalcalculation of local entropy generation in a methaneair burnerrdquoEnergy Conversion and Management vol 46 pp 1885ndash19192005

[33] N Lior W Sarmiento-Darkin and H S Al-Sharqawi ldquoTheexergy fields in transport processes their calculation and userdquoEnergy vol 31 no 5 pp 553ndash578 2006

[34] D Stanciu MMarinescu and A Dobrovicescu ldquoThe influenceof swirl angle on the irreversibilities in turbulent diffusionflamesrdquo International Journal of Thermodynamics vol 10 no 4pp 143ndash153 2007

[35] M Safari M R H Sheikhi M Janbozorgi and H MetghalchildquoEntropy transport equation in large eddy simulation for exergyanalysis of turbulent combustion systemsrdquo Entropy vol 12 no3 pp 434ndash444 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

as the kinetic energy Note that since 119901 is strictly convex119901(120588 | 120588) is nonnegative for every 120588 and 119901(120588 | 120588) = 0 if andonly if 120588 = 120588 Mellet and Vasseur [25] also showed that if theinitial total energy is finite that is the sum of the kinetic andpotential energy at time 0 satisfies

intR

[1

212058801199062

0+ 119901 (120588

0| 120588)] 119889119909 lt +infin (22)

then the following global-energy estimate on (minusinfininfin) times

[0 119879] holds uniformly with respect to all strong solutionsthat is for every 119879 gt 0 there exists a positive constant 119862(119879)depending only on 119879 120588

0(119909) and 119906

0(119909) such that

sup[0119879]

intR

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

0

intR

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119879)

(23)

holds for any strong solution (120588 119906) Correspondingly for 119877 gt0 0 lt 119905

1lt 119879 and some constant 119862(119877 119905

1 119879) we call

sup1199051le119905le119879

int

119877

minus119877

[1

21205881199062+ 119901 (120588 | 120588)] 119889119909

+ int

119879

1199051

int

119877

minus119877

120583 (120588)1003816100381610038161003816119906119909

10038161003816100381610038162

119889119909 119889119905 le 119862 (119877 1199051 119879)

(24)

the local-energy estimate on [minus119877 119877] times [1199051 119879] Note that the

global-energy estimate implies the local-energy estimateThe main result for the self-similar solutions of the

isentropic compressible Navier-Stokes equations is as follows

Theorem 1 Assume that 120583(120588) and 119875(120588) in (1) verify (2) Thenthe following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

For the self-similar solutions of the coupled system of thenonisentropic compressible Navier-Stokes equations with anentropy transport equation the main result is as follows

Theorem 2 Assume that 120583(120588 119904) and 119875(120588 119904) in (3)ndash(5) verify(6) Then the following statements are true

(1) There is no self-similar strong solution satisfying theglobal-energy estimate (23)

(2) If there is a forward (backward) self-similar strongsolution satisfying the local-energy estimate (24) thenits kinetic energy (21) blows up as 119905 darr 0+(119905 uarr 119879minus)

Theorem 1 is proved in Section 2 andTheorem 2 in Section 3

2 Proof of Theorem 1

Any self-similar solution of (1) is either forward or backwardsowefirst proveTheorem 1 for forward and then for backwardself-similar solutions

21 Forward Self-Similar Solutions

Lemma 3 If (119876 119880) solves (11)-(12) then the correspondingstrong solution (120588 119906) defined by (9) of (1) does not satisfy theglobal-energy estimate (23)

Proof From (11)

119876 (119910)119880 (119910) = 119888119910119876 (119910) + (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (25)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (1) le 120588 (119910 1) = 119876 (119910) le 120581 (1) forall119910 isin R (26)

Since 119886 ge 119888 gt 0 (25) and (26) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) =119888

2119910119876 (119910) + 119888

119910

2119876 (119910) + (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

ge119888

2119910119876 (119910) +

119888

2119910119876 (119910) + 119862

0

ge119888

2119910119876 (119910) +

119888

21198840120581 (1) + 119862

0

ge119888

2119910119876 (119910)

(27)

Thus from (9) and (21) for any 119905 gt 0

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot

1

11990521198871198802(119909

119905119888)119889119909

=1

2119905119886+2119887minus119888intR

1

119876 (119910)[119876 (119910)119880 (119910)]

2

119889119910

ge1

2119905119886+2119887minus119888int

+infin

1198840

1

119876 (119910)[119888

2119910119876 (119910)]

2

119889119910

ge1198882120581 (1)

8119905119886+2119887minus119888int

+infin

1198840

1199102119889119910

= +infin

(28)

This proves the lemma

Lemma 4 If (119876 119880) solves (11)-(12) and the correspondingstrong solution (120588 119906) defined by (9) of (1) satisfies the local-energy estimate (24) then as 119905 darr 0 the kinetic energy (21)mustblow up

Proof Similar to the proof of Lemma 3 for any 119905 gt 0 and119877 gt 119884

0119905119888gt 0

119864 (119905) ge1198882120581 (1)

8119905119886+2119887minus119888int

119877119905119888

1198840

1199102119889119910

=1198882120581 (1)

24119905119886+2119887minus119888(1198773

1199053119888minus 1198843

0)

=1198882120581 (1)

24119905119886+2119887+2119888[1198773minus (1198840119905119888)3

] 997888rarr +infin as 119905 darr 0

(29)

This proves the lemma

4 Journal of Applied Mathematics

22 Backward Self-Similar Solutions

Lemma 5 If (119876 119880) solves (13)-(14) then the correspondingstrong solution (120588 119906) defined by (10) of (1) does not satisfy theglobal-energy estimate (23)

Proof Fix 119879 gt 1 From (13)

119876 (119910)119880 (119910) = minus119888119910119876 (119910) minus (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (30)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (119879) le 120588 (119910 119879 minus 1) = 119876 (119910) le 120581 (119879) forall119910 isin R (31)

Since 119886 ge 119888 gt 0 (30) and (31) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) = minus119888

2119910119876 (119910) minus 119888

119910

2119876 (119910) minus (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

le minus119888

2119910119876 (119910)

(32)

Thus from (10) and (21) for any 119905 lt 119879

119864 (119905) = intR

1

2sdot

1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888) sdot

1

(119879 minus 119905)21198871198802

times (119909

(119879 minus 119905)119888)119889119909

ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

+infin

1198840

1199102119889119910

= +infin

(33)

This proves the lemma

Lemma 6 If (119876 119880) solves (13)-(14) and the correspondingstrong solution (120588 119906) defined by (10) of (1) satisfies the local-energy estimate (24) then as 119905 uarr 119879 the kinetic energy (21)mustblow up

Proof Recalling the proofs of Lemmas 4 and 5 for119877 gt 1198840(119879minus

119905)119888gt 0

119864 (119905) ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

119877(119879minus119905)119888

1198840

1199102119889119910

=1198882120581 (119879)

24(119879 minus 119905)119886+2119887+2119888

times [1198773minus 1198843

0(119879 minus 119905)

3119888] 997888rarr +infin as 119905 uarr 119879

(34)

This proves the lemma

Now Theorem 1 follows from the four lemmas above

3 Proof of Theorem 2

If (120588 119906 119904) solves (3)ndash(6) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)

120582119897119904 (120582119888119909 120582119889119905))

(35)

does so for any120582 gt 0 by setting 119886 = 1(120574minus120579) 119887 = (120574minus1)2(120574minus120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) 119889 = 1 and 119897 = 0 Note that from(6) 119886 gt 0 119887 ge 0 and 119888 gt 0The forward self-similar solutionshave the following form

120588 (119909 119905) =1

119905119886119876(

119909

119905119888)

119906 (119909 119905) =1

119905119887119880(

119909

119905119888)

119904 (119909 119905) = 119878 (119909

119905119888)

119905 gt 0

(36)

where119876(119910) = 120588(119910 1)119880(119910) = 119906(119910 1) and 119878(119910) = 119904(119910 1)Thebackward self-similar solutions are

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119876(

119909

(119879 minus 119905)119888)

119904 (119909 119905) = 119878 (119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(37)

where 119876(119910) = 120588(119910 119879 minus 1) 119880(119910) = 119906(119910 119879 minus 1) and 119878(119910) =119904(119910 119879 minus 1) for 119879 gt 1

Lions [29] investigated the coupled system of the Navier-Stokes equations with an entropy transport equation in a pureform and obtained the existence of weak solutions satisfying(19) and (23)

Proof of Theorem 2 Suppose that (120588 119906 119904) is a forward self-similar solution Inserting (36) into (3) one gets 119880(119910) = 119888119910and thus 119906(119909 119905) = 119888119909119905 Therefore for any 119905 gt 0 (36) (19)and (21) yield

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot (

119888119909

119905)

2

ge1198882120581 (1)

2119905119886+2intR

1199092119889119909 = +infin

(38)

Thismeans that the global-energy estimate (23) does not holdand that the kinetic energy (21) blows up as 119905 darr 0

The case of backward self-similar solutions can be provedsimilarly so Theorem 2 is proved

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 5

Acknowledgments

This work is partially supported by Zhejiang ProvincialNatural Science Foundation of China (no LQ13G030018) andNational Natural Science Foundation of China (nos 11001049and 11226184)

References

[1] J Leray ldquoSur le mouvement drsquoun liquide visqueux emplissantlrsquoespacerdquo Acta Mathematica vol 63 no 1 pp 193ndash248 1934

[2] G K Batchelor An Introduction to Fluid Dynamics CambridgeUniversity Press Cambridge UK 1999

[3] L I Sedov Similarity and Dimensional Methods in MechanicsCRC Press Boca Raton Fla USA 10th edition 1993

[4] O A Barraza ldquoSelf-similar solutions in weak 119871119901-spaces of theNavier-Stokes equationsrdquo Revista Matematica Iberoamericanavol 12 no 2 pp 411ndash439 1996

[5] M Cannone and F Planchon ldquoSelf-similar solutions forNavier-Stokes equations inR3rdquo Communications in Partial DifferentialEquations vol 21 no 1-2 pp 179ndash193 1996

[6] J Necas M Ruzicka and V Sverak ldquoOn Lerayrsquos self-similarsolutions of the Navier-Stokes equationsrdquo Acta Mathematicavol 176 no 2 pp 283ndash294 1996

[7] F Planchon ldquoAsymptotic behavior of global solutions to theNavier-Stokes equations inR3rdquo Revista Matematica Iberoamer-icana vol 14 no 1 pp 71ndash93 1998

[8] T-P Tsai ldquoOn Lerayrsquos self-similar solutions of theNavier-Stokesequations satisfying local energy estimatesrdquoArchive for RationalMechanics and Analysis vol 143 no 1 pp 29ndash51 1998

[9] T-P Tasi ldquoForward discrete self-similarsolutions of the Navier-Stokes equationsrdquo submitted httparxivorgabs12102783

[10] J R Miller M OrsquoLeary and M Schonbek ldquoNonexistenceof singular pseudo-self-similar solutions of the Navier-Stokessystemrdquo Mathematische Annalen vol 319 no 4 pp 809ndash8152001

[11] P G Lemarie-Rieusset Recent Developments in the Navier-Stokes Problem vol 431 ofChapmanampHallCRCResearchNotesin Mathematics Chapman amp HallCRC Boca Raton Fla USA2002

[12] D Chae ldquoNonexistence of asymptotically self-similar singular-ities in the Euler and the Navier-Stokes equationsrdquoMathematis-che Annalen vol 338 no 2 pp 435ndash449 2007

[13] D Chae ldquoNotes on the incompressible Euler and relatedequations on 119877119873rdquo Chinese Annals of Mathematics B vol 30 no5 pp 513ndash526 2009

[14] D Chae ldquoNonexistence of self-similar singularities in the idealmagnetohydrodynamicsrdquo Archive for Rational Mechanics andAnalysis vol 194 no 3 pp 1011ndash1027 2009

[15] L Brandolese ldquoFine properties of self-similar solutions of theNavier-Stokes equationsrdquo Archive for Rational Mechanics andAnalysis vol 192 no 3 pp 375ndash401 2009

[16] X-Y Jiao ldquoSome similarity reduction solutions to two-dimensional incompressible Navier-Stokes equationrdquo Commu-nications inTheoretical Physics vol 52 no 3 pp 389ndash394 2009

[17] I F Barna ldquoSelf-similar solutions of three-dimensionalNaviermdashstokes equationrdquo Communications in TheoreticalPhysics vol 56 no 4 pp 745ndash750 2011

[18] G Karch andD Pilarczyk ldquoAsymptotic stability of Landau solu-tions to Navier-Stokes systemrdquo Archive for Rational Mechanicsand Analysis vol 202 no 1 pp 115ndash131 2011

[19] V Sverak ldquoOn Landaursquos solutions of the Navier-Stokes equa-tionsrdquo Journal of Mathematical Sciences vol 179 no 1 pp 208ndash228 2011

[20] H Jia and V Sverak ldquoLocal-in-space estimates near initial timefor weak solutions of the Navier-Stokes equations and forwardself-similar solutionsrdquo Inventiones Mathematicae 2013

[21] Z Guo and S Jiang ldquoSelf-similar solutions to the isothermalcompressible Navier-Stokes equationsrdquo IMA Journal of AppliedMathematics vol 71 no 5 pp 658ndash669 2006

[22] Z Guo and Z Xin ldquoAnalytical solutions to the compressibleNavier-Stokes equations with density-dependent viscosity coef-ficients and free boundariesrdquo Journal of Differential Equationsvol 253 no 1 pp 1ndash19 2012

[23] M Yuen ldquoSelf-similar solutions with elliptic symmetry for thecompressible Euler and Navier-Stokes equations in R119873rdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 12 pp 4524ndash4528 2012

[24] H An andM Yuen ldquoSupplement to lsquoSelf-similar solutions withelliptic symmetry for the compressible Euler and Navier-Stokesequations in R119873rsquordquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1558ndash1561 2013

[25] A Mellet and A Vasseur ldquoExistence and uniqueness of globalstrong solutions for one-dimensional compressible Navier-Stokes equationsrdquo SIAM Journal on Mathematical Analysis vol39 no 4 pp 1344ndash1365 2008

[26] M Okada S Matusocircu-Necasova and T Makino ldquoFreeboundary problem for the equation of one-dimensional motionof compressible gas with density-dependent viscosityrdquo AnnalidellrsquoUniversita di Ferrara Nuova Serie Sezione VII ScienzeMatematiche vol 48 pp 1ndash20 2002

[27] H Teng C M Kinoshita S M Masutani and J ZhouldquoEntropy generation in multicomponent reacting flowsrdquo Jour-nal of Energy Resources Technology Transactions of the ASMEvol 120 no 3 pp 226ndash232 1998

[28] J O Hirschfelder C F Curtiss and R B BirdMolecularTheoryof Gases and Liquids John Wiley amp Sons New York NY USA1954

[29] P-L Lions Mathematical Topics in Fluid Mechanics vol 2of Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press New York NY USA 1996

[30] K Nishida T Takagi and S Kinoshita ldquoAnalysis of entropygeneration and exergy loss during combustionrdquo Proceedings ofthe Combustion Institute vol 29 pp 869ndash874 2002

[31] Z W Li S K Chou C Shu and W M Yang ldquoEntropy gener-ation during microcombustionrdquo Journal of Applied Physics vol97 no 8 Article ID 084914 2005

[32] H Yapıcı N Kayatas B Albayrak and G Basturk ldquoNumericalcalculation of local entropy generation in a methaneair burnerrdquoEnergy Conversion and Management vol 46 pp 1885ndash19192005

[33] N Lior W Sarmiento-Darkin and H S Al-Sharqawi ldquoTheexergy fields in transport processes their calculation and userdquoEnergy vol 31 no 5 pp 553ndash578 2006

[34] D Stanciu MMarinescu and A Dobrovicescu ldquoThe influenceof swirl angle on the irreversibilities in turbulent diffusionflamesrdquo International Journal of Thermodynamics vol 10 no 4pp 143ndash153 2007

[35] M Safari M R H Sheikhi M Janbozorgi and H MetghalchildquoEntropy transport equation in large eddy simulation for exergyanalysis of turbulent combustion systemsrdquo Entropy vol 12 no3 pp 434ndash444 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

22 Backward Self-Similar Solutions

Lemma 5 If (119876 119880) solves (13)-(14) then the correspondingstrong solution (120588 119906) defined by (10) of (1) does not satisfy theglobal-energy estimate (23)

Proof Fix 119879 gt 1 From (13)

119876 (119910)119880 (119910) = minus119888119910119876 (119910) minus (119886 minus 119888) int

119910

0

119876 (119911) 119889119911 + 1198620 (30)

where 1198620is an arbitrary constant From (19)

0 lt 120581 (119879) le 120588 (119910 119879 minus 1) = 119876 (119910) le 120581 (119879) forall119910 isin R (31)

Since 119886 ge 119888 gt 0 (30) and (31) imply that for 119910 ge 1198840gt 0

where 1198840is large enough

119876 (119910)119880 (119910) = minus119888

2119910119876 (119910) minus 119888

119910

2119876 (119910) minus (119886 minus 119888)

times int

119910

0

119876 (119911) 119889119911 + 1198620

le minus119888

2119910119876 (119910)

(32)

Thus from (10) and (21) for any 119905 lt 119879

119864 (119905) = intR

1

2sdot

1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888) sdot

1

(119879 minus 119905)21198871198802

times (119909

(119879 minus 119905)119888)119889119909

ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

+infin

1198840

1199102119889119910

= +infin

(33)

This proves the lemma

Lemma 6 If (119876 119880) solves (13)-(14) and the correspondingstrong solution (120588 119906) defined by (10) of (1) satisfies the local-energy estimate (24) then as 119905 uarr 119879 the kinetic energy (21)mustblow up

Proof Recalling the proofs of Lemmas 4 and 5 for119877 gt 1198840(119879minus

119905)119888gt 0

119864 (119905) ge1198882120581 (119879)

8(119879 minus 119905)119886+2119887minus119888

int

119877(119879minus119905)119888

1198840

1199102119889119910

=1198882120581 (119879)

24(119879 minus 119905)119886+2119887+2119888

times [1198773minus 1198843

0(119879 minus 119905)

3119888] 997888rarr +infin as 119905 uarr 119879

(34)

This proves the lemma

Now Theorem 1 follows from the four lemmas above

3 Proof of Theorem 2

If (120588 119906 119904) solves (3)ndash(6) then

(120588(120582) 119906(120582)) = (120582

119886120588 (120582119888119909 120582119889119905) 120582119887119906 (120582119888119909 120582119889119905)

120582119897119904 (120582119888119909 120582119889119905))

(35)

does so for any120582 gt 0 by setting 119886 = 1(120574minus120579) 119887 = (120574minus1)2(120574minus120579) 119888 = (120574 + 1 minus 2120579)2(120574 minus 120579) 119889 = 1 and 119897 = 0 Note that from(6) 119886 gt 0 119887 ge 0 and 119888 gt 0The forward self-similar solutionshave the following form

120588 (119909 119905) =1

119905119886119876(

119909

119905119888)

119906 (119909 119905) =1

119905119887119880(

119909

119905119888)

119904 (119909 119905) = 119878 (119909

119905119888)

119905 gt 0

(36)

where119876(119910) = 120588(119910 1)119880(119910) = 119906(119910 1) and 119878(119910) = 119904(119910 1)Thebackward self-similar solutions are

120588 (119909 119905) =1

(119879 minus 119905)119886119876(

119909

(119879 minus 119905)119888)

119906 (119909 119905) =1

(119879 minus 119905)119887119876(

119909

(119879 minus 119905)119888)

119904 (119909 119905) = 119878 (119909

(119879 minus 119905)119888)

0 lt 119905 lt 119879

(37)

where 119876(119910) = 120588(119910 119879 minus 1) 119880(119910) = 119906(119910 119879 minus 1) and 119878(119910) =119904(119910 119879 minus 1) for 119879 gt 1

Lions [29] investigated the coupled system of the Navier-Stokes equations with an entropy transport equation in a pureform and obtained the existence of weak solutions satisfying(19) and (23)

Proof of Theorem 2 Suppose that (120588 119906 119904) is a forward self-similar solution Inserting (36) into (3) one gets 119880(119910) = 119888119910and thus 119906(119909 119905) = 119888119909119905 Therefore for any 119905 gt 0 (36) (19)and (21) yield

119864 (119905) = intR

1

2sdot1

119905119886119876(

119909

119905119888) sdot (

119888119909

119905)

2

ge1198882120581 (1)

2119905119886+2intR

1199092119889119909 = +infin

(38)

Thismeans that the global-energy estimate (23) does not holdand that the kinetic energy (21) blows up as 119905 darr 0

The case of backward self-similar solutions can be provedsimilarly so Theorem 2 is proved

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Journal of Applied Mathematics 5

Acknowledgments

This work is partially supported by Zhejiang ProvincialNatural Science Foundation of China (no LQ13G030018) andNational Natural Science Foundation of China (nos 11001049and 11226184)

References

[1] J Leray ldquoSur le mouvement drsquoun liquide visqueux emplissantlrsquoespacerdquo Acta Mathematica vol 63 no 1 pp 193ndash248 1934

[2] G K Batchelor An Introduction to Fluid Dynamics CambridgeUniversity Press Cambridge UK 1999

[3] L I Sedov Similarity and Dimensional Methods in MechanicsCRC Press Boca Raton Fla USA 10th edition 1993

[4] O A Barraza ldquoSelf-similar solutions in weak 119871119901-spaces of theNavier-Stokes equationsrdquo Revista Matematica Iberoamericanavol 12 no 2 pp 411ndash439 1996

[5] M Cannone and F Planchon ldquoSelf-similar solutions forNavier-Stokes equations inR3rdquo Communications in Partial DifferentialEquations vol 21 no 1-2 pp 179ndash193 1996

[6] J Necas M Ruzicka and V Sverak ldquoOn Lerayrsquos self-similarsolutions of the Navier-Stokes equationsrdquo Acta Mathematicavol 176 no 2 pp 283ndash294 1996

[7] F Planchon ldquoAsymptotic behavior of global solutions to theNavier-Stokes equations inR3rdquo Revista Matematica Iberoamer-icana vol 14 no 1 pp 71ndash93 1998

[8] T-P Tsai ldquoOn Lerayrsquos self-similar solutions of theNavier-Stokesequations satisfying local energy estimatesrdquoArchive for RationalMechanics and Analysis vol 143 no 1 pp 29ndash51 1998

[9] T-P Tasi ldquoForward discrete self-similarsolutions of the Navier-Stokes equationsrdquo submitted httparxivorgabs12102783

[10] J R Miller M OrsquoLeary and M Schonbek ldquoNonexistenceof singular pseudo-self-similar solutions of the Navier-Stokessystemrdquo Mathematische Annalen vol 319 no 4 pp 809ndash8152001

[11] P G Lemarie-Rieusset Recent Developments in the Navier-Stokes Problem vol 431 ofChapmanampHallCRCResearchNotesin Mathematics Chapman amp HallCRC Boca Raton Fla USA2002

[12] D Chae ldquoNonexistence of asymptotically self-similar singular-ities in the Euler and the Navier-Stokes equationsrdquoMathematis-che Annalen vol 338 no 2 pp 435ndash449 2007

[13] D Chae ldquoNotes on the incompressible Euler and relatedequations on 119877119873rdquo Chinese Annals of Mathematics B vol 30 no5 pp 513ndash526 2009

[14] D Chae ldquoNonexistence of self-similar singularities in the idealmagnetohydrodynamicsrdquo Archive for Rational Mechanics andAnalysis vol 194 no 3 pp 1011ndash1027 2009

[15] L Brandolese ldquoFine properties of self-similar solutions of theNavier-Stokes equationsrdquo Archive for Rational Mechanics andAnalysis vol 192 no 3 pp 375ndash401 2009

[16] X-Y Jiao ldquoSome similarity reduction solutions to two-dimensional incompressible Navier-Stokes equationrdquo Commu-nications inTheoretical Physics vol 52 no 3 pp 389ndash394 2009

[17] I F Barna ldquoSelf-similar solutions of three-dimensionalNaviermdashstokes equationrdquo Communications in TheoreticalPhysics vol 56 no 4 pp 745ndash750 2011

[18] G Karch andD Pilarczyk ldquoAsymptotic stability of Landau solu-tions to Navier-Stokes systemrdquo Archive for Rational Mechanicsand Analysis vol 202 no 1 pp 115ndash131 2011

[19] V Sverak ldquoOn Landaursquos solutions of the Navier-Stokes equa-tionsrdquo Journal of Mathematical Sciences vol 179 no 1 pp 208ndash228 2011

[20] H Jia and V Sverak ldquoLocal-in-space estimates near initial timefor weak solutions of the Navier-Stokes equations and forwardself-similar solutionsrdquo Inventiones Mathematicae 2013

[21] Z Guo and S Jiang ldquoSelf-similar solutions to the isothermalcompressible Navier-Stokes equationsrdquo IMA Journal of AppliedMathematics vol 71 no 5 pp 658ndash669 2006

[22] Z Guo and Z Xin ldquoAnalytical solutions to the compressibleNavier-Stokes equations with density-dependent viscosity coef-ficients and free boundariesrdquo Journal of Differential Equationsvol 253 no 1 pp 1ndash19 2012

[23] M Yuen ldquoSelf-similar solutions with elliptic symmetry for thecompressible Euler and Navier-Stokes equations in R119873rdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 12 pp 4524ndash4528 2012

[24] H An andM Yuen ldquoSupplement to lsquoSelf-similar solutions withelliptic symmetry for the compressible Euler and Navier-Stokesequations in R119873rsquordquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1558ndash1561 2013

[25] A Mellet and A Vasseur ldquoExistence and uniqueness of globalstrong solutions for one-dimensional compressible Navier-Stokes equationsrdquo SIAM Journal on Mathematical Analysis vol39 no 4 pp 1344ndash1365 2008

[26] M Okada S Matusocircu-Necasova and T Makino ldquoFreeboundary problem for the equation of one-dimensional motionof compressible gas with density-dependent viscosityrdquo AnnalidellrsquoUniversita di Ferrara Nuova Serie Sezione VII ScienzeMatematiche vol 48 pp 1ndash20 2002

[27] H Teng C M Kinoshita S M Masutani and J ZhouldquoEntropy generation in multicomponent reacting flowsrdquo Jour-nal of Energy Resources Technology Transactions of the ASMEvol 120 no 3 pp 226ndash232 1998

[28] J O Hirschfelder C F Curtiss and R B BirdMolecularTheoryof Gases and Liquids John Wiley amp Sons New York NY USA1954

[29] P-L Lions Mathematical Topics in Fluid Mechanics vol 2of Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press New York NY USA 1996

[30] K Nishida T Takagi and S Kinoshita ldquoAnalysis of entropygeneration and exergy loss during combustionrdquo Proceedings ofthe Combustion Institute vol 29 pp 869ndash874 2002

[31] Z W Li S K Chou C Shu and W M Yang ldquoEntropy gener-ation during microcombustionrdquo Journal of Applied Physics vol97 no 8 Article ID 084914 2005

[32] H Yapıcı N Kayatas B Albayrak and G Basturk ldquoNumericalcalculation of local entropy generation in a methaneair burnerrdquoEnergy Conversion and Management vol 46 pp 1885ndash19192005

[33] N Lior W Sarmiento-Darkin and H S Al-Sharqawi ldquoTheexergy fields in transport processes their calculation and userdquoEnergy vol 31 no 5 pp 553ndash578 2006

[34] D Stanciu MMarinescu and A Dobrovicescu ldquoThe influenceof swirl angle on the irreversibilities in turbulent diffusionflamesrdquo International Journal of Thermodynamics vol 10 no 4pp 143ndash153 2007

[35] M Safari M R H Sheikhi M Janbozorgi and H MetghalchildquoEntropy transport equation in large eddy simulation for exergyanalysis of turbulent combustion systemsrdquo Entropy vol 12 no3 pp 434ndash444 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Acknowledgments

This work is partially supported by Zhejiang ProvincialNatural Science Foundation of China (no LQ13G030018) andNational Natural Science Foundation of China (nos 11001049and 11226184)

References

[1] J Leray ldquoSur le mouvement drsquoun liquide visqueux emplissantlrsquoespacerdquo Acta Mathematica vol 63 no 1 pp 193ndash248 1934

[2] G K Batchelor An Introduction to Fluid Dynamics CambridgeUniversity Press Cambridge UK 1999

[3] L I Sedov Similarity and Dimensional Methods in MechanicsCRC Press Boca Raton Fla USA 10th edition 1993

[4] O A Barraza ldquoSelf-similar solutions in weak 119871119901-spaces of theNavier-Stokes equationsrdquo Revista Matematica Iberoamericanavol 12 no 2 pp 411ndash439 1996

[5] M Cannone and F Planchon ldquoSelf-similar solutions forNavier-Stokes equations inR3rdquo Communications in Partial DifferentialEquations vol 21 no 1-2 pp 179ndash193 1996

[6] J Necas M Ruzicka and V Sverak ldquoOn Lerayrsquos self-similarsolutions of the Navier-Stokes equationsrdquo Acta Mathematicavol 176 no 2 pp 283ndash294 1996

[7] F Planchon ldquoAsymptotic behavior of global solutions to theNavier-Stokes equations inR3rdquo Revista Matematica Iberoamer-icana vol 14 no 1 pp 71ndash93 1998

[8] T-P Tsai ldquoOn Lerayrsquos self-similar solutions of theNavier-Stokesequations satisfying local energy estimatesrdquoArchive for RationalMechanics and Analysis vol 143 no 1 pp 29ndash51 1998

[9] T-P Tasi ldquoForward discrete self-similarsolutions of the Navier-Stokes equationsrdquo submitted httparxivorgabs12102783

[10] J R Miller M OrsquoLeary and M Schonbek ldquoNonexistenceof singular pseudo-self-similar solutions of the Navier-Stokessystemrdquo Mathematische Annalen vol 319 no 4 pp 809ndash8152001

[11] P G Lemarie-Rieusset Recent Developments in the Navier-Stokes Problem vol 431 ofChapmanampHallCRCResearchNotesin Mathematics Chapman amp HallCRC Boca Raton Fla USA2002

[12] D Chae ldquoNonexistence of asymptotically self-similar singular-ities in the Euler and the Navier-Stokes equationsrdquoMathematis-che Annalen vol 338 no 2 pp 435ndash449 2007

[13] D Chae ldquoNotes on the incompressible Euler and relatedequations on 119877119873rdquo Chinese Annals of Mathematics B vol 30 no5 pp 513ndash526 2009

[14] D Chae ldquoNonexistence of self-similar singularities in the idealmagnetohydrodynamicsrdquo Archive for Rational Mechanics andAnalysis vol 194 no 3 pp 1011ndash1027 2009

[15] L Brandolese ldquoFine properties of self-similar solutions of theNavier-Stokes equationsrdquo Archive for Rational Mechanics andAnalysis vol 192 no 3 pp 375ndash401 2009

[16] X-Y Jiao ldquoSome similarity reduction solutions to two-dimensional incompressible Navier-Stokes equationrdquo Commu-nications inTheoretical Physics vol 52 no 3 pp 389ndash394 2009

[17] I F Barna ldquoSelf-similar solutions of three-dimensionalNaviermdashstokes equationrdquo Communications in TheoreticalPhysics vol 56 no 4 pp 745ndash750 2011

[18] G Karch andD Pilarczyk ldquoAsymptotic stability of Landau solu-tions to Navier-Stokes systemrdquo Archive for Rational Mechanicsand Analysis vol 202 no 1 pp 115ndash131 2011

[19] V Sverak ldquoOn Landaursquos solutions of the Navier-Stokes equa-tionsrdquo Journal of Mathematical Sciences vol 179 no 1 pp 208ndash228 2011

[20] H Jia and V Sverak ldquoLocal-in-space estimates near initial timefor weak solutions of the Navier-Stokes equations and forwardself-similar solutionsrdquo Inventiones Mathematicae 2013

[21] Z Guo and S Jiang ldquoSelf-similar solutions to the isothermalcompressible Navier-Stokes equationsrdquo IMA Journal of AppliedMathematics vol 71 no 5 pp 658ndash669 2006

[22] Z Guo and Z Xin ldquoAnalytical solutions to the compressibleNavier-Stokes equations with density-dependent viscosity coef-ficients and free boundariesrdquo Journal of Differential Equationsvol 253 no 1 pp 1ndash19 2012

[23] M Yuen ldquoSelf-similar solutions with elliptic symmetry for thecompressible Euler and Navier-Stokes equations in R119873rdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 12 pp 4524ndash4528 2012

[24] H An andM Yuen ldquoSupplement to lsquoSelf-similar solutions withelliptic symmetry for the compressible Euler and Navier-Stokesequations in R119873rsquordquo Communications in Nonlinear Science andNumerical Simulation vol 18 no 6 pp 1558ndash1561 2013

[25] A Mellet and A Vasseur ldquoExistence and uniqueness of globalstrong solutions for one-dimensional compressible Navier-Stokes equationsrdquo SIAM Journal on Mathematical Analysis vol39 no 4 pp 1344ndash1365 2008

[26] M Okada S Matusocircu-Necasova and T Makino ldquoFreeboundary problem for the equation of one-dimensional motionof compressible gas with density-dependent viscosityrdquo AnnalidellrsquoUniversita di Ferrara Nuova Serie Sezione VII ScienzeMatematiche vol 48 pp 1ndash20 2002

[27] H Teng C M Kinoshita S M Masutani and J ZhouldquoEntropy generation in multicomponent reacting flowsrdquo Jour-nal of Energy Resources Technology Transactions of the ASMEvol 120 no 3 pp 226ndash232 1998

[28] J O Hirschfelder C F Curtiss and R B BirdMolecularTheoryof Gases and Liquids John Wiley amp Sons New York NY USA1954

[29] P-L Lions Mathematical Topics in Fluid Mechanics vol 2of Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press New York NY USA 1996

[30] K Nishida T Takagi and S Kinoshita ldquoAnalysis of entropygeneration and exergy loss during combustionrdquo Proceedings ofthe Combustion Institute vol 29 pp 869ndash874 2002

[31] Z W Li S K Chou C Shu and W M Yang ldquoEntropy gener-ation during microcombustionrdquo Journal of Applied Physics vol97 no 8 Article ID 084914 2005

[32] H Yapıcı N Kayatas B Albayrak and G Basturk ldquoNumericalcalculation of local entropy generation in a methaneair burnerrdquoEnergy Conversion and Management vol 46 pp 1885ndash19192005

[33] N Lior W Sarmiento-Darkin and H S Al-Sharqawi ldquoTheexergy fields in transport processes their calculation and userdquoEnergy vol 31 no 5 pp 553ndash578 2006

[34] D Stanciu MMarinescu and A Dobrovicescu ldquoThe influenceof swirl angle on the irreversibilities in turbulent diffusionflamesrdquo International Journal of Thermodynamics vol 10 no 4pp 143ndash153 2007

[35] M Safari M R H Sheikhi M Janbozorgi and H MetghalchildquoEntropy transport equation in large eddy simulation for exergyanalysis of turbulent combustion systemsrdquo Entropy vol 12 no3 pp 434ndash444 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of