On a generalisation of compactness by compensation in theLp–Lq setting

Marin Misur

email: pfobos@gmail.comDepartment of Mathematics, Faculty of Science

University of Zagreb

joint work with Darko Mitrovic, University of Montenegro and University of Bergen

October 11, 2013

Motivation - Maxwell’s equationsLet ⌦ ✓ R

3. Denote by E and H the electric and magnetic field, and by D andB the electric and magnetic induction. Let ⇢ denote the charge, and j thecurrent density. Maxwell’s system of equations reads:

@t

B+ rot E = G,

div B = 0,

@t

D+ j� rot H = F,

div D = ⇢.

Assume that properties of the material can be expressed by following linearconstitutive equations:

D = ✏E, B = µH.

The energy of electromagnetic field at time t is given by:

T (t) =1

2

Z

⌦

(D · E+ B · H)dx.

It’s natural to consider

D,B 2 L1([0, T ]; L2

div (⌦;R3)),

E,H 2 L1([0, T ]; L2

rot (⌦;R3)),

J 2 L1([0, T ]; L2(⌦;R3)), F,G 2 L2([0, T ]; L2(⌦;R3)).

Let us consider a family of problems:

@t

Bn + rot En = Gn,

@t

Dn + Jn � rot Hn = Fn,

with constitution equations:

Dn = ✏nEn, Bn = µnHn, Jn = �nEn.

What can we say about energy T (t) if we know

Tn(t) =1

2

Z

⌦

(Dn · En + Bn · Hn)dx.

Div-rot lemma in L2

Theorem 1. Assume that ⌦ is open and bounded subset of R3, and that itholds:

u

n

* u in L2(⌦;R3),

v

n

* v in L2(⌦;R3),

rotun

bounded in L2(⌦;R3), divvn

bounded in L2(⌦).

Thenu

n

· vn

* u · v

in the sense of distributions.

Quadratic theorem

Theorem 2. (Quadratic theorem) Assume that ⌦ ✓ R

d is open and that⇤ ✓ R

r is defined by

⇤ :=n

� 2 R

r : (9 ⇠ 2 R

d \ {0})d

X

k=1

⇠k

A

k� = 0o

,

where Q is a real quadratic form on R

r, which is nonnegative on ⇤, i.e.

(8� 2 ⇤) Q(�) > 0 .

Furthermore, assume that the sequence of functions (un

) satisfies

u

n

�* u weakly in L2

loc

(⌦;Rr) ,

⇣

X

k

A

k@k

u

n

⌘

relatively compact in H�1

loc

(⌦;Rq) .

Then every subsequence of (Q � un

) which converges in distributions to it’slimit L, satisfies

L > Q � u

in the sense of distributions.

IntroductionClassical results

L2 theoryResult by Panov

H-distributionsDefinitionLocalisation principle

Application to the parabolic type equation

The most general version of the classical L2 results has recently been proved byE. Yu. Panov (2011):

Assume that the sequence (un

) is bounded in Lp(Rd;Rr), 2 p < 1, andconverges weakly in D0(Rd) to a vector function u.Let q = p0 if p < 1, and q > 1 if p = 1. Assume that the sequence

⌫

X

k=1

@k

(Ak

u

n

) +

d

X

k,l=⌫+1

@kl

(Bkl

u

n

)

is precompact in the anisotropic Sobolev space W�1,�2;q

loc

(Rd;Rm), wherem⇥ r matrices Ak and B

kl have variable coe�cents belonging to L2q(Rd),q = p

p�2

if p > 2, and to the space C(Rd) if p = 2.

We introduce the set ⇤(x)

⇤(x) =

(

� 2 C

r

�

� (9⇠ 2 R

d \ {0}) : (1)

i

⌫

X

k=1

⇠k

A

k(x)� 2⇡

d

X

k,l=⌫+1

⇠k

⇠l

B

kl(x)

!

� = 0

m

)

,

and consider the bilinear form on C

r

q(x,�,⌘) = Q(x)� · ⌘, (2)

where Q 2 Lq

loc

(Rd; Symr

) if p > 2 and Q 2 C(Rd; Symr

) if p = 2.Finally, let q(x,u

n

,un

)* ! weakly in the space of distributions.

Result by Panov

The following theorem holds

Theorem 3. [P, 2011] Assume that (8� 2 ⇤(x)) q(x,�,�) � 0 (a.e. x 2 R

d)and u

n

* u, then q(x,u(x),u(x)) !.

The connection between q and ⇤ given in the previous theorem, we shall callthe consistency condition.

H-distributions

H-distributions were introduced by N. Antonic and D. Mitrovic (2011) as anextension of H-measures to the Lp � Lq context.

M. Lazar and D. Mitrovic (2012) extended and applied them on a velocityaveraging problem.

We need Fourier multiplier operators with symbols defined on a manifold Pdetermined by d-tuple ↵ 2 (R+)d:

P =n

⇠ 2 R

d :d

X

k=1

|⇠k

|l↵k = 1o

,

where l is the smallest number such that l↵k

> d for each k. In order toassociate an Lp Fourier multiplier to a function defined on P, we extend it toR

d\{0} by means of the projection

(⇡P

(⇠))j

= ⇠j

⇣

|⇠1

|l↵1 + · · ·+ |⇠d

|l↵d

⌘�1/l↵j, j = 1, . . . , d.

Theorem 4. [LM, 2012] Let (un

) be a bounded sequence in Ls(Rd), s > 1,and let (v

n

) be a bounded sequence of uniformly compactly supportedfunctions in L1(Rd). Then, after passing to a subsequence (not relabelled),for any s 2 (1, s) there exists a continuous bilinear functional B on

Ls

0(Rd)⌦ Cd(P) such that for every ' 2 Ls

0(Rd) and 2 Cd(P) it holds

B(', ) = limn!1

Z

Rd'(x)u

n

(x)�

A Pvn

�

(x)dx ,

where A P is the Fourier multiplier operator on R

d associated to � ⇡P

.

Corollary 1. [LM, 2012] The bilinear functional B defined in the previoustheorem can be extended by continuity to a continuous functional onLs

0(Rd;Cd(P)).

We need the following extension of the results given above.

Theorem 5. Let (un

) be a bounded sequence in Lp(Rd), p > 1, and let (vn

)be a bounded sequence of uniformly compactly supported functions in Lq(Rd),1/q + 1/p < 1. Then, after passing to a subsequence (not relabelled), for anys 2 (1, pq

p+q

) there exists a continuous bilinear functional B on

Ls

0(Rd)⌦ Cd(P) such that for every ' 2 Ls

0(Rd) and 2 Cd(P), it holds

B(', ) = limn!1

Z

Rd'(x)u

n

(x)�

A Pvn

�

(x)dx ,

where A P is the Fourier multiplier operator on R

d associated to � ⇡P

.The bilinear functional B can be continuously extended as a linear functionalon Ls

0(Rd;Cd(P)).

Localisation principle

Lemma

Assume that sequences (un

) and (vn

) are bounded in Lp(Rd;Rr) andLq(Rd;Rr), respectively, and converge toward u and 0 in the sense ofdistributions.Furthermore, assume that sequence (u

n

) satisfies:

G

n

:=d

X

k=1

@↵kk

(Ak

u

n

) ! 0 in W�1,p(⌦;Rm), (3)

where either ↵k

2 N, k = 1, . . . , d or ↵k

> d, k = 1, . . . , d, and elements ofmatrices Ak belong to Ls

0(Rd), s 2 (1, pq

p+q

).Finally, by µ denote a matrix H-distribution corresponding to subsequences of(u

n

) and (vn

� v). Then the following relation holds

⇣

d

X

k=1

(2⇡i⇠k

)↵kA

k

⌘

µ = 0.

Proof of the lemma

Assume, without loosing any generality, that v = 0. Denote by B

the Fouriermultiplier operator with the symbol

( � ⇡P

)(⇠)(1� ✓(⇠))

(|⇠1

|l↵1 + · · ·+ |⇠d

|l↵d)1/l.

According to [LM 2012, Lemma 5], for any 2 Cd(P) and any s > 1, themultiplier operator B

: L2(Rd) \ Ls(Rd) ! W↵;s(Rd), is bounded (with Ls

norm considered on the domain of operator B

and ↵ = (↵1

, . . . ,↵d

))Take a test function g

n

given by:

g

n

(x) = B

�

�vn

�

(x),

where 2 Cd(P) and � 2 C1c

(Rd) and multiply with G

n

.

Proof of the lemma - continued

We get

Z

RdG

n

· gn

dx =

Z

Rd

d

X

k=1

A

k

u

n

· A( �⇡P)(⇠)

(1�✓(⇠))(2⇡i⇠k)↵k

(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v

n

)dx

=

Z

Rd

d

X

k=1

A

k

u

n

· A( �⇡P)(⇠)

(2⇡i⇠k)↵k

(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v

n

)dx +

+

Z

Rd

d

X

k=1

A

k

u

n

· A( �⇡P)(⇠)

✓(⇠)(2⇡i⇠k)↵k

(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v

n

)dx.

Taking into account the preceeding theorem and strong precompactess ofsequence (G

n

), we get

⇣

d

X

k=1

(2⇡i⇠k

)↵kA

k

⌘

µ = 0.

Strong consistency condition

Introduce the set

⇤D =n

µ 2 Ls(Rd; (Cd(P))0)r :⇣

n

X

k=1

(2⇡i⇠k

)↵kA

k

⌘

µ = 0

m

o

,

where the given equality is understood in the sense of Ls(Rd; (Cd(P))0)m.

Definition

We say that set ⇤D, bilinear form q from (2) and matrixµ = [µ

1

, . . . ,µr

],µj

2 Ls(Rd; (Cd(P))0)r satisfy the strong consistencycondition if (8j 2 {1, . . . , r}) µ

j

2 ⇤D, and it holds

h�Q⌦ 1,µi � 0, � 2 Ls(Rd;R+

0

).

Compactness by compensation

Let us assume that coe�cients of the bilinear form q from (2) belong to spaceLt

loc

(Rd), where 1/t+ 1/p+ 1/q < 1.

Theorem 6. Assume that sequences (un

) and (vn

) are bounded inLp(Rd;Rr) and Lq(Rd;Rr), respectively, and converge toward u and v in thesense of distributions.Assume that (3) holds and that

q(x;un

,vn

)* ! in D0(Rd).

If the set ⇤D, the bilinear form (2), and matrix H-distribution µ,corresponding to subsequences of (u

n

� u) and (vn

� v), satisfy the strongconsistency condition, then

q(x;u,v) ! in D0(Rd).

Proof of the theorem

Let us abuse the notation by denoting u

n

= u

n

� u* 0 andv

n

= v

n

� v * 0.According to the theorem on existence of H-distributions, for any non-negative� 2 D(Rd) it holds

limn!1

Z

RdQu

n

· vn

� dx = h�Q⌦ 1,µi,

where µ is matrix H-distribution corresponding to (subsequences of)u

n

,vn

* 0. Since, according to the localisation principle, for every fixedk 2 {1, . . . , r}, the r-tuple µ

k

belongs to ⇤D, we conclude from the strongconsistency condition that

h�Q⌦ 1,µi � 0.

From the fact that

q(x;un

,vn

)* ! � q(x;u,v) � 0 in D0(Rd),

the statement of the theorem follows.

Application to the parabolic type equation

Now, let us consider the non-linear parabolic type equation

L(u) = @t

u� div div (g(t,x, u)A(t,x)),

on (0,1)⇥ ⌦, where ⌦ is an open subset of Rd. We assume that

u 2 Lp((0,1)⇥ ⌦), g(t,x, u(t,x)) 2 Lq((0,1)⇥ ⌦), 1 < p, q,

A 2 Ls

loc

((0,1)⇥ ⌦)d⇥d, where 1/p+ 1/q + 1/s < 1,

and that the matrix A is strictly positive definite, i.e.

A⇠ · ⇠ > 0, ⇠ 2 R

d \ {0}, (a.e.(t,x) 2 (0,1)⇥ ⌦).

Furthermore, assume that g is a Caratheodory function and non-decreasingwith respect to the third variable.

Then we have the following theorem.

Theorem 7. Assume that sequences (ur

) and g(·, ur

) are such thatur

, g(ur

) 2 L2(R+ ⇥R

d) for every r 2 N; assume that they are bounded inLp(R+ ⇥R

d), p 2 (1, 2], and Lq(R+ ⇥R

d), q > 2, respectively, where1/p+ 1/q < 1; furthermore, assume u

r

* u and, for some,f 2 W�1,�2;p(R+ ⇥R

d), the sequence

L(ur

) = fr

! f strongly in W�1,�2;p(R+ ⇥R

d).

Under the assumptions given above, it holds

L(u) = f in D0(R+ ⇥R

d).

References

[AM] N. Antonic, D. Mitrovic, H-distributions: An Extension of H-Measures to anLp � Lq Setting, Abstr. Appl. Anal. Volume 2011, Article ID 901084, 12pages.

[LM] M. Lazar, D. Mitrovic, Velocity averaging – a general framework, Dynamics ofPDEs 9 (2012) 239–260.

[P] E. J. Panov, Ultraparabolic H-measures and compensated compactness,Annales Inst. H.Poincare 28 (2011) 47–62.