stability of the kuramoto-sivashinsky and related systems

Stability of the Kuramoto-Sivashinsky and Related Systems

JONATHAN GOODMAN Courant Institute

Dedicated with admiration to Henry McKean, who certainly knows the solution of (3.7).

1. Introduction

The Kuramoto-Sivashinsky (KS) equation in one dimension,

(1.1) ur + uux = - u x x - ~ x x x x 9

is a model for nonlinear evolution of linearly unstable flame fronts; see [l I ] and [ 171. The right-hand side represents linearized behavior: disturbances of low wave number are amplified while high wave number disturbances are damped. The uux term on the left is thought of as a “nonlinear energy transfer mechanism” that can move energy from low to high wave numbers and thus prevent unbounded growth of low wave number modes. This nonlinear stabilization is observed in numerical computations with periodic boundary conditions,

(1 .2)

and has been proven mathematically by Nicolaenko, Scheurer, and Temam (NST) in [ 161 under the additional hypothesis that the solution is odd: u(x, t ) = 4 - x , t) . The main purpose of this work is to remove this oddness restriction, which we do with a slight generalization of the NST Liapounov function method.

u(x + R, t ) = u(x, t ) for all x and t ,

We will assume that the solution has mean value zero,

for all t. This is natural for two reasons. First, the derivation of (1.1) proceeds by first deriving

1 2 - 4, + j 4 x - - 4 x x - 4xxxx 9

and then taking u = + x . If 4 satisfies periodic boundary conditions then u will have mean value zero. Second, if (1.3) did not hold, we could reduce to the case of ( 1 -3) by subtracting a constant from u without affecting the dynamics. That constant would complicate the statement of

THEOREM 1. Solutions of the KS equation (1.1) with side Conditions (1.2) and (1.3) sarisfy the stability estimate

IIu(.,t)llL2 5 C2(R) for all t

Communications on Pure and Applied Mathematics, Vol. XLVII, 293-306 (1994) 0 1994 John Wiley & Sons, Inc. CCC 0010-3640/94/030293 -14

294 J. GOODMAN

if I Iu(.,O)IIL~ 5 C1(R) ,

where C2 is independent of t . Moreover, C I ( R ) 5 C2(R) 5 C - R5/2, where C is an absolute constant.

Numerical computations suggest that the best constants C2(R) and Cl(R) in Theorem 1 grow rather more slowly than R5I2; see [18]. However, it may be imprudent to conjecture small constants estimated by computer simulations. KS solutions are observed to be chaotic (see [15] and [18]) and chaotic systems have much in common with random systems. A random system might have typical behavior of one kind but be subject to occasional fits (large deviations) of different behavior. Such fits might be hard to detect in computer simulations either because they are exponentially rare or because they are not robust enough to survive numerical approximation.

To explore the sharpness of our proof, we considered other models to which it applies, in particular, the “Burgers-Sivashinsky” (BS) equation’ :

(1.4) Ut + uu, = u + u,, . This superficially seems to have much in common with the KS equation (1.1). It too has low wave number instability, high wave number damping, and nonlinear stabilization via energy transfer. In fact, the proof of Theorem 1 immediately leads to:

THEOREM 2. Solutions of the BS equation (1.4) with side conditions (1.2) and ( I .3) satisfy the stability estimate

IIu(.,t)llLz 5 C2(R) for all t

if I Iu( - ,O) I IL~ 5 C I ( R ) ,

where C2 is independent of t . Moreover, CI (R) 5 C2(R) 5 C . R3/2, where C is an absolute constant.

As with KS, the integral zero condition (1.3) is preserved under the evolution (although the integral itself need not be) and can be achieved by a normalization. If u(x, t ) satisfies (1.4) then so does ii(x, t ) = u(x + a(t), t ) - b(t), provided that b = b and u = b. We can choose b(t = 0) so that ii has mean value zero.

Despite the similarity between KS and BS, when R is large their solutions have different qualitative behavior. KS solutions are observed to have high dimensional chaos (see [ 151) while BS solutions just approach time independent steady states as t - 00. Whatever the recipe for chaos is, it must be subtle to distinguish so sharply between these two equations.

Equation (1.4) was not proposed by Burgers or Sivashinsky but it represents some composite of ( 1 . 1 ) with Burgers’ equation; see [12].

KURAMOTO-SIVASHINSKY AND RELATED SYSTEMS 295

The BS equation gives some insight into the shortcomings of our proof of Theorem 1. In several respects the proof is sharp for BS and reflects the behavior of BS solutions. To the extent that BS and KS solutions are different, then, the proof does not reflect KS behavior. For example, the steady solutions of BS have L2 norm of order R3/2 so the constants have the correct order of magnitude in Theorem 2. This makes it seem as though the proof of Theorem 1 cannot be improved to give C2 as small as observed in computations. Moreover, the proof of Theorem 1 makes use of an artificial length scale E = O( l/R). In the context of KS this may indeed be artificial; such small length scales are not seen in computations. However, the 1 / R length scale is prominent in the large time behavior of BS solutions (see Section 3), which explains its presence in the proof. This motivates the search for another stability proof for KS, one that does not apply to BS.

The BS equation (1.4) also provides an example that may be of interest to the study of finite dimensional manifolds for dissipative nonlinear PDE’s; see [3]. Inertial manifolds for BS would have dimension at least n 2 const. R for large R since the rest state u = 0 has an unstable manifold of that size. However this high dimensional manifold is irrelevent for large time behavior of BS solutions, which is trivial. In this respect, and in several others, the qualitative behavior of BS solutions resembles that of solutions of the Cahn-Allan equation in one space dimension; see 111.

2. Proof of Theorem 1 and Theorem 2

The proof uses the method of Liapounov. We exhibit a functional F(u) with the property that there is an F , so that if F ( u ) L F , and u satisfies either KS (1 . I ) or BS (1.4), then d F / d i 5 0. This implies that F ( u ( . , t ) ) 5 F , for all t provided that F ( u ( . , O)F(u(-, 0)) 5 F , and u remains regular. Theorem 1 and Theorem 2 will follow if F(u) 5 F , 3 llull~z. 5 C2 and IIuIIL2 I C1 F(u) 5 F , . In the examples below, we actually have F 5 C . F if F 2 F , . This implies the existence of an “absorbing ball”; see [3].

We illustrate the construction of F in an oversimplified model problem

(2.1) x - x - x y

y = -y + x 2 .

Here, x repersents the unstable low wave number modes and y represents the stable high wave numbers. The nonlinear terms by themselves conserve energy ( E = x2 + y 2 ) just as uu, conserves u2 in KS or BS. The nonlinear terms transfer energy from x to y in a way that prevents x - 00 as t - 00. In particular, y > 1 is good because it stabilizes the x equation. If increasing y is good, we should choose F to decrease with increasing y. This motivates trying

1 F = ; (x2 + y 2 ) - CY .

Taking c = 2 yields F = - (x’ + y2) + 2y .

296 J. GOODMAN

When (x,y) is large, the quadratic part dominates the linear part and leads to decrease in F. First, I2y I 5 $ (x2 + y2) + 4 so F 5 x2 + y2 + 4. Therefore,

if F 2 F , = 8, then L 2, so 2. y 5 dm. ,/- = x2 + y2, and F 5 0. Finally, F(x, y) 5 F , = 8 * x2 + y2 5 C2 = 48 and F(x, y) 5 8 if x2 + y2 5 C1 = 4. This proves the analogue of Theorem 1 for the model (2.1).

For application to KS or BS a reinterpretation of F due to NST is helpful. Note that F can be written

1 1 2

2 2 F(x, y) = - (2 + y2) - 2y = - ((x - x0l2 + (Y - Yo) ) + c

where (xo, yo) = (0,2) and C = -2. This motivates the NST ansatz

where s(x) is some well chosen function. This approach works perfectly when u is periodic and odd but it is unlikely to work here because the problem is translation invariant. If the special function s(x) were unique it would also be translation invariant and hence uselessly constant.

The NST idea can be rescued by generalizing it slightly. Let Y be a set of functions that is translation invariant aS a set: if s(x) E 9’ then s(x + E ) E Y for any constant E . Then the functional

2 F(u) = dist2(u,Y) = inf IIu - sIILz S € Y

is translation invariant. For us it will suffice to take Y = { s(x + E ) 1 0 5 E 5 R } for a very specific smooth periodic s(x) to be chosen later.

must satisfy the first-order optimality condition

The optimal translate, [*, may not be unique, but any optimal

Now s(x) is periodic and smooth, so ss’ = 0. Since u has mean value 0 (hypothesis (1.3)), the optimality condition implies that

for any constant a and optimal E * . The optimal [ * = E*( t ) will change with time but this need not be taken into

account when computing the time derivative of F(u(.,t)). If you want to know that

F(u(t + t’)) 5 F(u(t)) = dist2 (u(t), 9’) ,


it suffices to choose2 s , ( t ) E 9 with IIu(t) - s*(t)1I2 = dist2 ( u ( t ) , Y ) and show that

2 “u(t + t’) - s*( t ) l l 5 F(u(t)) + oV2) ,

i.e.. that

when u satisfies ( 1 . l ) or (1.4). For any particular time t we may normalize so that

Thus, the proof reduces to the following: Find a suitable smooth function, s(x), periodic with period R, and a constant F , ( R ) so that if so u(x, t ) s ’ (x)dx = 0 and s,” (u(x, t ) - ~ ( x ) ) ~ dx 2 F , then ;i; so (u(x, t ) - s(x))~ d x 5 0. To do this for KS, we calculate using (1.1):

< * ( t ) = 0.

R

d R

The inequality 2k2 d i k 4 + 2 for all k implies, via Fourier transform, that

Now write

1 2

- -s’(x) = -3 + b(x) (2.3)

to get (after some calculation with Cauchy Schwarz)

We will now make choices that force the right side to be negative if u is large enough. The main technical point in the stability proof is that we can choose b(x) so that the Poincark inequality

(2.5) J b(x)u(x) dx = o 3 J b(x)u2(x) d x 5 J u:(x> dx

holds. Note that the optimality condition (2.2) is exactly the orthogonality condition that is the hypothesis of (2.5).

It is consistent with (2.3) to choose a periodic function b(x) with the following properties:

(2.6) b(x) 2 0 for all x ,

* This seems to use the axiom of choice in a fundamental way.

J. GOODMAN

l R b ( x ) d x = 3 . R ,

298

(2.7)

(2.9)

(2.10)

9 R supb(x) = - . - ,

X 4 E

27 R2 [& b’(x)’ dx = - . -

2 1 3 .

It seems good to take b(x) to be a quadratic between in the interval -E 5 x 5 E .

PROPOSITION 1. If b(x) satis$es (2.6), (2.7), (2.8), and (2.9), and if E 5 B / R for su@ciently small 9, then (2.5) holds.

Proof For any x and y ,

4x1 = u(y) + Lx u’(z)dz .

Multiply this b(y) and integrate over y using the orthogonality condition

and (2.7) and (2.81, to get

u(x) = 1 Jz=x b(y)u’(z) d z d y .

After reversing the order of integration, this is, for 1x1 5 E ,

3R Y=--E z=y

(2.1 1)

with -1 Y & - 3R sI Y z b ( y ) d y if z 2 x

3~ JyY==fE b(y)dy if z d x

From (2.6) and (2.7) we have IK(x,z)I S 1 for 1x1 5 E and IzI 5 E . Thus the Hilbert-Schmidt norm of kernel K satisfies

K(x,z ) =


so if u is gotten from u’ by (2.11) then

u2(x)dx 5 IIKllis s’ ~ ’ ( x ) ~ dx 5 4~~ u’(xI2 dx , J: --E

so / b(x)u2(x) dx 5 sup I b(x)I . u ( x ) ~ dx 5 9&R/ u’(x)~ dx I x l 5 e L

This proves (2.5) if E = 1/(9R). To finish the argument, combine (2.4) with (2.5) to get

d R - 1 (u - s ) ~ dx S / (s (x ) + ~ ” ( x ) ) ~ dx - 1 u(x, t)2 dx . dt

If we choose Is(x)l 5 3 1x1 for 1x1 5 R/2 (as we may), then

21 R2 Finally, s s ’ ’ ( x ) ~ dx = s b’(xI2 dx = T . 7 < 9000 . R’. This proves Theorem 1.

Small modifications are needed for the BS Theorem 2. Using (4), we compute

where -fs’(x) = -2 + b(x). The terms [ . . . I are controlled using the Poincare inequality (2.5), which holds if E 5 1/(9R). The quantity (...} is negative if s u2 L const ’ R3.

3. Behavior of BS Solutions

We always assume that R, the length of the interval, is large. Some of the statements below may not hold in the case of moderate or small R. The mean zero hypothesis (1.3) is in force throughout. The facts are:

FACT 1. There exists a family of steady periodic solutions whose structure is given by asymptotic analysis, see Figures 1 and 2.

A periodic function, f (x) , has primitive period P if f has period P but f does not have period P/n for any n > 1.

FACT 2. Steady periodic solutions with primitive period R are stable to perturbations satisfying the mean zero condition (1.3); steady solutions with period R but smaller primitive period are all unstable to such perturbations.

300

20.0 I . . I , I

0.0

-10.0 -

-20 0

10.0 v -

5 J. GOODMAN

Steady period 40 BS solution

20.0

10.0

0.0

-10.0

-20.0

I

-

-

I

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

Figure 1. transitions are at x = 20 and x = -20.

Steady periodic solution of the BS equation with period R = 40. Shock

Shock transition in a steady BS solution

Figure 2. A blowup of the shock transition region near x = 20 from Figure 1. The solid line is the exact solution. The dashed line is leading term in its asymptotic expansion: ktanh[k(x - R/2) /2] with k = R / 2 = 20. Including the correction term gives a curve that is indistinguishable in this plot from the exact solution. Note the range of x values marked on the horizontal axis.

KURAMOTO-SIVASHINSKY AND RELATED SYSTEMS 30 1

FACT 3. The BS equation has the structure of a gradient system. A generic3 time dependent solution converges to a steady solution as t - x (no chaos). The global attractor consists of steady periodic solutions and heteroclinic orbits between them.

I will explain these facts without giving full technical proofs. Finding the structure of steady BS solutions with large primitive period is a simple exercise using the method of matched asymptotic expansions; see [9]. The steady profile equation, uu, = u + u,, has two asymptotic simplifications. One is uu, = u, or u = x + C . The other is uu, = u,, or u = -k tanh [k(x - x,)/2]. To get a periodic solution with large primitive period R, these pieces must be matched together as illustrated in Figures 1 and 2, where k = R/2 and the arbitrary choice C = 0 forces xo = R/2. This shows that the length scale l / k = O(l/R) arises naturally in the BS eqation: it is the width of the shock transition layer. Figure 1 has such a transition layer at x = 20. Figure 2 is a detail of this transition region. A phase plane analysis (see below) shows that all steady solutions with primitive period R have this form.

The method of matched asymptotic expansions also gives a family of time dependent BS solutions with moving fronts, using the ideas of [l], [5], [61, [lo], and [ 141. To place the moving front at x = x&), we take u(x, t ) = x for xo(t)-R < x < xo(t) and u(x, t ) = xo - R/2 - k tanh[k(x - xo(t))/2] in the transition region x - xg = O(l/R) (k = R/2 as before). We get an approximate a BS solution if $(XO - R/2) = xo - R/2. This solution is a viscous perturbation of the inviscid construction of moving shock solutions to the model duct equations; see [ 131 and [4]. The solution xo(t) = R/2 is the steady solution given before. See Figure 4.

There are certain exact time dependent BS solutions related to these but with different boundary conditions. Let $(x) be the steady BS solution having the structure: $(x) - x + R/2 as x - -00, +(x) - x - R/2 as x - co, and $(x) - k .tanh(kx/2) ( k = R/2 as before) for x = O( f ). Then u(x, t ) = $(x - e') + e' is an exact BS solution having the form of a moving viscous front.

In general these time dependent solutions violate the integral zero constraint (1.3). However, consider a steady solution with primitive period R/2 and shocks matched in at XO = R/4 and XI = -R/4. Then the unsteady solution with shocks matched in at xo(r) and xl( t ) where ;i;(xo(t) - XO) = xo(t) - 30 and z(xl(t) - X I 1 = x l ( t ) - XI will satisfy the integral constraint if xo(t) + xl(t) = 0. This shows that steady periodic solutions whose primitive period is large but less than R are unstable. Figure 3 is a numerical computation of this instability. It shows a primitive period R = 20 solution, having been slightly perturbed, going over to the period R = 40 solution. During the evolution, the front that starts at XO = 10 moves to the right while the front starting near .%I = -10 moves to the left. Eventually, two shock transitions of strength R/2 combine to form a single transition of strength R . This happens between time t = 16 and time t = 16.5 in Figure 3. This

d d

See Remark 3 below.

302 J. GOODMAN

Subhormonic instobility of o periodic BS solution Period 20 steady solution subject to a period 40 perturbation

20.0

10.0

0.0

-10.0

-20 0 -. ~

-30.0 -20.0 -10.0 0.0 10.0 2 0 0 30.0 ~ Times t = 0 , t = 13, t = 15.5, t = 17 _ _ - Times f = t i . t = 14, t = 16

Times t = 12, t = 15, t = 16.5

Figure 3. Period 40 instability of a period steady 20 BS solution. Plots of u(x, t ) as a function of x for t values listed in the legend. The final curve u(x, I = 16.5) is (almost exactly) the primitive period 40 steady solution.

Instability of a BS shock transition Blowup of the reyon around x = 10 showing the beginning 01 the instability

10.0

0.0

-10.0

-7nn I 10.0 - 10.5 11.0 11.5

~ limes I = 0. t = 11. t = 15. t = 14.5 _ _ _ limes I = 9. t = 12. t = 13.5 ~~~~ limes I = 10. t = 125. t = 14

9.5

Figure 4. described in the text. Note the range of x values listed on the horizontal axis.

Blowup from Figure 3. The curves seem to be translates of each other as


evolution does not depend on the form of the initial perturbation if it is small enough, except that the small transitions could have combined at x = 0.

It remains to show that steady solutions, u, with primitive period P = 0(1) are unstable for large R. For this, linearize about u and consider the eigenvalue problem,

(3.1) AV = LZ'V = v + vxx - ~ , ( u v ) ,

for eigenfunctions with period P (the same as the steady solution u). If we ignore the integral constraint then (3.1) has eigenpair

(3.2) A = 1 , v(x) = vo(x) = e$(') where 4, = u .

Of course, we have not ignored the integral constraint on u. Now consider eigenfunctions for (3.1) with primitive period R having the Bloch wave form v(v) - e2nix/R (v&) + f vl (x) +. . .). These solutions satisfy the integral constraint and have eigenvalue A = 1 - O( l /R) . Thus, all steady solutions with primitive period P < R are unstable to period R disturbances with mean zero.

To investigate the BS phase portrait more closely we use a variational formu- lation given by the Hopf-Cole transformation; see [12]. We again write u = ax+ so that (1.4) is a consequence of

(3.3)

Under the change of variable

4 = -2log($)

(3.3) itself follows from

(3.4)

This equation is not linear as it would have been for Burgers' equation, but it is still useful because of its variational structure. That is, (3.4) can be written

$r = $xx + $log($)

$r = - s @ w where 6 is the first variation, and S($) is the energy functional

with F'($) = $ log($), i.e., F($) = -I,!I~ log($)/2 + $2/4. This variational structure gives some information but it has the drawback that 8 is not bounded from below and the constraint 8($) 5 const does not restrict $ to a compact set in L2.

These difficulties can be overcome using a little renormalization trick. If $(x, t) = A(?)&, t ) then (3.4) becomes

pr = p x x + plog(p) + Ap 3

304 J. GOODMAN

with A(t) = log(A(t)) - A(t)/A(t). Clearly A ( t ) can be chosen essentially arbitrarily as long as it remains finite. This equation has the structure of a gradient flow, but one restricted to the surface

(3.5)

if A is chosen by the Lagrange multiplier formula

A = 2M (lR p: dx - 1" p2 log(p) d x ) .

It is an exercise in Sobolev inequalities to show that if %(p) and M are finite, and p is non-negative, then A is finite. Also, under the constraint (3.5), %(p) is bounded from below and the sets %(p) 5 const are compact.

The critical points of %(p) restricted to the surface (3.5) correspond to steady periodic BS solutions and satisfy

(3.7)

from which the structure of steady BS solutions can be determined. Fact 3 now follows from this variational structure together with general theory

about gradient systems; see [7]. The only complication is that the minimizer is not unique since the problem is rotationally invariant. The set of minimizers is the set of translates of any particular one. General theory assures us that the w-limit set of a solution will be contained in the set of minimizers but it does not guarantee that it will be a single point. There are counterexamples.

Our problem is not a counterexample because it is non-degenerate: the set of minimizers is locally exponentially attracting. To see this we study the linearization of the BS equation (1.4) about a steady solution, u. That is: vt = 2 ' v where v in an infinitesimal perturbation of u and 2 is the linearized operator defined in (3.1). The nondegeneracy comes from facts about the spectrum of 2'.

px = -.J. + Pp2 - p2 log(p)

(i) All eigenvalues of 2' are real. In the eigenvalue equation (3.1) write v = e@/2w where r$x = u and get hw = w,, + q(x)w, for some real periodic function q(x). The eigenvalues of this self adjoint problem are real.

(ii) The largest eigenvalue is A1 = 1 . The eigenfunction vo from (3.2) is strictly positive so its corresponding w = e-@/2vo is also. A positive eigenfunction of a Schrodinger operator must correspond to the largest eigenvalue and that eigenvalue is simple.

(iii) The next largest eigenvalue is A2 = 0. If there were another eigenvalue A2 > 0 then the linearized evolution problem vr = 2'v would be unstable even under the integral zero restriction (1.3). This is impossible because p = const . e@ (the Hopf-Cole variable) is a minimizer. The eigenfunction corresponding to rotation, v = u,, has eigenvalue zero.


(iv) The eigenvalue A2 = 0 is simple; A3 < 0. If not there is another eigenfunction, + with 2’+ = rC, + rclXX - (urC,), = 0, having period R. This is not possible. The Wronskian W(v, +) = uX+, - uXX@ satisfies W , = uW. If there is a linearly independent IC, we can choose W = ee with 4, = u again. This leads to u , + ~ = uXX@ + ee, from which it follows (by a tedious argument using the structure of u) that + can be chosen positive if it has period R. This is not possible since only the top eigenfunction can be positive.

Remark 1. It seems that this variational structure provides an alternative proof of something like Theorem 2, but it is not clear what constants C1 and C2 would emerge. On the other hand, this variational structure applies formally to the multidimensional problem 4, + f = 4 + A+ with periodic boundary conditions. It may be possible to prove stability for multidimensional BS in this way.

Remark 2. It seems likely that the closure of the unstable manifold of the constant solution (u(x ,r ) = 0) is the entire global attractor as it is for the Cahn- Hilliard equation in one space dimension; see [l].

Remark 3 . I believe that every trajectory, not just a generic one, converges to a steady solution as t - 00. This would follow from certain eigenvalue nondegeneracy conditions.

Remurk 4. Many of the conclusions about BS are uniform in the viscosity coefficient and also hold for viscosity weak solutions when the viscosity is set to zero. This and more was demonstrated in an elegant way by Brian Hays in [8].

Remark 5 . I recently learned of a proof by Collet, Eckmann, Epstein, and Stubbe (see [2]) of an independent stability theorem closely related and similar to Theorem 1.

Acknowledgements. It is a pleasure to acknowledge helpful conversations with Edriss Titi and Weinan E. In particular, Titi told me the “real space” interpretation of the NST argument which greatly simplifies the construction of s in [16].

This work was supported by the National Science Foundation and the Office of Naval Research and was completed while the author was visiting Stanford University.

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pp. 1-18.

pp. 2593-2602.

Received July 1992.

stability of the kuramoto-sivashinsky and related systems

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