New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations

New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations

Michael Goldman111LJLL, Université Paris Diderot, CNRS, UMR 7598, France, email: goldman@math.univ-paris-diderot.fr, Marc Josien222Ecole Polytechnique, Palaiseau, France, email: marc.josien@polytechnique.edu and Felix Otto333Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, email: otto@mis.mpg.de
Abstract

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from [14], at the same time slightly improving the result. The result in [14] relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of the results obtained in [7].

1 Introduction

1.1 The Kuramoto-Sivashinsky equation

We consider the one-dimensional Kuramoto-Sivashinsky equation:

(K-S)

This equation appears in many physical contexts, in particular in the modeling of surface evolutions. Sivashinsky used it to describe flame fronts [16], wavy flow of viscous liquids on inclined planes [17] and crystal growth [6]. Although the solutions of (K-S) are smooth and even analytic [10], they display a chaotic behavior for sufficiently large systems size (see [9] and Figure 1). The structure of the Kuramoto-Sivashinsky equation has some similarities with the Navier-Stokes equation. Therefore, it is sometimes possible to apply similar techniques to study both equations (see [15]).

Figure 1: Chaotic behavior of

For a given system size , we will consider -periodic solutions of (K-S). Since the spatial average is constant over time, and since the equation is invariant under the Galilean transformation:

it is not restrictive to assume that for all .
We can artificially cut the equation in two parts and consider separately the two mechanisms involved in (K-S):

(1)
(2)

The first equation (1) is linear and can be seen in Fourier space as:

(3)

The fourth partial derivative term decreases the short wavelength part of the energy spectrum whereas the second derivative term amplifies the long wavelength part. The second equation (2) corresponds to Burgers equation. It is nonlinear and develops shocks in finite time for non-trivial initial data. Nevertheless, as we will see later, this term has some mild regularizing effect. It is worth mentioning that for (2), the energy

is conserved. Therefore, one can intuitively say that in (K-S) the linear terms transport the energy from long wavelengths to short ones. Numerical simulations suggest (see the article of Wittenberg and Holmes [19]) that the time-averaged power spectrum

is independent of for . Moreover, this quantity is independent of and in the long wavelength regime and decays exponentially in the short wavelength regime . In line with this, numerical simulations suggest that for all :

This conjecture is supported by a universal bound on all stationary periodic solutions of (K-S) with mean , due to Michelson [12].

1.2 Known bounds

A first energy bound was obtained in the 80’s by Nicolaenko, Scheurer and Temam [13], who established by the “background flow method” that for every odd (in space) solution of (K-S):

with . This has been later generalized by Goodman [8], and Bronski and Gambill [2] and improved to . Using an entropy method, Giacomelli and the third author [5] improved this result by showing that:

The proof is based on the fact that the dispersion relation in (3) vanishes for and it implies that for every , we have:

by using the energy identity,

In a more recent paper [14], the third author proved that, for all ,

(4)

by using two ingredients: an a priori estimate for the capillary Burgers equation and an a priori estimate for the inhomogeneous Burgers equation, that is . More precisely, the result of [14] states that, for every solution of (K-S),

(5)

where denotes a Besov norm (see the appendix).

1.3 Main result

In this paper, we improve and simplify the result of the third author by showing that:

Theorem 1.1.

Let . For a smooth -periodic solution with zero average of the equation

there holds

(6)

This result is indeed slightly stronger than the previous one, since by (19), it implies an improvement of the exponent in (5) from to . However, this is not the main contribution of this paper. It is rather a simplified proof of the a priori estimate for inhomogeneous Burgers equation, which was one of the main tool for proving (5). For this purpose, we derive a modified Kármán-Howarth-Monin formula (see (14)). We also show how the proof of Golse and Perthame [7] (based on the kinetic formulation of Burgers equation) of a similar estimate for the homogeneous Burgers equation can be reinterpreted in this light. Since we work with slightly different Besov norms compared to [14], we need also to adapt most of the other steps to get (6). Besides Proposition 2.5, which we borrow directly from [14], we give here self-contained proofs.

The structure of the paper is the following: In Section 2, we enunciate the main theorem and give the structure of the proof. It has several ingredients: a Besov estimate for the inhomogeneous inviscid equation (Proposition 2.3), a regularity estimate for the capillary Burgers equation (Proposition 2.5) and an inverse estimate for Besov norms on solutions of (K-S) (Proposition 2.7). The following sections (i.e. Section 3, 4 and 5) are devoted to the proofs. In the appendix, we recall definitions and a few classical results regarding Besov spaces.

General notations

We denote by the finite-difference operator , by the space and for , by .
For an -periodic function , the spatial Fourier transform is defined by:

and for a Schwartz function :

For , we let (and similarly, ).

2 Main theorem and structure of the proof

In this section, we state the main theorem and the results on which it is based (see the appendix for the definition and main properties of Besov spaces).

2.1 Main theorem

Theorem 2.1.

Let . For a smooth -periodic solution with zero average of (K-S), there holds

(7)

From this theorem, we derive by interpolation (57) the following corollary:

Corollary 2.2.

Let . For a smooth -periodic solution with zero average of (K-S) and for indices , and related by

we have

2.2 Structure of the proof

The proof of Theorem 2.1 uses four important ingredients: a regularity result for Burgers equation (Proposition 2.3), a higher regularity estimate for the capillary Burgers equation (Proposition 2.5), an energy estimate (Lemma 2.6), and a result which allows us to “increase” the index of Besov spaces (Proposition 2.7). Let us now sketch the proof, discarding lower-order terms (in particular all the terms containing ) and taking borderline exponents in the estimates555Let us stress that we cannot reach these exponents since some of the constants (in particular the one in (8)) explode.. The strategy is graphically represented in Figure 2. The starting point is Proposition 2.3, which for , , and (recall also (59)), roughly says that

Using then the interpolation inequality (57), we get

Proposition 2.5 for , , and therefore , indicates that

(8)

Using the interpolation inequality (57) once again, we find

(9)

From Lemma 2.6, we obtain

Legend:

Interpolation

Energy estimate

Higher regularity

Increase of the index

Estimate of the inhomogeneous Burgers equation

Figure 2: Strategy of the proof

At this point, we see that we could have buckled the estimates if in (9), the Besov norm was replaced by the stronger norm . Unfortunately, this seems not doable with our method of proof. Therefore, we need Proposition 2.7 in order to control by . It is at this last stage that we lose a logarithm since (19) gives

Putting all these estimates together, we find

which is (7).

As mentioned, the first ingredient is an estimate for the inhomogeneous Burgers equation. A similar estimate was obtained in [14, Prop. 1]. A related inequality for the homogeneous Burgers equation has been recently derived in [7].

Let us consider the following inhomogeneous Burgers equation:

(10)
Proposition 2.3 (Besov estimate for the inhomogeneous Burgers equation).

Let , be smooth -periodic functions. Then, for any smooth -periodic solution of (10), there holds: For , verifying and , there exists a constant just depending on such that:

(11)

Therefore, taking the time-space average, it holds:

(12)

The proof of (11) is based on a modified Kármán-Howarth-Monin identity:

Lemma 2.4.

Let be a smooth -periodic function and let be a smooth periodic solution with zero average of

(13)

then for ,

(14)

The usual Kármán-Howarth-Monin identity [4] states that

(15)

This formula can be easily checked by using equation (13) and the periodicity. The main difference between (15) and (14) is that in the latter, the coercive term replaces the non-coercive term .
We will give two proofs of (14). The first is by a direct computation and the second uses the kinetic formulation of Burgers equation following ideas of [7]. Therefore, this second proof gives a new, and hopefully interesting, interpretation of the arguments of [7].

The second ingredient is a higher regularity result for the capillary Burgers equation (see [14, Prop. 2, p. 14]).

Proposition 2.5 (Higher regularity).

Let , satisfying:

and

Then, there exists such that, if , are smooth, -periodic in and satisfy

the following estimate holds:

(16)

Proposition 2.5 allows to jump from higher derivatives to smaller ones in Besov spaces. The proof, which we will not provide, is based on a narrow-band Littlewood-Paley decomposition.

The third ingredient is an elementary energy estimate, which directly bounds the norm of a solution of the inhomogeneous capillary Burgers equation.

Lemma 2.6 (Energy estimate).

Let be a smooth solution of:

(17)

Then, the following estimate holds:

(18)
Proof.

Since the proof is straightforward, we give it now. Integrating over the equation (17) over , we get:

Therefore by (58)

taking then the time-space average yields the result. ∎

As already mentioned, these three estimates will not be sufficient to conclude. We will also need an estimate relating Besov norms with different exponents . One can easily see that if , then (it is a consequence of convexity inequality). In fact, it is possible to reverse the inequality for solutions of (K-S), but this comes with a price: a logarithm of the spatial period appears.

Proposition 2.7 (Increasing the index ).

There exists such that, for all , solution of (K-S), the following estimate holds:

(19)

3 Proof of Theorem 2.1

In this section, we derive the main theorem from the above propositions. We now consider the rescaled Besov norm as a point in the space . All the norms involved in our problem lie in the rectangle of defined by666See Figure 2 in Section 3.2 which represents the strategy in .:

Proof of Theorem 2.1.

Let be a solution of (K-S). It is convenient to introduce the abbreviation:

Notice that and . With this notation, interpolation inequality (57) takes the form

(20)

for and .

Letting , and in (12) and using (59) for , (12) can be rewritten as

(21)

for . In turn, (16) with , and (which implies ) gives

(22)

for and . Finally, (18) is equivalent to

(23)

and (19) to

(24)

Our first goal is to argue that for , we can replace in the above estimates all the Besov norms involving by . By (59) and (57),

Hence, in view of (23), Young’s inequality and since by (59), , it will be enough to prove that

(25)

for (which reduces the use of (22) to ). We can indeed prove more generally that for , and any , there holds

(26)

Thanks to Jensen’s inequality, we have and . By monotonicity of the Besov norms with respect to the last index, there also holds and . Therefore, we are left with proving that

By definition of the Besov norms, for ,

which after taking the average over time and space, finishes the proof of (26).

To sum up, we now have that (21), (22) and (23) together with (25) imply

(27)

for ,

(28)

for and and

(29)

We now gather the above estimates in order to bound . Passing to the logarithm in the above inequalities, we see that optimizing the parameters to get the best power of is equivalent to a linear programming problem. Its solution thus lie at the boundaries of the admissible domain. It is not hard to see that in particular, we want to take with as close as possible to . Let be such that

so that is close to and is close to . Thanks to (20),

(30)

Since we can assume that , we get from (27), (30) and (28),

where in the last inequality, we used (29). From (30), (29) and (24), we deduce

Dividing by this inequality and noticing that for close to , is close to , we obtain that if , then

which gives finally

and thus the result since . ∎

4 Proof of Proposition 2.3

For the reader’s convenience, let us recall the statement of Proposition 2.3. Let be a smooth solution of the following inhomogeneous Burgers equation:

(31)
Proposition.

Let , be smooth -periodic functions. Then, for any smooth -periodic solution of (31), there holds: For , verifying and , there exists a constant just depending on such that:

Before proceeding further, let us remark that, by approximation, this applies to any (possibly non smooth) entropy solution of Burgers equation (31). Indeed, if we consider a solution of

then it is a smooth solution of (31) with and . A careful inspection of the proof of Proposition 2.3 shows that for it extends to , yielding

that is

Combining this with the energy inequality: , gives

which passes to the limit as . The indices are optimal in the light of the result of De Lellis and Westdickenberg [3] which states that we cannot hope to have more regularity, in the sense that the Besov index cannot be better than .

As already pointed out, the proof of the aimed estimate is based on a modified Kármán-Howarth-Monin identity:

Lemma 4.1.

Let be a smooth -periodic function and let be a smooth periodic solution with zero average of

(32)

then for ,

(33)
Proof.

By periodicity, (33) will be a direct consequence of the following pointwise identity:

(34)

For simplicity, let us introduce the notation (so that ). Using (32) we get:

It remains to prove that

(35)

We start with

and

to get