New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations
We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from , at the same time slightly improving the result. The result in  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 .
1.1 The Kuramoto-Sivashinsky equation
We consider the one-dimensional Kuramoto-Sivashinsky equation:
This equation appears in many physical contexts, in particular in the modeling of surface evolutions. Sivashinsky used it to describe flame fronts , wavy flow of viscous liquids on inclined planes  and crystal growth .
Although the solutions of (K-S) are smooth and even analytic , they display a chaotic behavior for sufficiently large systems size (see  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 ).
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):
The first equation (1) is linear and can be seen in Fourier space as:
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 ) 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 :
1.2 Known bounds
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 , the third author proved that, for all ,
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  states that, for every solution of (K-S),
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:
Let . For a smooth -periodic solution with zero average of the equation
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 
(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 ,
we need also to adapt most of the other steps to get (6). Besides Proposition 2.5, which we borrow directly from ,
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.
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
Let . For a smooth -periodic solution with zero average of (K-S), there holds
From this theorem, we derive by interpolation (57) the following corollary:
Let . For a smooth -periodic solution with zero average of (K-S) and for indices , and related by
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
Using the interpolation inequality (57) once again, we find
From Lemma 2.6, we obtain
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 .
Let us consider the following inhomogeneous Burgers equation:
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:
Therefore, taking the time-space average, it holds:
The proof of (11) is based on a modified Kármán-Howarth-Monin identity:
Let be a smooth -periodic function and let be a smooth periodic solution with zero average of
then for ,
The usual Kármán-Howarth-Monin identity  states that
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 . Therefore, this second proof gives a new, and hopefully interesting, interpretation of the arguments of .
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:
Then, there exists such that, if , are smooth, -periodic in and satisfy
the following estimate holds:
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:
Then, the following estimate holds:
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:
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
for and .
for . In turn, (16) with , and (which implies ) gives
for and . Finally, (18) is equivalent to
and (19) to
for (which reduces the use of (22) to ). We can indeed prove more generally that for , and any , there holds
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).
for and and
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),
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:
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
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  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:
Let be a smooth -periodic function and let be a smooth periodic solution with zero average of
then for ,