Simplified a priori Estimate for the Time Periodic Burgers' Equation

We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgers equation with irregular forcing terms (with Sobolev regularity -1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity -1/2 in time.


Introduction
The study of the Burgers' equation has a long history starting with the seminal papers by Burgers [1], Cole [2] and Hopf [7] where the Cole-Hopf transformation was introduced. The Cole-Hopf transformation transforms the homogeneous Burgers' equation into the heat equation.
More recently there have been several articles dealing with the forced Burgers' equation: The vast majority treats the initial value problem in time with homogeneous Dirichlet or periodic space boundary conditions (see for instance [9]).
Only recently has the question of the time-periodic forced Burgers' equation been tackled ( [8,3,10,4]). In most cases [8,3] the authors are chiefly interested in the inviscid limit (the limit when the viscosity ν tends to zero).
The closest related work to ours is that of Jauslin, Kreiss and Moser [8] in which the authors show existence and uniqueness of a space and time periodic solution of the Burgers' equation for a space and time periodic forcing term which is smooth.

Definitions
In this section we recall some well known facts and fix some general notations. We will be concerned will time-periodic solutions, of the Burgers' equation so we will use the following notation for the one-dimensional torus T: and similarly: Simple computations then give: Notice that if H is a function space then H maps real functions to real functions.
The following properties will be useful in the sequel.
where ℜ denotes the real part of the expression.

Fractional Sobolev Spaces.
We define fractional Sobolev spaces in the following manner, for any s ∈ R: H) is then a Hilbert space with the following scalar product: The following classical result holds: H (s) (T, H) * = H (−s) (T, H * ).

Anisotropic Fractional Sobolev Spaces.
We will now describe some useful Sobolev spaces defined on the interval We will also fix a non-negative real number s: Let H (s) (I) denote the usual fractional Sobolev space of real-valued s-times differentiable functions on I. H We will also use the following notations, for α, β nonnegative real numbers: and We also introduce H In the following sections we will use the following notations. The main working space will be H ( 1 2 ,1) 0 (T × I), for which we use the more concise notation: More generally, we will drop the dependence of the spaces on the domain T × I, so that L p stands for L p (T × I), H (α,β) stands for H (α,β) (T × I), and so on.
We will also use the following notations, for u ∈ H

Interpolation and regularity
If s k (ξ) is the Fourier transform s k (ξ) =û(k, ξ) of a distribution u defined on T × R, we have the following Hölder inequality for any θ ∈ [0, 1]: From this Hölder inequality we deduce  (2)): It turns out that the second definition makes sense because of the embedding Figure 1). We now prove the main result of this article, namely an a priori estimate on the solutions of the family of equations: In the proof we use techniques similar to those in [6]. The proof is simpler (but the result is weaker) than the one obtained in [5]. The a priori estimate above may now be used to prove existence of solutions by a (nonlinear, compact) degree argument using the Leray-Schauder Theorem (cf. [5]).

Existence and Uniqueness of the time-periodic solutions
A weaker result of the main result proved in [5] is In this section we briefly sketch the proof of that result, the full proof being available in [5].
The main ingredient of the proof is the Cole-Hopf transformation, which is essentially defined by: In our case there are complications due to the fact that u ∈ H ( 1 2 ,1) 0 , so u is timeperiodic. This change of variable will transform the periodicity problem into an eigenvalue problem (because the Cole-Hopf transformation linearises the Burgers' equation). After working out the details one shows that the uniqueness problem is equivalent to the uniqueness of the ground state eigenvalue problem: is K = 0 and ϕ = 1 if and only if T u = T v implies u = v (that is, the solution to the original Burger's equation is unique).
The proof of that proposition essentially hinges on the embedding properties exposed in section 2 (see Figure 2).
The remaining part of the proof is concerned with the eigenvalue problem of the Proposition above. One first shows that the eigenvalue is zero using a weaker version of the Perron-Frobenius theorem. The second step is to show that the remaining eigenvalue problem is non degenerate, namely that the dimension of the eigenspace must be one. This last step makes use of the a priori estimate proved in Theorem 1.
The details of that part of the proof are too lengthy to be exposed here in depth so the interested reader is referred to [5].