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International Journal of Differential Equations
Volume 2011 (2011), Article ID 408704, 24 pages
http://dx.doi.org/10.1155/2011/408704
Research Article

Global Existence of Solutions to the Fowler Equation in a Neighbourhood of Travelling-Waves

Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier 2, Place Eugène Bataillon, CC 051, 34095 Montpellier, France

Received 30 June 2011; Accepted 27 August 2011

Academic Editor: Khalil Ezzinbi

Copyright © 2011 Afaf Bouharguane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.

1. Introduction

The study of mechanisms that allow the formation of structures such as sand dunes and ripples at the bottom of a fluid flow plays a crucial role in the understanding of coastal dynamics. The modeling of these phenomena is particularly complex since we must not only solve the Navier-Stokes or Saint-Venant equations with equation for sediment transport, but also take into account the evolution of the bottom. Instead of solving the whole system fluid flow, free surface and free bottom, nonlocal models of fluid flow interacting with the bottom were introduced in [1, 2]. Among these models, we are interested in the following nonlocal conservation law [1, 3]: where is any given positive time, represents the dune height (see Figure 1), and is a nonlocal operator defined as follows: for any Schwartz function and any ,

408704.fig.001
Figure 1: Domain considered for the Fowler model: is the depth water, the free surface, and the sea bottom.

Equation (1.1) is valid for a river flow over an erodible bottom with slow variation and describes both accretion and erosion phenomena [4]. See [4, 5] for numerical results on this equation.

The nonlocal term can be seen as a fractional power of order of the Laplacian with the bad sign. Indeed, it has been proved [4] where with positive constants, denotes the Fourier transform defined in (1.7), and denotes the Euler function. One simple way to establish this fact is the derivation of a new formula for the operator , see Proposition 2.5.

Remark 1.1. For causal functions (i.e., for ), this operator is up to a multiplicative constant, the Riemann-Liouville fractional derivative operator which is defined as follows [6]:

Therefore, the Fowler model has two antagonistic terms: a usual diffusion and a nonlocal fractional anti-diffusive term of lower order. This remarkable feature enabled to apply this model for signal processing. Indeed, the diffusion is used to reduce the noise whereas the nonlocal anti-diffusion is used to enhance the contrast [7].

Recently, some results regarding this equation have been obtained, namely, existence of travelling-waves where and represents wave velocity, the global well-posedness for -initial data, the failure of the maximum principle, and the local-in-time well-posedness in a subspace of [4, 8]. Notice that the travelling-waves are not necessarily of solitary type (see [8]) and therefore may not belong to , the space where a global well-posedness result is available. In [8], the authors prove local well-posedness in a subspace of but fail to obtain global existence.

To prove the existence of travelling-waves solutions of the Fowler equation, the authors used the implicit function theorem on suitable Banach spaces [8]. Much work has been devoted to investigate existence, uniqueness, and regularity of travelling-waves for integral differential equations, see for instance [9] and references therein.

An interesting topic is to know if the shape of this travelling-wave is maintained when it is perturbed. This raises the question of the stability of travelling-waves. But before interesting ourselves in this problem, we have to show first the global existence of perturbations around these travelling-waves. Hence in this paper, we prove the global well-posedness in an -neighbourhood of a regular travelling-wave, namely . To prove this result, we consider the following Cauchy problem: where is an initial perturbation and is any given positive time.

To prove the existence and uniqueness results, we begin by introducing the notion of mild solution (see Definition 2.1) based on Duhamel’s formula (2.1), in which the kernel of appears. The use of this formula allows to prove the local-in-time existence with the help of a contracting fixed point theorem. The global existence is obtained thanks to an energy estimate (4.68). This approach is classical: we refer for instance to [4, 10].

The plan of this paper is organised as follows. In the next section, we define the notion of mild solution to (1.6) and we give some properties on the kernel of that will be needed in the sequel. Sections 3 and 4 are, respectively, devoted to the proof of the uniqueness and the existence of a mild solution for (1.6). Section 5 contains the proof of the regularity of the solution.

Notations
(i) The norm of a measurable function is written for .
(ii) We denote by the Fourier transform of which is defined by the following: for all and denotes the inverse of Fourier transform.
(iii) The Schwartz space of rapidly decreasing functions on is denoted by .
(iv) We write .
(v) We denote by the space of all bounded continuous real-valued functions on with the norm .
(vi) We write for any ,
(vii) We denote by the space of test functions on and denotes the distribution space.

Here is our main result.

Theorem 1.2. Let and. There exists a unique mild solution of (1.6) (see Definition 2.1) which satisfies Moreover, if then and satisfies on , in the classical sense or equivalently, is a classical solution of (1.1).

2. Duhamel Formula and Main Properties of

Definition 2.1. Let and . We say that is a mild solution to (1.6) if for any : where is the kernel of the operator , and is defined in (1.4).

The expression (2.1) is the Duhamel formula and is obtained using the spatial Fourier transform.

Proposition 2.2 (main properties of , [4]). The kernel satisfies , and , ,, such that , such that .

Remark 2.3. An interesting property for the kernel is the non-positivity (see Figure 2), and the main consequence of this feature is the failure of maximum principle [4]. We use again this property to show that the constant solutions of the Fowler equation are unstable [11].

408704.fig.002
Figure 2: Evolution of the kernel for (red) and s (blue).

Remark 2.4. Using Plancherel formula, we have for any and any where .

Proposition 2.5 (integral formula for ). For all and all ,

Proof. The proof is based on simple integrating by parts. The regularity and the rapidly decreasing of ensure the validity of the computations that follow. We have There is no boundary term at infinity (resp., at zero) because is a rapidly decreasing function on (resp., is smooth).

Remark 2.6. Using integral formula (2.3), [4, 8] proved that Notice that which is consistent with Remark 1.1: up to a multiplicative constant is.

Proposition 2.7. Let and . Then and we have

Proof. For all and all , we have, using Remark 2.6

Remark 2.8. From the previous proposition, we deduce that for all and all , . In particular, using the Sobolev embedding , we deduce that is a bounded linear operator.

Proposition 2.9 (Duhamel formula (2.1) is well defined). Let , and . Then, the function is well defined and belongs to (being extended at by the value ).

Proof. From Proposition 2.2, it is easy to see that is well defined and that for any , . Indeed, so by Young inequalities exists, and using the estimates on the gradient (item 3 and 4 of Proposition 2.2) we deduce that is well defined and .
Let us prove the continuity of . By the second item of Proposition 2.2, we have that the function is continuous and it is extended continuously up to by the value . We define the function Now, we are going to prove that is uniformly continuous. For any , Young inequalities imply where are such that . Since , the dominated convergence theorem implies that Moreover, using the estimates on the gradient (items 3 and 4 of Proposition 2.2), we have the following inequality: where is a positive constant and .
From (2.10), we obtain that . Hence, the function is continuous and this completes the proof of the continuity of .

Remark 2.10. Using the semi group property of the kernel , we have for all and all , [4]

3. Uniqueness of a Solution

Let us first establish the following Lemma.

Lemma 3.1. Let and . For , let and define as in Proposition 2.9 by Then,

Proof. For all , we have Hence with the help of Young inequalities, we get It then follows that Using again the estimates of the gradient of (see Proposition 2.2), we conclude the proof of this Lemma.

Proposition 3.2. Let and . There exists at most one which is a mild solution to (1.6).

Proof. Let be two mild solutions to (1.6) and . Using the previous Lemma, we get Since, with , then Therefore, on for any satisfying . Since and are continuous with values in , we have that on where is the positive solution of the following: that is, . To prove that on , let us define and we assume that . By continuity of and , we have that . Using the semigroup property, see Remark 2.10, we deduce that are mild solutions to (1.6) with the same initial data which implies, from the first step of this proof, that for . Finally, we get a contradiction with the definition of and we infer that . This completes the proof of this proposition.

4. Global-In-Time Existence of a Mild Solution

Proposition 4.1 (local-in-time existence). Let . There exists that only depends on and such that (1.6) admits a unique mild solution . Moreover, satisfies

Proof. For , we consider the following norm: and we define the affine space It is readily seen that endowed with the distance induced by the norm is a complete metric space. For , we define the function From Proposition 2.9, and satisfies . Step 1 (). Since the dominated convergence theorem implies that for any , Therefore, the function is continuous and since is an isometry of , we deduce that is continuous. We have then established that is continuous. Moreover, from Proposition 2.2, we have Let denote the function Let us now prove that . We first have Using Young inequalities and Proposition 2.2, we get We then obtain where and , being the beta function defined by As then We then deduce that is in and so for all .
Let us now prove that is continuous on with values in . For and , we define Since and then Proposition 2.9 implies that is continuous. Moreover, we have for any and , It then follows that We next infer that because it is a local uniform limit of continuous functions. Hence, we have established that . To prove that , it remains to show that . Using (4.7) and (4.11), we have Finally, we have .
Step 2. We begin by considering a ball of of radius centered at the origin where . Take and let us now prove that maps into itself. We have By Remark 2.4, we get where . Moreover, since and with the help of Proposition 2.2, we get Using (4.17) and (4.21), we deduce that Therefore, for sufficiently small such that we get that .
To finish with, we are going to prove that is a contraction.
For , we have for any and since, we get Moreover And since then Therefore, we obtain Finally, using (4.26) and (4.30), we get
Step 3 (conclusion). For any sufficiently small such that (4.23) holds true and is a contraction from into itself. The Banach fixed point theorem then implies that admits a unique fixed point which is a mild solution to (1.6).

Lemma 4.2 (regularity of ). Let and . There exists that only depends on and such that (1.6) admits a unique mild solution . Moreover, satisfies

Proof. To prove this result, we use again a contracting fixed point theorem. But this time, it is the gradient of the solution which is searched as a fixed point.
From Proposition 4.1, there exists which depends on and such that is a mild solution to (1.6). Since , we can consider the gradient of for any . Let then and . We consider the same complete metric space defined in the proof of Proposition 4.1 and we take the norm defined in (4.2): with the initial data .
We now wish to apply the fixed point theorem at the following function: where . First, we leave the reader to verify that maps into itself. The proof is similar to the one given in Proposition 4.1.
For any , we have from Young inequalities and Remark 2.4 and from Proposition 2.2, we get Differentiating with respect to the space variable, we obtain and developing, we get Now, from Young inequalities, we have Finally, from Proposition 2.2, we obtain In other words, we have for all where . Hence, using (4.37) and (4.42), we get for some positive constant which depends on and .
We next leave reader to verify that: for any , where is a positive constant which depends on and .
Then, if satisfies is a contraction, where is ball of of radius centered at the origin. Using a contracting point fixed theorem, there exists a unique fixed point, which we denote by . But it is easy to see that taking into account the space derived from the Duhamel formulation (2.1). Thanks to a uniqueness argument, we deduce that and thus that , which completes the proof of this lemma.

Let us now prove the global-in-time existence of mild solution .

Proposition 4.3 (global-in-time existence). Let , and . Then, there exists a (unique) mild solution to (1.6). Moreover, satisfies the PDE (1.6) in the distribution sense.

Proof. Step 1 ( is a distribution solution). Taking the Fourier transform with respect to the space variable in (2.1), we get for all and all , Define Classical results on ODE imply that is differentiable with respect to the time with Let us now prove that all terms in (4.48) are continuous with values in . Since, then