H4-Solutions for the Olver–Benney equation

The Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation.


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On the initial datum, we assume From a physical point of view, (1.1) was derived in the context of water waves by Olver [16,17] (see also [10,18]), using Hamiltonian perturbation theory, with further generalization given by Craig and Groves [8], while, under Assumption (1.3), (1.1) was derived by Benney [1] as a model to describe the interaction effects between short and long waves. [17] shows that (1.1) is a particular case of the following equation: If q = = = = 0 in (1.6), (1.6) becomes the Korteweg-de Vries equation [11], whose the well-posedness is studied in [4]. Instead, if = = 0 , (1.6) becomes the Kawahara-Korteweg-de Vries type equation, which was derived by Kawahara [9] to describe small-amplitude gravity capillary waves on water of a finite depth when the Weber number is close to 1/3 (see [15]). In [2], the well-posedness of the Cauchy problem for the Kawahara-Korteweg-de Vries type equation is studied.
Moreover, assuming = q = = 0 and (1.2), (1.6) reads It is a particular case of the Kudryashov-Sinelshchikov equation [5,12], which describes pressure waves in liquids with gas bubbles taking into account heat transfer and viscosity. In [5], the existence of solutions of the Cauchy problem is proven. From a mathematical point of view, under suitable assumptions on , q, , , , , the existence of the travelling waves solutions for (1.6) is proven in [14,20], while a method to find exact solutions of (1.6) is given in [13]. Instead, in [19], the local wellposedness of the Cauchy problem of (1.1) is proven.
The main result of this paper is the following theorem. Since we are able to prove estimates only on the spatial derivatives of (1.1), the proof of Theorem 1.1 is based on the Aubin-Lions lemma (see [3,6,7,21]), which requires only the H −1 boundedness of the time derivative.
One of the main point of our argument is the invariance of the energy space H 4 (see Lemma 2.4). The key point in that direction is the H 2 regularity. Assumptions (1.2), (1.5) u 0 (x) ∈ H 4 (ℝ).
(1.8) u ∈ L ∞ (0, T;H 4 (ℝ)), T ≥ 0. So we can say that the assumptions on the constants are needed for the H 2 regularity of u and only indirectly for the H 4 one.
Observe again that, (1.5) is the same assumption to prove the well-posedness of the Cauchy problem for the Kawahara equation (see [2]). [2,Appendix A] shows that, for the Kawahara equation, it is possible also to assume on the initial datum also and obtain the well-posedness of the classical solution (see [
Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.1).
Fix a small number 0 < < 1 and let u = u (t, x) be the unique classical solution of the following problem [2,4,19]: where u ,0 is a C ∞ approximation of u 0 such that Let us prove some a priori estimates on u . We denote with C 0 the constants which depend only on the initial data, and with C(T), the constants which depend also on T. Assume (1.2). For each t ≥ 0,
x ∈ ℝ, Proof Multiplying  The proof of the previous lemma is based on the regularity of the functions u and the following result.

Lemma 2.3 For each t ≥ 0 , we have that
Proof We begin by observing that, thanks to the regularity of u and the Hölder inequality, .
In particular, we have The proof of the previous lemma is based on the regularity of the functions u and the following result. (2.20) (2.21) Moreover, since we have that It follows from (2.21), (2.22) and an integration of (2.20) that
Proof To prove (2.38), we rely on the Aubin-Lions lemma (see [3,6,7,21]). We recall that where the first inclusion is compact and the second one is continuous. Owing to the Aubin-Lions lemma [21], to prove (2.38), it suffices to show that We prove (2.40 (2.39) u k → u in L 2 (K) and a.e.
Let us prove some a priori estimates on u . (1.3). For each t ≥ 0, Proof Multiplying (2.1) by 2u , an integration on ℝ give Proof Let 0 ≤ t ≤ T . We begin by observing that, thanks to the Hölder inequality, Again by the Hölder inequality,

Lemma 3.1 Assume
Therefore, by (3.5) and ( Since 0 < < 1 , we have that . Integrating on (0, t), by ( Proof Let 0 ≤ t ≤ T . Consider two real constants F, G , which will be specified later. Multiplying (2.1) by we have that Observe that ≤ C(T).

We search F, G such that that is
Since is the unique solution of (3.16), it follows from (3.15) that that is Due to the Young inequality, where D 1 is a positive constant, which will be specified later. (3.20) Consequently, by (3.20) and (3.21),

It follows from (3.22) that
Choosing we obtain that which gives (3.8).
Let us prove some a priori estimates on u .
Proof Let 0 ≤ t ≤ T . We begin by observing that, thanks to the Hölder inequality, Hence, by (4.1), Moreover, by the Hölder inequality and the Young one, where D 2 is positive constant, which will be specified later. Since 0 < < 1 , it follows from (4.4), (4.5) and the Young inequality, Therefore, Choosing we have that . . (4.6) .
Consequently, by (1.4), (3.12) with H, I instead of F, G , (3.14) with H, I instead of F, G , (4.15) and an integration on ℝ of (4.14), we have that We search H, I, L such that that is Since is the unique solution of (4.17), it follows from (4.16) that  , � � � �