In this work we are interested in developing approximate models for water waves equation. We present the derivation of the new equations uses approximation of the phase velocity that arises in the linear water wave theory. We treat the (KdV) equation and similarly the C-H equation. Both of them describe unidirectional shallow water waves equation. At the same time, together with the (BBM) equation we propose, we provide the best approximation of the phase velocity for small wave numbers that can be obtained with second and third-order equations. We can extend the results of [3,4]. A comparison between the methods is mentioned in this article.

Abstract:In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution to the Cauchy problem for Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given.

Abstract: We deal with the Camassa-Holm equation   possesses a global continuous semigroup of weak conservative solutions for initial data. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure µ with . The total energy is preserved by the solution.

We introduce the model of equation that is closely related to the following equation (1.1). We study the global existence of week solution for this class of equation. Using the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Logarithmic Quantum Mechanics. We can extend the results of [1, 2].

Abstract:In this paper we study the initial boundary value problem for logarithmic Higher Order Wave equation. Introducing the Logarithmic Sobolev inequality and using the combination of Galerkin method, we consider the theorem of existence of a global weak solution to problem for the initial boundary value problem of the logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for logarithmic Higher Order Wave equation. The proof of the main theorem is given.