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Review of some modified generalized Korteweg - de Vries - Kuramoto-Sivashinsky equations (mgKdV-KS)

Marie-Thérèse Aimar, Abdelkader Intissar

Abstract

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and the non homogeneous boundary value problem for KdV-KS equation in quarter plane.

Review of some modified generalized Korteweg - de Vries - Kuramoto-Sivashinsky equations (mgKdV-KS)

Abstract

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and the non homogeneous boundary value problem for KdV-KS equation in quarter plane.
Paper Structure (8 sections, 27 theorems, 113 equations)

This paper contains 8 sections, 27 theorems, 113 equations.

Table of Contents

  1. Elementary properties of Sobolev's spaces
  2. Some technicalities inequalities
  3. Bonna-Smith results on KDV equation (1975)
  4. Aimar's results on K.S equation (1982)
  5. Presentation of KdV-KS equation (1996)
  6. On travelling wave solution of the Korteweg-de Vries-Kuramoto-Sivashinsky equation
  7. Existence and Uniquness of KdV-KS equation on domain $[0, T]\times [0, 2\pi] , T > 0$
  8. A study of the nonhomogeneous boundary value problem for KdV-KS equation in quarter plane: (2016)

Key Result

Theorem 1.3

(continuous representation) (i) Let $u \in L_{loc}^{1}(I)$, if ${\int_{I}u\varphi = 0 \quad \forall \varphi \quad \in C_{c}(I)}$ then ${u = 0 \,\, \text{ almost everywhere on} \,\, I }$ (ii) Let $u \in L_{loc}^{1}(I)$ which satisfies ${\int_{I}u \varphi^{'} = 0 \quad \forall \,\, \varphi \in C

Theorems & Definitions (45)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 35 more