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Global weak solutions of the Serre-Green-Naghdi equations with surface tension

Billel Guelmame

Abstract

We consider in this paper the Serre--Green--Naghdi equations with surface tension. Smooth solutions of this system conserve an $H^1$-equivalent energy. We prove the existence of global weak dissipative solutions for any relatively small-energy initial data. We also prove that the Riemann invariants of the solutions satisfy a one-sided Oleinik inequality.

Global weak solutions of the Serre-Green-Naghdi equations with surface tension

Abstract

We consider in this paper the Serre--Green--Naghdi equations with surface tension. Smooth solutions of this system conserve an -equivalent energy. We prove the existence of global weak dissipative solutions for any relatively small-energy initial data. We also prove that the Riemann invariants of the solutions satisfy a one-sided Oleinik inequality.
Paper Structure (11 sections, 25 theorems, 234 equations, 2 figures)

This paper contains 11 sections, 25 theorems, 234 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of previous results
  3. Main results
  4. An approximated system
  5. Riccati-type equations
  6. The choice of the approximated system
  7. Uniform estimates
  8. Precompactness
  9. The global weak solutions
  10. Some classical lemmas
  11. The energy equation

Key Result

Theorem 2.1

Let $\gamma>0$, $\bar{h}>0$ and $s \geqslant 2$, then, for any $(h_0-\bar{h},u_0) \in H^s(\mathds{R})$ satisfying $\inf_{x\, \in\, \mathds{R}}h_0(x)>0$ there exists $T>0$ and $(h-\bar{h},u) \in C([0,T],H^s(\mathds{R})) \cap C^1([0,T],H^{s-1}(\mathds{R}))$ a unique solution of SGNxi such that Moreover, the solution satisfies the conservation of the energy

Figures (2)

  • Figure 1: Fluid domain.
  • Figure 2: Characteristics.

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 5.1
  • ...and 20 more