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Long time existence for a class of weakly transverse Boussinesq systems

Qi Li, Jean-Claude Saut, Li Xu

Abstract

We prove the existence on long time scales of the solutions to the Cauchy problem for a version of weakly transverse Boussinesq systems arising in the modeling of surface water waves. This system is much more complicated than the isotropic Boussinesq systems because dispersion is only present in the x-direction, leading to anisotropic eigenvalues in the linearized system. This anisotropic character leads to loss of y-derivatives for the solutions. To overcome this main difficulty our strategy is to symmetrize the system by introducing suitable good unknowns in the sense of [3].

Long time existence for a class of weakly transverse Boussinesq systems

Abstract

We prove the existence on long time scales of the solutions to the Cauchy problem for a version of weakly transverse Boussinesq systems arising in the modeling of surface water waves. This system is much more complicated than the isotropic Boussinesq systems because dispersion is only present in the x-direction, leading to anisotropic eigenvalues in the linearized system. This anisotropic character leads to loss of y-derivatives for the solutions. To overcome this main difficulty our strategy is to symmetrize the system by introducing suitable good unknowns in the sense of [3].

Paper Structure

This paper contains 24 sections, 19 theorems, 470 equations.

Table of Contents

  1. Introduction
  2. The general setting
  3. Derivation of the new system
  4. Main result
  5. Main idea of the proof
  6. Preliminary
  7. Definitions and notations
  8. Technical lemmas
  9. Proof of Theorem \ref{['main theorem']} for case $a=f=g=0, b=d=e=\frac{1}{3},c=-\frac{1}{3}$
  10. The reduction of (\ref{['WTB 2']}) for case 1
  11. Symmetrization of system \ref{['system for p theta']}.
  12. Good unknowns.
  13. Symmetrization of \ref{['system for p theta']}.
  14. The proof of Theorem \ref{['main theorem']} for case 1
  15. Ansatz for continuity arguments.
  16. ...and 9 more sections

Key Result

Proposition 1.3

Assuming that $(a,b,c,d,e,f,g)$ satisfies constraint, the system WTB case a is consistent with the system WTB 1 in the sense of Definition def of consistent with the precision $O(\varepsilon^2)$.

Theorems & Definitions (45)

  • Remark 1.1
  • Definition 1.2: see Definition 1 of BCL
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 35 more