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Local well-posedness for a generalized sixth-order Boussinesq equation

Long Zhong, Shenghao Li

Abstract

A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} - (u^2)_{xxxx} - (uu_{xx})_{xx} - (u^3)_{xx} = 0.$$ Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE), $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} = 0.$$ The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, $X^{s,b}$, framework. The multi-linear estimates for $(u^2)_{xx}$, $(u^2)_{xxxx}$, $(uu_{xx})_{xx}$ and $(u^3)_{xx}$ are given, the local wellposedness of the gSOBE is established for $s>\frac{1}{2}$.

Local well-posedness for a generalized sixth-order Boussinesq equation

Abstract

A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, Such equation is analogous to original Boussinesq equation but with higher order approximation which may ensure a more accuracy description on a long time scale. Moreover, through a rigorous derivation from Boussiensq systems, it has redeemed all the non-linear terms neglected in the sixth order Boussinesq equation (SOBE), The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, , framework. The multi-linear estimates for , , and are given, the local wellposedness of the gSOBE is established for .
Paper Structure (8 sections, 10 theorems, 89 equations)

This paper contains 8 sections, 10 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Derivation of the gSOBE
  3. Multi-linear estimate
  4. Preliminary
  5. Proof of Proposition \ref{['bi0']}
  6. Proof of Estimate \ref{['bi13']}
  7. Proof of Estimate \ref{['tri1']}
  8. Proof of (\ref{['bi']})

Key Result

Theorem 1.2

Let $s>\frac{1}{2}$ be given, there exists $b = b(s) \in (\frac{1}{2}, 1)$, we can find $T = T(s, b)>0$ such that if then the IVP(1.6) admits a unique solution $u \in X^{s,b}_T,$ and the corresponding solution map is real analytic.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 1 more