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A new highly nonlinear equation modelling shallow-water waves with constant vorticity

Yu Liu, Xingxing Liu, Min Li

Abstract

In this paper we apply the approach of formal asymptotic expansions and perturbation theory to derive a new highly nonlinear shallow-water model from the full governing equations for two dimensional incompressible fluid with constant vorticity. This approximate model is generated by introduction of a larger scaling than the Camassa-Holm one, which is shown to be optimal in the sense that there are no non-local terms appearing in free surface equation. Moreover, we establish the local well-posedness of the Cauchy problem in Besov spaces, and give a blow-up criterion, which improve the previous corresponding results in Sobolev spaces.

A new highly nonlinear equation modelling shallow-water waves with constant vorticity

Abstract

In this paper we apply the approach of formal asymptotic expansions and perturbation theory to derive a new highly nonlinear shallow-water model from the full governing equations for two dimensional incompressible fluid with constant vorticity. This approximate model is generated by introduction of a larger scaling than the Camassa-Holm one, which is shown to be optimal in the sense that there are no non-local terms appearing in free surface equation. Moreover, we establish the local well-posedness of the Cauchy problem in Besov spaces, and give a blow-up criterion, which improve the previous corresponding results in Sobolev spaces.
Paper Structure (9 sections, 7 theorems, 87 equations)

This paper contains 9 sections, 7 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model equation
  3. Nondimensionalisation and scaling
  4. A new highly nonlinear equation
  5. Local well-posedness
  6. Well-posedness in $B_{p,r}^s$ with $s>\max({1+\frac{1}{p}},\frac{3}{2}),1\leq p, r\leq +\infty$
  7. Well-posedness in $B_{2,1}^{\frac{3}{2}}$
  8. Blow-up scenario
  9. Appendix

Key Result

Theorem 3.1

Let $s,p,r$ satisfy the condition $s>\max({1+\frac{1}{p}},\frac{3}{2}), 1\leq p,r\leq +\infty,$ or $s=\frac{3}{2},p=2,r=1$. If $u_0\in B^{s}_{p,r}$, then there exists a time $T>0$, such that the initial value problem (equation) has a unique solution $u\in E^{s}_{p,r}(T)$. Moreover, the solution depe

Theorems & Definitions (12)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Definition 5.1
  • Definition 5.2
  • Proposition 5.1
  • Lemma 5.1
  • Lemma 5.2
  • ...and 2 more