Geometrical demonstration for persistence properties for a bi-Hamiltonian shallow water system

Persistence and asymptotic analysis of solutions of nonlinear wave equations

We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai's, the Camassa-Holm, and the Dullin-Gottwald-Holm equations, as well as some recent shallow water equations with the Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentione…

Global persistence of geometrical structures for the boussinesq equation with no diffusion

Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{ö}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial…

Least action principle for an integrable shallow water equation

For an integrable shallow water equation we describe a geometrical approach showing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.

Conserved quantities, continuation and compactly supported solutions of some shallow water models

A proof that strong solutions of the Dullin-Gottwald-Holm equation vanishing on an open set of the (1+1) space-time are identically zero is presented. In order to do it, we use a geometrical approach based on the conserved quantities of the equation to prove a unique continuation result for its solutions. We show that this idea can be applied to a large class of equations of the Camassa-Holm type.

准地转浅水方程涡旋斑块边界的正则性

The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

研究准地转浅水方程中涡旋斑块边界光滑性的保持，并证明当Rossby半径参数趋于0时解在little Hölder空间中局部收敛到Euler方程解。

Imperfect bifurcation for the quasi-geostrophic shallow-water equations

We study analytical and numerical aspects of the bifurcation diagram of simply-connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length $\EE^{-1}$, which enters in the relation between streamfunction and (potential) vorticity. The Euler equations are recovered in the limit $\EE\to 0$. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However …