Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

2019 ◽  
Vol 18 (02) ◽  
pp. 333-358
Author(s):  
Ben Duan ◽  
Zhen Luo ◽  
Yan Zhou
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Force Term ◽  
The Cauchy Problem ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

scholarly journals On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jingjing Xu ◽  
Zaihong Jiang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Propagation Speed ◽  
One Dimensional ◽  
Dissipative Term ◽  
The Cauchy Problem ◽  
Infinite Propagation Speed

We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations

1991 ◽  
Vol 227 ◽  
pp. 393-406 ◽  
Author(s):  
Darryl D. Holm
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Hamiltonian Dynamics ◽  
Relative Equilibria ◽  
Hamiltonian Mechanics ◽  
Lagrangian Coordinates ◽  
Vortex Solutions ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

scholarly journals Variational integrator for the rotating shallow‐water equations on the sphere

2019 ◽  
Vol 145 (720) ◽  
pp. 1070-1088 ◽  
Author(s):  
Rüdiger Brecht ◽  
Werner Bauer ◽  
Alexander Bihlo ◽  
François Gay‐Balmaz ◽  
Scott MacLachlan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Variational Integrator ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations
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