Existence of global strong solutions for the shallow-water equations with large initial data

TL;DR

This paper proves the existence of global strong solutions for a viscous shallow-water system with large initial data, using a new effective velocity to handle coupling and damping effects in critical spaces.

Contribution

Introduces a new effective velocity to simplify the shallow-water equations, enabling proof of global solutions with large initial data in critical function spaces.

Findings

Existence of global strong solutions for large initial data.

Effective velocity simplifies coupling in the equations.

Friction induces damping, aiding solution stability.

Abstract

This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when $N\geq 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{Hprepa} a new unknown,\textit{a effective velocity} $v=u+\mu\n\ln h$ ($u$ is the classical velocity and $h$ the depth variation of the fluid) with $\mu$ the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity $u$ and the depth variation $h$. We obtain then the existence of global strong solution if $m_{0}=h_{0}v_{0}$ is small in $B^{\N-1}_{2,1}$ and $(h_{0}-1)$ large in $B^{\N}_{2,1}$. In particular it implies that the classical momentum $m_{0}^{'}=h_{0} u_{0}$ can be large in $B^{\N-1}_{2,1}$, but small when we project $m_{0}^{'}$ on the divergence field. These solutions are in some sense \textit{purely compressible}. We would like to point out that the friction term term has a fundamental role in our work inasmuch as coupling with the pressure term it creates a damping effect on the effective velocity.