Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part

Boris Haspot

doi:10.1016/j.jde.2017.01.010

Abstract

We are interested in studying the Cauchy problem for the viscous shallow-water system in dimension N≥2, we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations. More precisely our smallness assumption on the initial data is supercritical for the scaling of the equations. It allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension N=2. To do this, we introduce the notion of quasi-solutions which consists in solving the pressureless viscous shallow water system. We can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density. This smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the “quasi-solutions”. Indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations. To finish we prove the existence of global strong solution with large initial data when N≥2 provided that the Mach number is sufficiently large.

Highlights

The motion of a general barotropic compressible fluid is described by the following system: ∂tρ + div(ρu) =∂t(ρu) + div(ρu ⊗ u) − div(2μ(ρ)D(u)) − ∇(λ(ρ)divu) + ∇P (ρ) = 0, (ρ, u)/t=0 (ρ0, u0). (1.1) 0.1Here u = u(t, x) ∈ RN stands for the velocity field, ρ = ρ(t, x) ∈ R+ is the density and D(u)
In [16] we introduce a notion of effective velocity in high frequencies which allows us to cancel out the coupling between the velocity and the pressure
We have proved of the form

Summary

Introduction

The motion of a general barotropic compressible fluid is described by the following system:. R1 Remark 1 Let us mention than the main interest of this theorem is to prove the existence of global strong solution with large initial data for the scaling of the equation which is new up to our knowledge. We have: the solution is unique if Remark 7 The main interest of this theorem is to prove the existence of global strong solution for any large initial data provided that K is sufficiently small with P (ρ) = Kρ Remark 9 Compared with the theorem 1.1, we do not need any assumption of smallness on the density ρ10, it corresponds to a result of global strong solution for large initial data when N ≥ 2.

Littlewood-Paley theory and Besov spaces

Hybrid Besov spaces

A linear model with convection

Proof of the existence

Let us deal now with hybrid the