Local existence and blow-up criterion of the ideal density-dependent flows

Fangyi He,Jishan Fan,Yong Zhou

doi:10.1186/s13661-016-0610-y

Abstract

In this paper, we consider two ideal density-dependent flows in a bounded domain, the Euler and magnetohydrodynamics equations. We prove the local existence and a blow-up criterion for each system.

Highlights

We consider the following D density-dependent Euler system:∂tρ + u · ∇ρ =, ( . )ρ∂tu + ρ(u · ∇)u + ∇π =, div u =,u · n = on ∂ × (, ∞),(ρ, u)(·, ) = (ρ, u ) in ⊂ R .Here is a bounded domain with smooth boundary ∂ ∈ C∞, n is the outward unit normal to ∂ ; the unknowns are the fluid velocity field u = u(x, t), the pressure π = π(x, t), and the density ρ = ρ(x, t).Beirão da Veiga and Valli [, ] and Valli and Zajaczkowski [ ] proved the unique solvability, local in time, in some supercritical Sobolev spaces and Hölder spaces in bounded domains
The first aim of this paper is to prove the local existence and a blow-up criterion of problem ( . )-( . ) in the Lp frame work
The second aim of this paper is to prove the local well-posedness of problem ( . )-( . ) without any smallness condition; we will prove a regularity criterion

Summary

Introduction

We consider the following D density-dependent Euler system:. (ρ, u)(·, ) = (ρ , u ) in ⊂ R. We consider the following D density-dependent Euler system:. The first aim of this paper is to prove the local existence and a blow-up criterion of problem We consider the following ideal density-dependent MHD system:. ). When := R , Zhou and Fan [ ] proved the local well-posedness of problem ) and proved the local unique solvability with the main condition that. The second aim of this paper is to prove the local well-posedness of problem There exists a positive time T∗ > such that problem We will use the following well-known Osgood lemma in [ ]. ), we derive d Dsρ dx = – Ds(u · ∇ρ) – u · ∇Dsρ Dsρ dx dt

Dsρ L

Hs u

Hs b