Vol. 292, No. 2, 2018

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ISSN: 0030-8730
Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping

Fei Hou, Ingo Witt and Huicheng Yin

Vol. 292 (2018), No. 2, 389–426
Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D irrotational compressible Euler equation with time-dependent damping

tρ + div(ρu) = 0, t(ρu) + div(ρu u + pI3) = α(t)ρu, ρ(0,x) = ρ̄ + ερ0(x),u(0,x) = εu0(x),

where x 3, the frictional coefficient α(t) = μ(1 + t)λ with μ > 0 and λ 0, ρ̄ > 0 is a constant, ρ0,u0 C0(3), (ρ0,u0)0, ρ(0,x) > 0, curlu0 0, and ε > 0 is sufficiently small. For 0 λ 1, we show that there exists a global C([0,) × 3)-smooth solution (ρ,u) by introducing and establishing some uniform time-weighted energy estimates of (ρ,u), while for λ > 1, in general, the smooth solution (ρ,u) blows up in finite time. Therefore, λ = 1 appears to be the critical value for the global existence of small amplitude smooth solution (ρ,u).

Keywords
compressible Euler equations, damping, time-weighted energy inequality, Klainerman–Sobolev inequality, blowup, hypergeometric function
Mathematical Subject Classification 2010
Primary: 35L70
Secondary: 35L65, 35L67, 76N15
Milestones
Received: 31 December 2016
Accepted: 2 May 2017
Published: 17 October 2017
Authors
Fei Hou
Department of Mathematics and IMS
Nanjing University
Nanjing
China
Ingo Witt
Mathematical Institute
University of Göttingen
Göttingen
Germany
Huicheng Yin
School of Mathematical Sciences
Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems
Nanjing Normal University
Nanjing
China