Nonlinear Thermal Instability in Compressible Viscous Flows Without Heat Conductivity

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Abstract

We investigate the thermal instability of a smooth equilibrium state, in which the density function satisfies Schwarzschild’s (instability) condition, to a compressible heat-conducting viscous flow without heat conductivity in the presence of a uniform gravitational field in a three-dimensional bounded domain. We show that the equilibrium state is linearly unstable by a modified variational method. Then, based on the constructed linearly unstable solutions and a local well-posedness result of classical solutions to the original nonlinear problem, we further construct the initial data of linearly unstable solutions to be the one of the original nonlinear problem, and establish an appropriate energy estimate of Gronwall-type. With the help of the established energy estimate, we finally show that the equilibrium state is nonlinearly unstable in the sense of Hadamard by a careful bootstrap instability argument.

Keywords

Compressible Navier–Stokes–Fourier equations Bénard problem thermal instability Hadamard sense 

Mathematics Subject Classification

Primary 76E06 Secondary 76E19 
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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceFuzhou UniversityFuzhouChina

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