Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2008; 45(3): 645-681

Printed May 1, 2008

Copyright © The Korean Mathematical Society.

Existence result for heat-conducting viscous incompressible fluids with vacuum

Yonggeun Cho and Hyunseok Kim

Chonbuk National University, Sogang University

Abstract

The Navier-Stokes system for heat-conducting incompressible fluids is studied in a domain $\Omega \subset \mathbf{R}^3$. The viscosity, heat conduction coefficients and specific heat at constant volume are allowed to depend smoothly on density and temperature. We prove local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. For the strong regularity, we do not assume the positivity of initial density; it may vanish in an open subset (vacuum) of $\Omega$ or decay at infinity when $\Omega$ is unbounded.

Keywords: heat-conducting incompressible Navier-Stokes equations, strong solutions, vacuum

MSC numbers: Primary 35A05, 76D03

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