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November  2003, 9(6): 1587-1606. doi: 10.3934/dcds.2003.9.1587

A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow

Gui-Qiang Chen 1, and  Bo Su 2,

1. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, United States
2. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, United States

Received  June 2002 Revised  September 2002 Published  September 2003

We study a nonlinear system of partial differential equations that is a viscous approximation for a multidimensional unsteady Euler potential flow governed by the conservation of mass and the Bernoulli law. The system consists of a transport equation for the density and the viscous nonhomogeneous Hamilton-Jacobi equation for the velocity potential. We establish the existence and regularity of global solutions for the nonlinear system with arbitrarily large periodic initial data. We also prove that the density in our global solutions has a positive lower bound, that is, our solutions always stay away from the vacuum, as long as the initial density has a positive lower bound.
Keywords: Viscous approximation, Euler flow, potential flow, multidimension, existence, regularity., Navier-Stokes equations, density bounds, large data, unsteady flow, compressible fluids.
Mathematics Subject Classification: Primary: 35B20, 35M20, 35B35, 76N1.
Citation: Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587
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