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2003, Volume 9, Issue 6: 1587-1606. Doi: 10.3934/dcds.2003.9.1587
This issue Previous Article Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions Next Article Global and local complexity in weakly chaotic dynamical systems

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

  • 1.

    Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730

  • 2.

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

Received:  May 31, 2002
Revised:  August 31, 2002
Published:  October 2003
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    Abstract
  • 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.
    Mathematics Subject Classification: Primary: 35B20, 35M20, 35B35, 76N10.

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