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

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  • 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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