Hyperbolic Navier-Stokes equations I: Local well-posedness

Reinhard Racke; Jürgen Saal

doi:10.3934/eect.2012.1.195

Article Contents

2012, Volume 1, Issue 1: 195-215. Doi: 10.3934/eect.2012.1.195

This issue Previous Article On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities Next Article Hyperbolic Navier-Stokes equations II: Global existence of small solutions

Hyperbolic Navier-Stokes equations I: Local well-posedness

Reinhard Racke^1, and
Jürgen Saal^2,

1.
Department of Mathematics, University of Konstanz, 78457 Konstanz
2.
Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 32, 64287 Darmstadt

Received: September 2011

Revised: December 2011

Published: June 2012

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Abstract

We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different approach via viscosity solutions is developed in order to prove higher regularity energy estimates for the linearized system. Surprisingly, this method yields stronger results than previous methods, by the fact that we can relax the regularity assumptions on the coefficients to a minimum. This leads to a short and elegant proof of a local-in-time existence result for the corresponding first order quasilinear system, hence also for the original hyperbolicly perturbed Navier-Stokes equations.

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Mathematics Subject Classification: Primary: 35L72, 35Q30, 76D05.

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References

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