Evolution Equations and Control Theory (EECT)

Hyperbolic Navier-Stokes equations I: Local well-posedness

Pages: 195 - 215, Volume 1, Issue 1, June 2012      doi:10.3934/eect.2012.1.195

       Abstract        References        Full Text (426.3K)       Related Articles       

Reinhard Racke - Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany (email)
Jürgen Saal - Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 32, 64287 Darmstadt, Germany (email)

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.

Keywords:  Navier-Stokes, well-posedness, Cattaneo law, Fourier law, Oldroyd.
Mathematics Subject Classification:  Primary: 35L72, 35Q30, 76D05.

Received: September 2011;      Revised: December 2011;      Available Online: March 2012.