Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy

Details Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy

2009-04-22

arxiv.org/abs/0904.3572v1

Mathematics

Analysis of PDEs

29 pages, submitted

Scientific paper

Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier-Stokes system.

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