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

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Abstract

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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Author notes

  1. Ning Jiang

    Present address: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

Authors and Affiliations

  1. Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Ning Jiang

  2. Department of Mathematics and Institute for Physical Science and Technology (IPST), University of Maryland, College Park, MD, 20742-4015, USA

    C. David Levermore

Authors
  1. Ning Jiang
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  2. C. David Levermore
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Correspondence to C. David Levermore.

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Communicated by C.M. Dafermos

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Jiang, N., Levermore, C.D. Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy. Arch Rational Mech Anal 201, 377–412 (2011). https://doi.org/10.1007/s00205-010-0361-3

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  • DOI: https://doi.org/10.1007/s00205-010-0361-3

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