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Center for Nonlinear Studies |
Hyperbolic-parabolic systems have spatially homogeneous stationary solutions. When the dissipation is weak, one can derive weakly onlinear-dissipative approximations that govern perturbations of these stationary solutions. These approximations are quadratically nonlinear. Up to a linear transformation, they are independent of the dependent variables used to express the original system. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that under a mild structural hypothesis, this approximation has global weak solutions for all initial data in that Hilbert space. This theory applies to the compressible Navier-Stokes system. The resulting approximate system is an incompressible Navier-Stokes system coupled toequations that govern the acoustic modes. The solution of this approximate system is unique if the incompressible modes are uniquely determined.