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Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a parallel flow. (English) Zbl 1282.76159
Summary: The initial-boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of \({\mathbf{R}^{2}}\). It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large-time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
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