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Asymptotic profiles for the compressible Navier-Stokes equations in the whole space. (English) Zbl 1358.35092
In this paper the authors investigate the initial value problem for the compressible Navier-Stokes equations in \(\mathbb R^n\), \(n \geq 3\). They are concerned with the asymptotic properties of strong solutions of the original problem around the constant stationary solution \((\bar \rho, 0)\). In former papers it was shown [S. Kawashima et al., Commun. Math. Phys. 70, 97–124 (1979; Zbl 0449.76053); D. Hoff and K. Zumbrun, Indiana Univ. Math. J. 44, No. 2, 603–676 (1995; Zbl 0842.35076)] that the perturbation of the constant state is time-asymptotic to a solution of the linearized problem. This leads to a first-order asymptotic profile. In the paper under consideration a second-order asymptotic profile of the solution, which is generated by the nonlinear effect, is derived in a very detailed manner.
The bibliography contains 8 items and concentrates to the essential papers.

MSC:
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35D35 Strong solutions to PDEs
76N99 Compressible fluids and gas dynamics, general
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[1] Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier analysis and nonlinear partial differential equation, (2011), Springer Heidelberg
[2] Friedman, A., Partial differential equations, (1969), Holt, Rinehert and Winston New York
[3] Hoff, D.; Zumbrun, K., Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44, 604-676, (1995) · Zbl 0842.35076
[4] Kawashima, S.; Matsumura, A.; Nishida, T., On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70, 97-124, (1979) · Zbl 0449.76053
[5] Kobayashi, T.; Shibata, Y., Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equation, Pacific J. Math., 207, 199-234, (2002) · Zbl 1060.35104
[6] Matsumura, A.; Nishida, T., The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55, 337-342, (1979) · Zbl 0447.76053
[7] Matsumura, A.; Nishida, T., The initial value problems for the equation of motion of compressible viscous and heat-conductive gases, J. Math. Kyoto Univ., 20, 67-104, (1980) · Zbl 0429.76040
[8] Okita, M., On the convergence rates for the compressible Navier-Stokes equations with potential force, Kyushu J. Math., 68, 261-286, (2014) · Zbl 1314.35085
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