Stationary waves to viscous heat-conductive gases in half-space: existence, stability and convergence rate. (English) Zbl 1213.35104
The authors study large-time behaviour of a solution to an initial-boundary value problem for the compressible Navier-Stokes equations with one spatial variable \(x\geq 0\). The gas is considered ideal, polytropic, compressible, viscous, and heat-conductive. The equations are written in Eulerian coordinates. Unknowns are density, velocity, and temperature. Initial distribution is asymptotically constant and positive. Dirichlet boundary conditions are given for \(x=0\), and the velocity there is negative. This corresponds to the outflow problem and guarantees well-posedness of the problem. Using the center manifold theory, the authors prove the existence of the stationary solution under smallness assumption of the boundary data. Also stability of this solution, provided that boundary data and initial perturbation are small in the Sobolev space, is proved. The convergence rate of the solution to the stationary solution is obtained.
Reviewer: Ilya A. Chernov (Petrozavodsk)
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 76N15 | Gas dynamics (general theory) |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L50 | Initial-boundary value problems for first-order hyperbolic systems |
Keywords:
compressible Navier-Stokes equation; ideal polytropic gas; outflow problem; boundary layer solution; weighted energy method; Eulerian coordinates; center manifold theoryReferences:
| [1] | Aoki, K.et al., Phys. Fluids A3, 2260 (1991), DOI: 10.1063/1.857907. |
| [2] | Aoki, K.Sone, Y.Yamada, T., Phys. Fluids A2, 1867 (1990), DOI: 10.1063/1.857661. |
| [3] | Carr, J., Applications of Centre Manifold Theory, 1981, Springer-Verlag · Zbl 0464.58001 |
| [4] | Friedman, A., Partial Differential Equations of Parabolic Type, 1964, Prentice Hall · Zbl 0144.34903 |
| [5] | Il’in, A. M.Oleĭnik, O. A., Dokl. Akad. Nauk SSSR120, 25 (1958). · Zbl 0082.08901 |
| [6] | Kagei, Y.Kawashima, S., Comm. Math. Phys.266, 401 (2006), DOI: 10.1007/s00220-006-0017-1. |
| [7] | Kawashima, S.Kurata, K., J. Funct. Anal.257, 1 (2009), DOI: 10.1016/j.jfa.2009.04.003. |
| [8] | Kawashima, S.Matsumura, A., Comm. Math. Phys.101, 97 (1985), DOI: 10.1007/BF01212358. |
| [9] | Kawashima, S.Nishibata, S.Zhu, P., Comm. Math. Phys.240, 483 (2003). · Zbl 1038.35057 |
| [10] | Liu, T.-P.Matsumura, A.Nishihara, K., SIAM J. Math. Anal.29, 293 (1998), DOI: 10.1137/S0036141096306005. |
| [11] | Matsumura, A., Methods Appl. Anal.8, 645 (2001). · Zbl 1161.76555 |
| [12] | Matsumura, A.Nishihara, K., Comm. Math. Phys.165, 83 (1994), DOI: 10.1007/BF02099739. |
| [13] | Nakamura, T.Nishibata, S., SIAM J. Math. Anal.41, 1757 (2009), DOI: 10.1137/090755357. |
| [14] | Nakamura, T.Nishibata, S.Yuge, T., J. Diff. Eqns.241, 94 (2007), DOI: 10.1016/j.jde.2007.06.016. |
| [15] | Nishihara, K., Jpn. J. Appl. Math.2, 27 (1985), DOI: 10.1007/BF03167037. |
| [16] | Nishikawa, M., Funkcial. Ekvac.41, 107 (1998). · Zbl 1140.35520 |
| [17] | Sone, Y.et al., Eur. J. Mech. B/Fluids17, 277 (1998), DOI: 10.1016/S0997-7546(98)80260-4. |
| [18] | Nakamura, T.; Ueda, Y.; Kawashima, S., Convergence rate toward degenerate stationary wave for compressible viscous gases, Proc. of Int. Conf. on Nonlinear Analysis and Convex Analysis · Zbl 1227.35072 |
| [19] | Ueda, Y.; Nakamura, T.; Kawashima, S., Arch. Rational Mech. Anal. |
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