Abstract
We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier–Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L 2-energy method with the aid of the Poincaré type inequality.
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References
Friedman, A.: Partial Differential Equations of Parabolic Type. Englewood Cliffs, NJ: Prentice Hall, Inc., 1964
Il'in, A., Oleinik, O.: Asymptotic behavior of the solutions of Cauchy problems for certain quasilinear equations for large time (Russian). Mat. Sb. 51, 191–216 (1960)
Kawashima, S., Nishibata, S.: Stationary waves for the discrete Boltzmann equation in the half space with the reflective boundaries. Commun. Math. Phys. 211, 183–206 (2000)
Kawashima, S., Nishida, T.: Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases. J. Math. Kyoto Univ. 21(4), 825–837 (1981)
Kawashima, S., Zhu, P.: Asymptotic stability of the rarefaction waves of compressible Navier–Stokes equations in the half space. To appear
Kawashima, S., Zhu, P.: Asymptotic stability of the superposition of nonlinear waves of compressible Navier–Stokes equations in the half space. To appear
Liu, T., Matsumura, A., Nishihara, K.: Behaviors of solutions for the Burgers equations with boundary corresponding to rarefaction waves. SIAM J. Math. Anal. 29, 293–308 (1998)
Liu, T., Nishihara, K.: Asymptotic behavior for scalar viscous conservation laws with boundary effect. J. Diff. Eq. 133, 296–320 (1997)
Matsumura, A.: Inflow and outflow problems in the half space for a one-dimensional isentropic model system for compressible viscous gas. In: Proceedings of IMS Conference on Differential Equations from Mechanics. Hong Kong, 1999
Matsumura, A., Mei, M.: Asymptotics toward viscous shock profile for solution of the viscous p–System with boundary effect. Arch. Rat. Mech. Anal. 146, 1–22 (1999)
Matsumura, A., Nishihara, K.: Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional isentropic model system for compressible viscous gas. Commun. Math. Phys. 222, 449–474 (2001)
Nishibata, S.: Asymptotic stability of traveling waves to a certain discrete velocity model of the Boltzmann equation in the half space. SIAM J. Math. Anal. 34, 555–572 (2002)
Nikkuni, Y., Kawashima, S.: Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions. Kyushu J. Math. 54, 233–255 (2000)
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Communicated by P. Constantin
The second author's work was supported in part by Grant-in-Aid for Scientific Research (C)(2) 14540200 of the Ministry of Education, Culture, Sports, Science and Technology and the third author's work was supported by JSPS postdoctoral fellowship under P99217.
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Kawashima, S., Nishibata, S. & Zhu, P. Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space. Commun. Math. Phys. 240, 483–500 (2003). https://doi.org/10.1007/s00220-003-0909-2
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DOI: https://doi.org/10.1007/s00220-003-0909-2