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Keywords:
asymptotic behavior of solutions; one-dimensional motion of the viscous gas; compressible viscous gas
Summary:
We study the one-dimensional motion of the viscous gas represented by the system $v_t-u_x = 0$, $u_t+ p(v)_x = \mu(u_x/v)_x + f \left( \int_0^xv\dd x,t \right)$, with the initial and the boundary conditions $(v(x,0), u(x,0)) = (v_0(x), u_0(x))$, $u(0,t) = u(X,t) = 0$. We are concerned with the external forces, namely the function $f$, which do not become small for large time $t$. The main purpose is to show how the solution to this problem behaves around the stationary one, and the proof is based on an elementary $L^2$-energy method.
References:
[1] H. Beirão da Veiga: An $L^p$-theory for the n-dimensional, stationary, compressible, Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions. Comm. Math. Physics 109 (1987), 229-248. DOI 10.1007/BF01215222 | MR 0880415
[2] H. Beirão da Veiga: Long time behavior for one-dimensional motion of a general barotropic viscous fluid. Arch. Rat. Mech. Anal 108 (1989), 141-160. DOI 10.1007/BF01053460 | MR 1011555
[3] N. Itaya: The existence and uniqueness of the solution of the equations describing compressible viscous fluid flow. Proc. Jpn. Acad. 46 (1970), 379-382. DOI 10.3792/pja/1195520358 | MR 0364914 | Zbl 0207.39902
[4] N. Itaya: A survey on the generalized Burger's equation with pressure model term. J. Math. Kyoto Univ. 16 (1976), 223-240. MR 0402303
[5] Ya. Kaneľ: On a model system of equations of one-dimensional gas motion. Diff. Eqns. 4 (1968), 374-380.
[6] A. V. Kazhikhov: Correctness "in the large" of initial-boundary-value problem for model system of equations of a viscous gas. Din. Sploshnoi Sredy 21 (1975), 18-47. (In Russian.)
[7] A. V. Kazhikhov, V. B. Nikolaev: On the correctness of boundary value problems for the equations of a viscous gas with a non-monotonic function of state. Chislennye Metody Mekh. Sploshnoi Sredy 10 (1979), 77-84. (In Russian.) MR 0558830
[8] A. V. Kazhikhov, V. B. Nikolaev: On the theory of the Navier-Stokes equations of a viscous gas with nonmonotone state function. Soviet Math. Dokl. 20 (1979), 583-585. Zbl 0424.35074
[9] A. V. Kazhikhov, V. V. Shelukhin: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41 (1977)), 273-282. DOI 10.1016/0021-8928(77)90011-9 | MR 0468593 | Zbl 0393.76043
[10] A. Matsumura: Large time behavior of the solutions of a one-dimensional barotropic model of compressible viscous gas. (preprint).
[11] A. Matsumura, T. Nishida: Periodic solutions of a viscous gas equation. Lec. Notes in Num. Appl. Anal. 10 (1989), 49-82. MR 1041375 | Zbl 0697.35015
[12] V. A. Solonnikov, A. V. Kazhikhov: Existence theorems for the equations of motion of a compressible viscous fluid. Ann. Rev. Fluid Mech. 13 (1981), 79-95. DOI 10.1146/annurev.fl.13.010181.000455 | Zbl 0492.76074
[13] A. Tani: A survey on the one-dimensional compressible isentropic Navier-Stokes equations in a field of external forces. (unpublished).
[14] S. Yanagi: Global existence for one-dimensional motion of non-isentropic viscous fluids. Math. Methods in Appl. Sci. 16 (1993), 609-624. DOI 10.1002/mma.1670160902 | MR 1240450 | Zbl 0780.35082
[15] A. A. Zlotnik: On equations for one-dimensional motion of a viscous barotropic gas in the presence of a body force. Sibir. Mat. Zh. 33 (1993), 62-79.

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