Abstract:In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and, we prove the weak solution (cQ(x, t), u(x, t)) behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if θ ∈ (0,

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<p style='text-indent:20px;'>In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].</p>

The aim of this paper is to study the global unique solvability on Sobolev solution perturbated around diffusion waves to the Cauchy problem of conservative form of Hsieh's equations. Furthermore, convergence rates are also obtained as one of the diffusion parameters goes to zero. The difficulty is created due to conservative nonlinearity to enclose the uniform (in diffusion parameter) higher order energy estimates. However this kind of difficulty will not occur for both the nonconservative nonlinearity and fixed diffusion parameter. The more subtle mathematical analysis needs to be introduced to overcome the difficulties.

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