214 research outputs found
Asymptotic Behavior of a Viscous Liquid-Gas Model with Mass-Dependent Viscosity and Vacuum
In this paper, we consider two classes of free boundary value problems of a
viscous two-phase liquid-gas model relevant to the flow in wells and pipelines
with mass-dependent viscosity coefficient. The liquid is treated as an
incompressible fluid whereas the gas is assumed to be polytropic. We obtain the
asymptotic behavior and decay rates of the mass functions ,\
when the initial masses are assumed to be connected to vacuum both
discontinuously and continuously, which improves the corresponding result about
Navier-Stokes equations in \cite{Zhu}.Comment: 24 page
Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant
This paper is concerned with the asymptotic behavior of the solution to the
Euler equations with time-depending damping on quadrant , \begin{equation}\notag \partial_t v
-
\partial_x u=0, \qquad \partial_t u
+
\partial_x p(v)
=\displaystyle
-\frac{\alpha}{(1+t)^\lambda} u, \end{equation} with null-Dirichlet boundary
condition or null-Neumann boundary condition on . We show that the
corresponding initial-boundary value problem admits a unique global smooth
solution which tends time-asymptotically to the nonlinear diffusion wave.
Compared with the previous work about Euler equations with constant coefficient
damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156,
439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918),
we obtain a general result when the initial perturbation belongs to the same
space. In addition, our main novelty lies in the facts that the cut-off points
of the convergence rates are different from our previous result about the
Cauchy problem. Our proof is based on the classical energy method and the
analyses of the nonlinear diffusion wave
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