System Of Compressible Adiabatic Flow Research Articles

Half space problem of the system of compressible adiabatic flow through porous media in multi-D

In this paper, we consider the half space problem for the system of compressible adiabatic flow through porous media in 2-D and 3-D. Combining energy method, Green's function method and bootstrap method, one establishes the decaying energy estimates which lead to global existence and optimal L2 decaying rate of the solution. Finally, for more large time information especially in spatial variables, one applies Green's function, solution representation obtained by Duhamel's principle and a priori estimate from energy method to obtain the space time pointwise structure of the solution.

Convergence rates to the damped system of compressible adiabatic flow through porous media with boundary effect

In this paper, we consider convergence rates to solutions for the damped system of compressible adiabatic flow through porous media with boundary effect. Compared with the results obtained by Pan, the better convergence rates are obtained in this paper. Our approach is based on the technical time-weighted energy estimates.

Large time behavior for the system of compressible adiabatic flow through porous media in R3

The Cauchy problem of the system of compressible adiabatic flow through porous media in R 3 is considered. We obtain the global existence and large time behavior of the solution when the initial data is close to its equilibrium in H 3 -norm. This is a continuing work of [31] , where due to non-dissipative property of the entropy s , an additional assumption that the initial data is bounded in L 1 -norm plays a key role in closing the a priori energy estimates and hence establishing the global existence of the solution. In this paper, to remove this assumption, we first decompose the solution U into high frequency part U h and low frequency part U l . Then by constructing delicate energy functionals for U h and optimal decay rates for U l , we can get the desired energy estimates to close the a priori assumption. Furthermore, we get the optimal L 2 − L 2 decay rate of the solution and show that the velocity decays faster than the density.

Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping

This paper is concerned with large-time behavior of solutions forthe system of compressible adiabatic flow through porous media with nonlinear damping. For the nonlinear damping case, i.e.$\beta \neq 0,$ results for the linear damping case are extended tothe case of nonlinear damping. Compared with the results obtained byMarcati and Pan, better decay estimates are obtained in this paper.

Global existence and large time behavior for the system of compressible adiabatic flow through porous media in [formula omitted

The system of compressible adiabatic flow through porous media is considered in R3 in the present paper. The global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium. We also show that the pressure of the system converges to its equilibrium state at the same L2-rate (1+t)−34 as the Navier–Stokes equations without heat conductivity, but the velocity of the system decays at the L2-rate (1+t)−54, which is faster than the L2-rate (1+t)−34 for the Navier–Stokes equations without heat conductivity (Duan and Ma, 2008 [3]).

Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant

We realize the best asymptotic profile for the solutions to the non-isentropic p-system with damping on quadrant is a particular solution of the IBVP for the corresponding nonlinear parabolic equation with special initial data, and we further show the convergence rates to this particular asymptotic profile. This rates are same to that for the isentropic case obtained by H. Ma and M. Mei (J. Differential Equations 249 (2010), 446-484).

Convergence Rates to Nonlinear Diffusion Waves for Solutions to the System of Compressible Adiabatic Flow through Porous Media

We realize the best asymptotic profile for the solutions to the Cauchy problem of the nonisentropic p-system with damping is a particular solution to the corresponding nonlinear parabolic equation with special initial data, and we further show the convergence rates to this particular asymptotic profile. This new rates are much better than that obtained by Nishihara and Nishikawa [15].

Global existence and blow-up phenomena of classical solutions for the system of compressible adiabatic flow through porous media

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small’ solution.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE COMPRESSIBLE ADIABATIC FLOW THROUGH POROUS MEDIA WITH BOUNDARY EFFECTS

The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy' laws on [0,1] × [0, + ∞) are considered respectively. The global existence of smooth solutions to the IBVP problems for two systems are proved, and their large-time behavior is analyzed. The time-asymptotic equivalence of these two systems are investigated, the decay rate of the difference of solutions between these two systems are shown to be determined explicitly by the initial perturbations and boundary effects. It is found that the oscillation of the specific volume can be cancelled by that of entropy, i.e., the oscillation of the specific volume and entropy is not required to be small.

Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media

Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media

Asymptotic Behavior of Solutions to the System of Compressible Adiabatic Flow through Porous Media

Hsiao and Serre in [Chinese Ann. Math. Ser. B, 16B (1995), pp. 1--14] showed the solution to the system $$ \left\{\begin{array}{@{}ll} v_t-u_x = 0, & (t,x) \in R_+ \times R, \\[3pt] u_t+p(v,s)_x =-\alpha u, &\alpha > 0, \\[3pt] s_t = 0 \end{array}\right. $$ with initial data $$ (v,u,s)(0,x) = (v_0,u_0,s_0)(x) \rightarrow (\underline{v},u_{\pm}, \underline{s})\qquad \text{ as } x \rightarrow \pm\infty $$ tends to the following nonlinear parabolic equation time-asymptotically: $$ \left\{\begin{array}{@{}ll} \tilde{v}_t = -\frac 1{\alpha} p(\tilde{v},s_0)_{xx}, & (t,x) \in R_+ \times R, \\[3pt] \tilde{u} = -\frac 1{\alpha} p(\tilde{v},s_0)_{x}. \end{array}\right. $$ In this paper we find its convergence rate, which will be optimal.

On the Diffusive Profiles for the System of Compressible Adiabatic Flow through Porous Media

We study the Cauchy problem for the system of one dimensional compressible adiabatic flow through porous media and the related diffusive problem. We introduce a new approach which combines the usual energy methods with special L1 -estimates and use the weighted Sobolev norms to prove the global existence and large time behavior for the solutions of the problems. The asymptotic states for the solutions are given by either stationary solutions or similarity solutions depending on the behavior of the initial data when $|x|\rightarrow \infty$. Our estimates provide asymptotic time decay rates.

Initial Boundary Value Problem for the System of Compressible Adiabatic Flow Through Porous Media

We study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition. Under the restriction on the oscillations in the initial data, we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods. The time-asymptotic states for the solution are found and the exponential convergence to the asymptotic states is proved. It is also shown that this problem can be approximated very well time-asymptotically by an initial boundary problem for the related diffusion equations obtained from the original hyperbolic system by Darcy's law, provided that the initial total mass is the same. The difference between these two solutions tends to zero exponentially fast as time goes to infinity in the sense of H1. The diffusive phenomena caused by damping mechanism with boundary effects are thus observed.

Nonlinear Diffusive Phenomena of Solutions for the System of Compressible Adiabatic Flow through Porous Media

Nonlinear Diffusive Phenomena of Solutions for the System of Compressible Adiabatic Flow through Porous Media

Global Existence of Solutions for the System of Compressible Adiabatic Flow through Porous Media

Consider the quasilinear hyperbolic system \[(1)\qquad \left\{ \begin{gathered} v_t - u_x = 0, \hfill \\ u_t + p(v,s)_x = - \alpha u,\quad \alpha > 0,\quad p_v < 0{\text{ for }}v > 0, \hfill \\ s_t = 0 \hfill \\ \end{gathered} \right.\] with initial data \[ (2)\qquad u(x,0) = u_0 (x),\quad v(x,0) = \bar v + v_o (x),\quad s(x,0) = \bar s + s_0 (x), \] where $\bar v > 0$, $\bar v$ and $\bar s$ are constants, $(u_0 (x),v_0 (x)) \in C^1 $ with a compact support, and $s_0 (x) \in C^2 $ with a compact support. It is proved in this paper that there exists a globally defined classical solution for the Cauchy problem if the $C^1 $-norm of $(u_0 (x),v_0 (x))$ and the $C^2 $-norm of $s_0 (x)$ are small.