Abstract:
The inhomogeneous plane waves, their mean energy flux vectors and mean energy dissipation rates in a liquid-saturated incompressible viscoelastic porous medium are investigated intensively under the assumption of the infinitesimal deformation of the solid skeleton. The general solutions for the plane longitudinal and transverse waves are obtained with the Helmholtz's resolution. The dependences of the wave vectors, phase velocities and attenuation coefficients on the material parameters of the solid and liquid phases are expressed explicitly, and the relation between the propagation vector and attenuation one are discussed. Numerical results show that the influences of the interaction between the solid skeleton and pore fluid as well as the viscosity of the solid on the wave velocities, attenuation coefficients, etc. are remarkable. An energy equation of mechanical power is derived, and the energy flux vector and energy dissipation rate are obtained in the general form. The explicit expressions of the mean energy flux vectors and the mean energy dissipation rates are presented for the inhomogeiieous plane waves.

Abstract:
This article concerns the time growth of Sobolev norms of classical solutions to the 3D incompressible isotropic elastodynamics with small initial displacements.

Abstract:
We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic elastodynamics in two space dimensions admits a uniqueness global classical solution.

Abstract:
We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/{\epsilon}]$, for initial data of the form $\epsilon U_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/{\epsilon}]$, provided that $\epsilon$ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted $L^2$ estimates and the ghost weight introduced by Alinhac.

Abstract:
A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view. Some estimates on the second fundamental form and velocity of the free surface are also obtained.

Abstract:
The equations of motion in a macroscopically inhomogeneous porous medium saturated by a fluid are derived. As a first verification of the validity of these equations, a two-layer rigid frame porous system considered as one single porous layer with a sudden change in physical properties is studied. A simple wave equation is derived and solved for this system. The reflection and transmission coefficients are calculated numerically using a wave splitting-Green's function approach (WS-GF). The reflected and transmitted wave time histories are also simulated. Experimental results obtained for materials saturated by air are compared to the results given by this approach and to those of the classical transfer matrix method (TMM).

Abstract:
We establish the global existence and the asymptotic behavior for the 2D incompressible isotropic elastodynamics for sufficiently small, smooth initial data in the Eulerian coordinates formulation. The main tools used to derive the main results are, on the one hand, a modified energy method to derive the energy estimate and on the other hand, a Fourier transform method with a suitable choice of $Z$- norm to derive the sharp $L^\infty$- estimate. This paper improves the almost global existence result of Lei-Sideris-Zhou in the Eulerian coordinates formulation. We mention that the global existence of the same system but in the \emph{Lagrangian} coordinates formulation was recently obtained by Lei. Our goal is to improve the understanding of the behavior of solutions in different coordinates using a different approach and from the point of view in frequency space.

Abstract:
We prove that the initial-value problem for the motion of a certain type of elastic body has a solution for all time if the initial data are sufficiently small. The body must fill all of three space, obey a ``neo-Hookean'' stress-strain law, and be incompressible. The proof takes advantage of the delayed singularity formation which occurs for solutions of quasi-linear hyperbolic equations in more than one space dimension. It turns out that the curl of the displacement of the body obeys such an equation. Thus using Klainerman's inequality, one derives the necessary estimates to gaurantee that solutions persist for all time.

Abstract:
we formulate conservation laws governing steam injection in a linear porous medium containing water. heat losses to the outside are neglected. we find a complete and systematic description of all solutions of the riemann problem for the injection of a mixture of steam and water into a water-saturated porous medium. for ambient pressure, there are three kinds of solutions, depending on injection and reservoir conditions. we show that the solution is unique for each initial data.

Abstract:
We formulate conservation laws governing steam injection in a linear porous medium containing water. Heat losses to the outside are neglected. We find a complete and systematic description of all solutions of the Riemann problem for the injection of a mixture of steam and water into a water-saturated porous medium. For ambient pressure, there are three kinds of solutions, depending on injection and reservoir conditions. We show that the solution is unique for each initial data.