Julien Diaz, Abdelaâziz Ezziani
We are interested in the modeling of wave propagation in an infinite bilayered acoustic/poroelastic media. We consider the biphasic Biot's model in the poroelastic layer. The first part is devoted to the calculation of analytical solution in two dimensions, thanks to Cagniard de Hoop method. In this second part we consider the 3D case.
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Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
本文在分层李群上，利用凹性论证和庞加莱不等式，证明了非线性多孔介质方程与伪抛物方程正解的整体存在性与爆破。
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is \emph{stratification} as opposed to previous stability…
We consider a linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic and a characteristic time of processes is small enough. Such kind of models may describe, for example, hydraulic fracturing or acoustic or seismic waves propagation. As the results, we derive different types of homogenized equations …
We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stratified steady states of IPM.
Sharp stability and instability of stratified steady states for the incompressible porous media equation
本文研究二维不可压缩多孔介质方程分层稳态的稳定性，证明周期通道上均匀递减稳态在小扰动下非线性稳定，而具有正导数上确界的稳态则非线性不稳定，并给出收敛速率与不稳定性机制。
Well-Posedness of the Free Boundary Incompressible Porous Media Equation
该研究针对自由边界不可压缩多孔介质方程，确定了初始自由边界的稳定性条件，证明该条件下分层密度的小局部扰动存在索伯列夫空间中的唯一局部时间解，证明过程用到了具独立价值的驯顺分数阶索伯列夫估计工具。
A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results, w…
Acoustic and Filtration Properties of Thermo-elastic porous medium: Biot's Equations of Thermo- Poroelasticity
本文针对含热流体的周期性热弹性多孔介质，采用Nguetseng两尺度收敛方法完成均匀化论证，推导得到不同形式的类Biot热孔隙弹性等方程组，解决了含振荡系数线性微分系统的处理难题。
In this paper, we consider a confined physical scenario to prove global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media (IPM) equation. The result is proved using the stability of stratified solutions, combined with an additional structure of our initial perturbation, which allows us to get rid of the boundary terms in the energy estimates.
We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace of the scattered field generated by an impulsive source localized just below the surface. From this relation it is, for example, clear that the eigenfrequencies can be recovered from the autocorrelat…
Homogenization and Filtration and Seismic Acoustic Problems in Thermo-elastic Porous Media
本文针对热弹性多孔介质的线性耦合运动问题，采用Nguetseng的两尺度收敛方法，在孔隙尺寸趋于零的周期结构下严格论证均质化过程，推导得到三类不同的热相关耦合或解耦方程组。
Rigorous justification of the hydrostatic-incompressible approximation for weakly stratified isothermal flow
本文针对弱分层等温流动，在小马赫数和小纵横比极限下，通过分析能量与声波演化建立可压缩Navier-Stokes系统解的一致存在唯一性，证明其解收敛到不可压缩原始方程的解，严格论证了静水压不可压缩近似。
Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure
本文利用能量结构直接分析二维周期通道中多孔介质方程与Stokes输运系统的非线性渐近稳定性，通过势能二阶导数的退化强制性结构改进扰动正则性假设与收敛速率。
Sharp asymptotic stability of the incompressible porous media equation
本文研究不可压缩多孔介质方程的渐近稳定性，证明其在稳定分层密度附近对$H^k$（$k>2$）初始扰动稳定，建立了高阶Sobolev空间的精确稳定性阈值，证明依赖势能泛函衰减的长时间收敛性分析与精细交换子估计。
Well-posedness and potential-based formulation for the propagation of hydro-acoustic waves and tsunamis
研究分层自由表面海洋中水声波与海啸传播的线性模型，通过分析函数空间和算子提出更易数学研究和高效离散化的新公式化，证明两种公式化的适定性，并用谱元法模拟海底地震和滑坡引发的海啸。
In this paper we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodicity hypothesis on the coefficients of the governing equations, we determine the macroscopic equivalent medium. To achieve our goal, we use some very recent tools about the sigma convergence of convolution sequences.
Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation
本文证明二维不可压缩多孔介质方程在临界索伯列夫空间$W^{1,\infty}$中对稳定剖面小扰动具有弱不适定性，且结果适用于任意垂直分层密度剖面，包括$g'(x_2)<0$的稳定构型，首次揭示此类构型的意外不稳定性。
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