scholarly journals STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA EQUATIONS WITH JUMPS

2009 ◽  
Vol 12 (03) ◽  
pp. 413-426 ◽  
Author(s):  
VIOREL BARBU ◽  
CARLO MARINELLI
Keyword(s):  
Porous Media ◽  
Strong Solutions ◽  
Strong Sense ◽  
Well Posedness ◽  
Independent Increments ◽  
Integrable Martingale ◽  
Porous Media Equations

We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by a square integrable martingale with stationary independent increments.

scholarly journals Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

2006 ◽  
Vol 31 (2) ◽  
pp. 277-291 ◽  
Author(s):  
G. Da Prato ◽  
Michael Röckner ◽  
B. L. Rozovskii ◽  
Feng-yu Wang
Keyword(s):  
Porous Media ◽  
Strong Solutions ◽  
Porous Media Equations

CONCENTRATION OF INVARIANT MEASURES FOR STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS

2007 ◽  
Vol 10 (03) ◽  
pp. 397-409 ◽  
Author(s):  
MICHAEL RÖCKNER ◽  
FENG-YU WANG
Keyword(s):  
Porous Media ◽  
Large Class ◽  
Invariant Measures ◽  
Strong Solutions ◽  
Probability Measures ◽  
Infinitesimal Generators ◽  
Bernstein Functions ◽  
Porous Media Equations ◽  
Concentration Properties

By using Bernstein functions, existence and concentration properties are studied for invariant measures of the infinitesimal generators associated to a large class of stochastic generalized porous media equations. In particular, results derived in Ref. 4 are extended to equations with non-constant and stronger noises. Analogous results are also proved for invariant probability measures for strong solutions.

Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organized criticality

2020 ◽  
pp. 2150029
Author(s):  
Marius Neuss
Keyword(s):  
Porous Media ◽  
Energy Functional ◽  
Physical Models ◽  
Well Posedness ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Self Organized ◽  
Porous Media Equations ◽  
Self Organized Criticality ◽  
Continuous Noise

We consider a class of generalized stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organized criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analyzed in detail.

scholarly journals L q -Functional Inequalities and Weighted Porous Media Equations

2007 ◽  
Vol 28 (1) ◽  
pp. 35-59 ◽  
Author(s):  
Jean Dolbeault ◽  
Ivan Gentil ◽  
Arnaud Guillin ◽  
Feng-Yu Wang
Keyword(s):  
Porous Media ◽  
Functional Inequalities ◽  
Porous Media Equations

scholarly journals ORLICZ–POINCARÉ INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 529-543 ◽  
Author(s):  
Feng-Yu Wang
Keyword(s):  
Porous Media ◽  
Probability Measure ◽  
Poincaré Inequality ◽  
Dirichlet Forms ◽  
Sobolev Inequalities ◽  
Young Function ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Porous Media Equations

AbstractCorresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

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