We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for $σ$-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called $N$-functions in the theory of Orlicz spaces.
A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.
The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.
In this paper we state the variational principle for the weighted porous media equation. It extends V.I. Arnold's approach to the description of Euler flows as a geodesics on some manifold, i.e. as a critical points of some energy functional.
One proves that the stochastic porous media equation in 3-D has a unique nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal O)$ if the nonlinearity is monotone and has polynomial growth.
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at infinity.
In this paper, we establish the Freidlin-Wentzell type large deviation principles for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $Δ(|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.
We prove new $L^2$-estimates and regularity results for generalized porous media equations "shifted by" a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of "$ζ$-monotonicity" for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.
Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.
Using the generalized variational framework, the strong/weak existence and uniqueness of solutions are derived for a class of distribution dependent stochastic porous media equations on general measure spaces, which also extends the classical well-posedness result of quasilinear SPDE to the distribution dependent case.
The purpose of this paper consists in proposing a generalized solution for a porous media type equation on a half-line with Neumann boundary condition and prove a probabilistic representation of this solution in terms of an associated microscopic diffusion. The main idea is to construct a stochastic differential equation with reflection which has a solution in law and whose marginal law densities provide the unique solution of the porous media type equation.
The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way the existence and uniqueness of entropy solutions is shown.
The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.
The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space R^1 perturbed by a multiplicative coloured noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion.
A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,ν)$ where $ν$ is an infinitesimally excessive measure. Then a weak solution is constructed.
The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.
We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution.
We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.
Stochastic Porous Media Equations With Nonhomogeneous Dirichlet Boundary Conditions
École d’actuariat, Université Laval（拉瓦尔大学精算学院）；School of Mathematics and Statistics, Shandong University（山东大学数学与统计学院）；Normandie University, INSA de Rouen Normandie, LMI (EA 3226-FR CNRS 3335)（诺曼底大学鲁昂国立应用科学学院）；Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）
By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the $L^p$-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.
We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by square integrable martingales with stationary independent increments.
We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.
The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising in self-organised criticality. Based on the so-called lower-bound method, the existence and uniqueness of an invariant measure is proved.
The paper deals with homogenization of a model problem describing an immiscible compressible two-phase flow in random statistically homogeneous porous media. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments.
We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $Δ(|u|^{m-1}u)$ for all $m\in(1,\infty)$, and Hölder continuous diffusion nonlinearity with exponent $1/2$.
We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised 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 analysed in detail.
We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding random attractor has infinite fractal dimension.
The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states.
