Stochastic Porous Media Equations Research Articles

Poisson stable solutions for stochastic PDEs driven by Lévy noise

This paper is mainly concerned with the existence, uniqueness and Poisson stability (including stationarity, periodicity, almost periodicity and almost automorphy) of solutions for a class of stochastic partial differential equations driven by Lévy noise, where the involved coefficients are assumed to be strictly monotone. Based on the variational method, we establish the well-posedness of L2-bounded solution and then prove that it has the same characters of periodicity, almost periodicity and almost automorphy as the coefficients of the equation. Moreover, we also investigate the additive noise case under strong monotone condition. In particular, we illustrate our results by applying to concrete models such as stochastic reaction-diffusion equations, stochastic porous media equations and stochastic p-Laplace equations.

The obstacle problem for stochastic porous media equations

We study an obstacle problem for stochastic porous media equations, and show that it has a unique entropy solution with a method of penalty.

Stochastic Integral Evolution Equations with Locally Monotone and Non-Lipschitz Coefficients

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms.

On state‐constrained porous‐media systems with gradient‐type multiplicative noise

AbstractThe aim of the present paper is to provide necessary and sufficient conditions to maintain a stochastic coupled system with porous media components and gradient‐type noise in a prescribed set of constraints by using internal controls. This work is a complementary contribution to the results obtained by the same authors, also on the viability problem associated to the porous media equation, but with Lipschitz noise. Second, the present paper provides a different framework in which the quasi‐tangency condition can be obtained with optimal speed. In comparison with the aforementioned result, and from a technical point of view, here, we transform the stochastic system into a random‐PDE one, via the rescaling approach, and then we study the viability of random sets. As an application, (stronger) conditions for the stabilization of the stochastic porous media equations are obtained. These are illustrated on a simple example.

Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

Distribution-dependent stochastic porous media equations

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.

Optimal regularity in time and space for stochastic porous medium equations

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be H\"older continuous and the cases of smooth coefficients of at most linear growth as well as $\sqrt{u}$ are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor 2020] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual $L^1$ context to the natural $L^2$ setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use $L^2$ based a priori bounds to treat the stochastic term.

Strong convergence rate of the averaging principle for a class of slow–fast stochastic evolution equations

ABSTRACT We prove a strong convergence rate of the averaging principle for general two-time-scales stochastic evolution equations driven by cylindrical Wiener processes. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic reaction–diffusion equations, stochastic p-Laplace equations, stochastic porous media equations, and so on.

Weak pullback mean random attractors for stochastic evolution equations and applications

In this paper, we investigate the existence and uniqueness of weak pullback mean random attractors for abstract stochastic evolution equations with general diffusion terms in Bochner spaces. As applications, the existence and uniqueness of weak pullback mean random attractors for some stochastic models such as stochastic reaction–diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations are established.

Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities

We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Lévy noise on a \(\sigma \)-finite measure space \((E,{\mathcal {B}}(E),\mu )\), and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient \(\Psi \) is only assumed to satisfy the increasing Lipschitz nonlinearity assumption, without the restriction \(r\Psi (r)\rightarrow \infty \) as \(r\rightarrow \infty \) for \(L^2(\mu )\)-initial data. We also extend the state space, which avoids the transience assumption on L or the boundedness of \(L^{-1}\) in \(L^{r+1}(E,{\mathcal {B}}(E),\mu )\) for some \(r\ge 1\). Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e., \(L=-(-\Delta )^\alpha ,\ \alpha \in (0,1]\), generalized \(\mathrm Schr\ddot{o}dinger\) operators, i.e., \(L=\Delta +2\frac{\nabla \rho }{\rho }\cdot \nabla \), and Laplacians on fractals.

Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate smash {O(t^frac{1}{2})} whereas the support of solutions to the deterministic PME grows only with rate smash {O(t^{frac{1}{m{+}1}})}. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

The long time behaviour of solutions to generalized stochastic porous media equations on bounded intervals with zero Dirichlet boundary conditions is studied. We focus on a degenerate form of nonlinearity arising in self-organized criticality. Based on the so-called lower bound technique, the existence and uniqueness of an invariant measure is proved.

Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

<p style='text-indent:20px;'>In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.</p>

On stochastic porous-medium equations with critical-growth conservative multiplicative noise

<p style='text-indent:20px;'>First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.

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

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.

Large deviations for stochastic porous media equation on general measure spaces

In this paper, we establish a large deviation principle for stochastic porous media equations driven by time-dependent multiplicative noise on a σ-finite measure space (E,B(E),μ), and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient Ψ is only assumed to satisfy the increasing Lipschitz nonlinearity assumption without the restriction rΨ(r)→∞ as r→∞ for L2(μ)-initial data. This paper also gets rid of the compact embedding assumption on the associated Gelfand triple. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e. L=−(−Δ)α,α∈(0,1], generalized Schro¨dinger operators, i.e. L=Δ+2∇ρρ⋅∇, and Laplacians on fractals.

Large deviation principle of occupation measures for non-linear monotone SPDEs

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises.

Ergodicity for Stochastic Porous Media Equations with Multiplicative Noise

The long time behavior of solutions to stochastic porous media equations with nonlinear multiplicative noise on bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-es...

Small time asymptotics for SPDEs with locally monotone coefficients

<p style='text-indent:20px;'>This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

Nonlinear diffusion equations with nonlinear gradient noise

We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.