Entropy solutions for stochastic porous media equations

Abstract

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.

Nonlinear diffusion equations with nonlinear gradient noise

“…There, we claim that (4.9) holds with α = 1. This follows if one reproduces the proof of [DGG18, Theorem 4.1, (4.8) and (4.18) therein] (with m = 1) with only one difference: In order to estimate the term D 1 (see [DGG18,(4.13)]), one uses that sup n sup r |a ′ n (r)| < ∞ to obtain the estimate |D 1 | δ 2 |u −ũ|, ≤εN E ∂ x [a n ](u) 2 L 2 (Q T ) + E (a n (u)) −1 2…”

“…in place of[DGG18, (4.16)]. Proceeding then as in the proof of Theorem 4.1 one obtains (4.36) with m = 1,κ = 1, and β = 1.…”

“…In particular, this generalization allows the application to the stochastic mean curvature flow. The proof of stability relies on entropy techniques and a careful control of the errors arising in the corresponding doubling the variables argument which was initiated in [DGG18] and is disjoint from the kinetic techniques put forward in [FG18b].…”

Ergodicity for Stochastic Porous Media Equations with Multiplicative Noise

“…x (which is rarely the case; see, for example, the discussion in [DGG19]), it is unclear how to prove stability without a smallness assumption on the Lipschitz constant of the coefficients \sigma k .…”

“…In the case of periodic boundary data, the existence of such a weight or an appropriate replacement is not clear. Second, we extend the framework of entropy solutions in [DGG19] where the periodic case was considered. Downloaded 09/30/20 to 129.11.76.232.…”