Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method

Abstract

The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong-Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brzeźniak and coworkers. Three-species Maxwell-Stefan systems and n-species biofilm models are examples that satisfy the general assumptions.

Global martingale solutions for a stochastic Shigesada-Kawasaki-Teramoto population model

“…The aim of this work is to prove the existence of global nonnegative martingale solutions to system (1). The paper is a continuation of our previous works [14,15]. The work [14] was concerned with a SKT-type system, in which the coefficients of the associated diffusion matrix depend quadratically on the densities (and not linearly as in (1)).…”

“…This allowed us to work in a Hilbert space framework, leading to a novel result through a standard approach. The paper [15] exploited the entropy structure of a general class of cross-diffusion systems with volume filling, leading to bounded martingale solutions. The L ∞ bound follows from the entropy structure and implies higher moment estimates, which were further used to establish the tightness of laws.…”

“…In the stochastic setting, we face some technical obstacles due to Itô's lemma and the treatment of the multiplicative noise. Our idea, first used in [15], is to replace the Wiener process by a Wong-Zakai approximation and to discretize the equations by a stochastic Galerkin method. We apply a variant of the boundedness-by-entropy method [21] to the resulting system of differential equations, providing the positivity of the approximate population densities and a priori estimates uniform in the Galerkin dimension.…”

“…The difference in the strategy of our proof, compared to [15], becomes apparent in the final steps. First, we need higher moment estimates, which followed in [15] from the L ∞ (O) bound, but here we have less regularity.…”

“…The difference in the strategy of our proof, compared to [15], becomes apparent in the final steps. First, we need higher moment estimates, which followed in [15] from the L ∞ (O) bound, but here we have less regularity. Second, we are proving a new fractional time regularity result, needed to show the tightness of the laws in L 2 (O).…”