Random perturbations of nonlinear parabolic systems

TL;DR

This paper explores how random perturbations influence the regularity and singularity formation in nonlinear parabolic systems, extending known regularity results to stochastic cases and analyzing the effects of noise.

Contribution

It extends Kalita's regularity result to stochastic parabolic systems and investigates the impact of noise on singularity development in these systems.

Findings

Extended regularity results to stochastic systems

Expected value of solutions remains non-singular

Uncertainty about noise preventing singularities in nonlinear cases

Abstract

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions are in fact more regular, and examples of systems with weak solutions which develop singularities in finite time. Our main result is the extension of a regularity result due to Kalita to the stochastic case. Concerning the examples with singular solutions (outside the setting of Kalita's regularity result), we do not know whether stochastic noise may prevent the emergence of singularities, as it happens for easier PDEs. We can only prove that, for a linear stochastic parabolic system with coefficients outside the previous regularity theory, the expected value of the solution is not singular.