Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one

TL;DR

This paper proves existence and uniqueness of solutions for a stochastic nonlinear heat equation with an $L^2$-norm constraint, employing a modified Galerkin scheme and novel Itô formula techniques.

Contribution

It introduces a new proof of an Itô formula for the $L^p$-norm and establishes well-posedness for a constrained stochastic heat equation in arbitrary dimensions.

Findings

Existence of martingale solutions in bounded smooth domains.

Pathwise uniqueness and strong solution existence via Yamada-Watanabe theorem.

Novel Itô formula for the $L^p$-norm of solutions.

Abstract

In this work, we investigate the well-posedness of a stochastic heat equation with an arbitrary (but polynomial) nonlinearity in any dimension $d\geq 1$ perturbed by a multiplicative white noise in the Stratonovich form, subject to an $L^2-$norm constraint on the solution. In bounded smooth domains, we establish the existence of a martingale solution taking values in $H_0^1 \cap L^p$ for arbitrary $2 \le p < \infty$, using a modified Faedo-Galerkin scheme. By utilizing a sequence of self-adjoint operators which are bounded in $L^p$ for any $2 \le p < \infty$, we provide a novel proof of an It\^o formula for the $L^p-$norm of the solution. Together with pathwise uniqueness of the martingale solution, the Yamada-Watanabe result then yields the existence of a strong solution and uniqueness in law.