A regularity theory for stochastic generalized Burgers' equation driven by a multiplicative space-time white noise

TL;DR

This paper establishes the existence, uniqueness, and regularity properties of solutions to a stochastic Burgers' equation driven by space-time white noise, revealing how nonlinearities and noise influence solution regularity.

Contribution

It provides a comprehensive regularity theory for stochastic generalized Burgers' equations with multiplicative noise, including maximal Hölder regularity results under various nonlinearities.

Findings

Solutions exist uniquely with specified regularity properties.

Hölder regularity depends on the nonlinearity parameter and noise growth.

Regularity results are robust under different noise and nonlinearity conditions.

Abstract

We introduce the uniqueness, existence, $L_p$-regularity, and maximal H\"older regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + \bar b|u|^\lambda u_{x} + \sigma(u)\dot W,\quad (t,x)\in(0,\infty)\times\mathbb{R}; \quad u(0,\cdot) = u_0, $$ where $\lambda > 0$. The function $\sigma(u)$ is either bounded Lipschitz or super-linear in $u$. The noise $\dot W$ is a space-time white noise. The coefficients $a,b,c$ depend on $(\omega,t,x)$, and $\bar b$ depends on $(\omega,t)$. The coefficients $a,b,c,\bar{b}$ are uniformly bounded, and $a$ satisfies ellipticity condition. The random initial data $u_0 = u_0(\omega,x)$ is nonnegative. We have the maximal H\"older regularity by employing the H\"older embedding theorem. For example, if $\lambda \in(0,1]$ and $\sigma(u)$ has Lipschitz continuity, linear growth, and boundedness in $u$, for $T<\infty$ and $\varepsilon>0$, $$u \in C^{1/4 - \varepsilon,1/2 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad(a.s.). $$ On the other hand, if $\lambda\in(0,1)$ and $\sigma(u) = |u|^{1+\lambda_0}$ with $\lambda_0\in[0,1/2)$, for $T<\infty$ and $\varepsilon>0$, $$u \in C^{\frac{1/2-(\lambda -1/2) \vee \lambda_0}{2} - \varepsilon,1/2-(\lambda -1/2) \vee \lambda_0 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad (a.s.).$$ It should be noted that if $\sigma(u)$ is bounded Lipschitz in $u$, the H\"older regularity of the solution is independent of $\lambda$. However, if $\sigma(u)$ is super-linear in $u$, the H\"older regularities of the solution are affected by nonlinearities, $\lambda$ and $\lambda_0.$