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

Abstract

We introduce the uniqueness, existence, $L_p$-regularity, and maximal Hölder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise:

$$\begin{aligned} u_t = au_{xx} + bu_{x} + cu + {{\bar{b}}}|u|^\lambda u_{x} + \sigma (u)\dot{W},\quad (t,x)\in (0,\infty )\times {\mathbb {R}}; \quad u(0,\cdot ) = u_0, \end{aligned}$$

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. To establish the $L_p$-regularity theory, we impose an algebraic condition on $\lambda$ depending on the nonlinearity of the diffusion coefficient $\sigma (u)$. For example, if $\sigma (u)$ has Lipschitz continuity, linear growth, and boundedness in u, $\lambda$ is assumed to be less than or equal to 1; $\lambda \in (0,1]$. However, if $\sigma (u) = |u|^{1+\lambda _0}$ with $\lambda _0\in [0,1/2)$, $\lambda$ is taken to be less than 1; $\lambda \in (0,1)$. Under those conditions, the uniqueness, existence, and regularity of the solution are obtained in stochastic $L_p$ spaces. Also, we have the maximal Hölder regularity by employing the Hölder 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 $\varepsilon >0$,

$$\begin{aligned} u \in C^{1/4 - \varepsilon ,1/2 - \varepsilon }_{t,x}([0,T]\times {\mathbb {R}})\quad (a.s.). \end{aligned}$$

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 $\varepsilon >0$,

$$\begin{aligned} u \in C^{\frac{1/2-(\lambda -1/2) \vee \lambda _0}{2} - \varepsilon ,1/2-(\lambda -1/2) \vee \lambda _0 - \varepsilon }_{t,x}([0,T]\times {\mathbb {R}})\quad (a.s.). \end{aligned}$$

It should be noted that if $\sigma (u)$ is bounded Lipschitz in u, the Hölder regularity of the solution is independent of $\lambda$. However, if $\sigma (u)$ is super-linear in u, the Hölder regularities of the solution are affected by nonlinearities, $\lambda$ and $\lambda _0.$