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 $$\lambda _0.$$