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: $$u_t=a{u}_{xx}+b{u}_x+cu+\bar{b}|u{|}^{\lambda }{u}_x+\sigma (u)\dot{W},(t,x)\in (0,\infty )\times \mathbb{R};u(0,\cdot )=u_0,$$ 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 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 $\epsilon >0$, $$u\in C_{t,x}^{1/4-\epsilon ,1/2-\epsilon }([0,T]\times \mathbb{R})(a.s.).$$ 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 $\epsilon >0$, $$u\in C_{t,x}^{\frac{1/2-(\lambda -1/2)\vee {\lambda }_0}{2}-\epsilon ,1/2-(\lambda -1/2)\vee {\lambda }_0-\epsilon }([0,T]\times \mathbb{R})(a.s.).$$ 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 ${\lambda }_0.$

arXiv comment33 pages. arXiv admin note: text overlap with arXiv:2111.03302