Paper detail

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

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: $$ u_t = au_{xx} + bu_{x} + cu + \bar b|u|^λu_{x} + σ(u)\dot W,\quad (t,x)\in(0,\infty)\times\mathbb{R}; \quad u(0,\cdot) = u_0, $$ where $λ> 0$. The function $σ(u)$ 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 $(ω,t,x)$, and $\bar b$ depends on $(ω,t)$. The coefficients $a,b,c,\bar{b}$ are uniformly bounded, and $a$ satisfies ellipticity condition. The random initial data $u_0 = u_0(ω,x)$ is nonnegative. We have the maximal Hölder regularity by employing the Hölder embedding theorem. For example, if $λ\in(0,1]$ and $σ(u)$ 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 $λ\in(0,1)$ and $σ(u) = |u|^{1+λ_0}$ with $λ_0\in[0,1/2)$, for $T<\infty$ and $\varepsilon>0$, $$u \in C^{\frac{1/2-(λ-1/2) \vee λ_0}{2} - \varepsilon,1/2-(λ-1/2) \vee λ_0 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad (a.s.).$$ It should be noted that if $σ(u)$ is bounded Lipschitz in $u$, the Hölder regularity of the solution is independent of $λ$. However, if $σ(u)$ is super-linear in $u$, the Hölder regularities of the solution are affected by nonlinearities, $λ$ and $λ_0.$