$L_p$-solvability and H\"older regularity for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise

Abstract

We present the $L_p$-solvability for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise: $$ \partial_t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^i u u_{x^i} + \partial_t^\beta\int_0^t \sigma(u)dW_t,\,t>0;\,\,u(0,\cdot) = u_0, $$ where $\alpha\in(0,1)$, $\beta < 3\alpha/4+1/2$, and $d< 4 - 2(2\beta-1)_+/\alpha$. The operators $\partial_t^\alpha$ and $\partial_t^\beta$ are the Caputo fractional derivatives of order $\alpha$ and $\beta$, respectively. The process $W_t$ is an $L_2(\mathbb{R}^d)$-valued cylindrical Wiener process, and the coefficients $a^{ij}, b^i, c$ and $\sigma(u)$ are random. In addition to the existence and uniqueness of a solution, we also suggest the H\"older regularity of the solution. For example, for any constant $T<\infty$, small $\varepsilon>0$, and almost sure $\omega\in\Omega$, we have $$ \sup_{x\in\mathbb{R}^d}|u(\omega,\cdot,x)|_{C^{[ \frac{\alpha}{2}( ( 2-(2\beta-1)_+/\alpha-d/2 )\wedge1 )+\frac{(2\beta-1)_{-}}{2} ]\wedge 1-\varepsilon}([0,T])}<\infty \quad\text{and}\quad \sup_{t\leq T}|u(\omega,t,\cdot)|_{C^{( 2-(2\beta-1)_+/\alpha-d/2 )\wedge1 - \varepsilon}(\mathbb{R}^d)} < \infty. $$ The H\"older regularity of the solution in time changes behavior at $\beta = 1/2$. Furthermore, if $\beta\geq1/2$, then the H\"older regularity of the solution in time is $\alpha/2$ times the one in space.