$L_p$-solvability and Hölder 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: $${\partial }_t^{\alpha }u=a^{ij}{u}_{x^i{x}^j}+b^i{u}_{x^i}+cu+{\bar{b}}^{i}u{u}_{x^i}+{\partial }_t^{\beta }{\int }_0^t\sigma (u)d{W}_t,t>0;u(0,\cdot )=u_0,$$ 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 Hölder regularity of the solution. For example, for any constant $T<\infty$, small $\epsilon >0$, and almost sure $\omega \in \Omega$, we have $$\underset{x\in {\mathbb{R}}^d}{\sup }|u(\omega ,\cdot ,x){|}_{C^{[\frac{\alpha }{2}((2-(2\beta -1{)}_{+}/\alpha -d/2)\wedge 1)+\frac{(2\beta -1{)}_{-}}{2}]\wedge 1-\epsilon }([0,T])}<\infty \text{and}\underset{t\le T}{\sup }|u(\omega ,t,\cdot ){|}_{C^{(2-(2\beta -1{)}_{+}/\alpha -d/2)\wedge 1-\epsilon }({\mathbb{R}}^d)}<\infty .$$ The Hölder regularity of the solution in time changes behavior at $\beta =1/2$. Furthermore, if $\beta \ge 1/2$, then the Hölder regularity of the solution in time is $\alpha /2$ times the one in space.

arXiv comment41 pages, typo corrected, conditions on initial data is changed, results of the Hölder embedding theorems and the Hölder continuity of solutions are improved