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

TL;DR

This paper establishes the existence, uniqueness, and H"older regularity of solutions to stochastic time fractional Burgers' equations driven by multiplicative space-time white noise, extending understanding of such equations with fractional derivatives.

Contribution

It provides the first $L_p$-solvability results and detailed regularity properties for stochastic time fractional Burgers' equations with multiplicative noise.

Findings

Proves existence and uniqueness of solutions.

Derives H"older regularity in space and time.

Shows regularity behavior changes at $eta=1/2$.

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.