$L_p$-solvability and Hölder regularity for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise
英文摘要：
We present the $L_p$-solvability for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise: $$ \partial_t^αu = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^i u u_{x^i} + \partial_t^β\int_0^t σ(u)dW_t,\,t>0;\,\,u(0,\cdot) = u_0, $$ where $α\in(0,1)$, $β< 3α/4+1/2$, and $d< 4 - 2(2β-1)_+/α$. The operators $\partial_t^α$ and $\partial_t^β$ are the Caputo fractional derivatives of order $α$ and $β$, 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 $σ(u)$ 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 $\varepsilon>0$, and almost sure $ω\inΩ$, we have $$ \sup_{x\in\mathbb{R}^d}|u(ω,\cdot,x)|_{C^{[ \fracα{2}( ( 2-(2β-1)_+/α-d/2 )\wedge1 )+\frac{(2β-1)_{-}}{2} ]\wedge 1-\varepsilon}([0,T])}<\infty \quad\text{and}\quad \sup_{t\leq T}|u(ω,t,\cdot)|_{C^{( 2-(2β-1)_+/α-d/2 )\wedge1 - \varepsilon}(\mathbb{R}^d)} < \infty. $$ The Hölder regularity of the solution in time changes behavior at $β= 1/2$. Furthermore, if $β\geq1/2$, then the Hölder regularity of the solution in time is $α/2$ times the one in space.
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