$$L_p$$ -solvability and Hölder regularity for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise

Abstract

This paper establishes \(L_p\)-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise:

$$\begin{aligned} \partial _t^\alpha u = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + {\bar{b}}^i u u_{x^i} + \partial _t^\beta \int _0^t \sigma (u)dW_t,\quad t>0;\quad u(0,\cdot ) = u_0, \end{aligned}$$

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 \(a^{ij}, b^i, c, {\bar{b}}^{i}\) and \(\sigma (u)\) are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant \(T<\infty \), small \(\varepsilon >0\), and almost sure \(\omega \in \varOmega \),

$$\begin{aligned} \sup _{x\in \mathbb {R}^d}|u(\omega ,\cdot ,x)|_{C^{\left[ \frac{\alpha }{2}\left( \left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 \right) +\frac{(2\beta -1)_{-}}{2} \right] \wedge 1-\varepsilon }([0,T])}<\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{t\le T}|u(\omega ,t,\cdot )|_{C^{\left( 2-(2\beta -1)_+/\alpha -d/2 \right) \wedge 1 - \varepsilon }(\mathbb {R}^d)} < \infty . \end{aligned}$$

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 that in space.