Lp-solvability and Holder 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:,partial derivative(alpha)(t)u = a(ij)u(x)i(x)j + b(i)u(x)i + cu + (b) over tilde (-i)uu(x)i+partial derivative(beta)(t)integral(t)(0)sigma(u)dW(t), t > 0; u(0, center dot) = u0,,where alpha is an element of (0, 1), beta < 3 alpha/4+1/2, and d < 4 - -2(2 beta-1)+/alpha. The operators partial derivative(alpha)(t) and partial derivative(beta)(t) are the Caputo fractional derivatives of order alpha and beta, respectively. The process W-t is an L-2(R-d)-valued cylindrical Wiener process, and the coefficients a(ij), b(i), c, (b) over tilde (-i) and sigma(u)are random. In addition to the uniqueness and existence of a solution, the Holder regularity of the solution is also established. For example, for any constant T < infinity, small epsilon > 0, and almost sure omega is an element of Omega,,sup(x is an element of Rd)(vertical bar u(omega,center dot,x& #4