Maximal \(L^p\)-regularity for stochastic evolution equations (Q2902725)

From the authors' abstract: We prove maximal \(L^p\)-regularity for the stochastic evolution equation NEWLINE\[NEWLINE\begin{aligned} dU(t) + A U(t)\, dt &= F(t,U(t))\,dt + B(t,U(t))\,dW_H(t),\quad t\in [0,T],\\ U(0)& = u_0\end{aligned}NEWLINE\]NEWLINE under the assumption that \(A\) is a sectorial operator with a bounded \(H^\infty\) calculus of angle less than \(1/2\pi\) on a space \(L^q(O, \mu)\). We prove existence of a unique strong solution with trajectory in \(L^p\) space. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain \(O\) with \(d\) bigger than two. For the latter, the existence of a unique strong local solution with values in \(H^{1, q}(O)^d\) is shown.