Maximal $L^p$-regularity for stochastic evolution equations

We prove maximal $L^p$-regularity for the stochastic evolution equation \{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}. under the assumption that $A$ is a sectorial operator with a bounded $H^{\mathrm{\infty }}$-calculus of angle less than $\frac{1}{2}\pi$ on a space $L^q(\mathcal{O},\mu )$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\mathrm{\infty })$ and $q\in [2,\mathrm{\infty })$ and initial conditions $u_0$ in the real interpolation space $\text{XAp}$ we prove existence of unique strong solution with trajectories in $$L^p(0,T;\text{Dom}(A))\cap C([0,T];\text{XAp}),$$ provided the non-linearities $F:[0,T]\times \text{Dom}(A)\to L^q(\mathcal{O},\mu )$ and $B:[0,T]\times \Dom(A) \to \g(H,\Dom(A^{\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. 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 $\text{OO}\subseteq {\mathbb{R}}^d$ with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\text{OO}){)}^d$ is shown.