Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Abstract

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$:

$$\begin{aligned} {\left\{ \begin{array}{ll} dX(t) = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T],\\ ~ X(0) = x\in H, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{aligned}$$

are measurable maps, $L_2(U,H)$ is the space of Hilbert–Schmidt operators from U to H and W is a U-cylindrical Wiener process. Such SPDEs include many interesting models in applied fields like fluid dynamics etc. In this paper, we establish the well-posedness of the above SPDEs under fully local monotonicity condition solving a longstanding open problem. The conditions on the diffusion coefficient $B(t,\cdot )$ are allowed to depend on both the H-norm and V-norm. In the case of classical SPDEs, this means that $B(\cdot ,\cdot )$ could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.