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Consider stochastic partial differential equations (SPDEs) with fully localmonotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$:\begin{align*} \left\{ \begin{aligned}  dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], \\  X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*}  A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrowL_2(U,H) \end{align*} are measurable maps, $L_2(U,H)$ is the space ofHilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wienerprocess. Such SPDEs include many interesting models in applied fields likefluid dynamics etc. In this paper, we establish the well-posedness of the aboveSPDEs under fully local monotonicity condition solving a longstanding openproblem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowedto 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 thesolution. The well-posedness is obtained through a combination ofpseudo-monotonicity techniques and compactness arguments.

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