论文标题
具有完全局部单调系数的随机部分微分方程的适合性
Well-posedness of stochastic partial differential equations with fully local monotone coefficients
论文作者
论文摘要
考虑在Gelfand Triple $ v \ subseteq h \ subseteq v^*$：\ begin {align*} \ left {allign \ lew {aligned} dx（aligned} dx（t）dx（t）＆= a（t）d t，t，t，x（t）d（t）d（t）d（t）d（t）d（t）d（t）d（t）d（t）d（t）d（t）d（ t \ in（0，t]，x（0）＆= x \ in h，\ end {parigned} \ right。可衡量的地图，$ L_2（U，H）$是Hilbert-Schmidt运营商的空间，从$ U $到$ h $，$ w $是$ u $ u $ cylindrical wiener过程$ b（t，\ cdot）$可以依赖于$ h $ norm和$ v $ norm，这意味着$ b（\ cdot，\ cdot）$也可以取决于解决方案的梯度。
Consider stochastic partial differential equations (SPDEs) with fully local monotone 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 \rightarrow L_2(U,H) \end{align*} 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.
