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

By Michael Röckner, Shijie Shang, Tusheng Zhang. Mathematische Annalen, 2022

52 citations recorded by the source corpus.

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple V⊆H⊆V∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\subseteq H \subseteq V^*$$\end{document}: dX(t)=A(t,X(t))dt+B(t,X(t))dW(t),t∈(0,T],X(0)=x∈H,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} dX(t) = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T],\\ ~ X(0) = x\in H, \end{array}\right. } \end{aligned}$$\end{document}where A:[0,T]×V→V∗,B:[0,T]×V→L2(U,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{aligned}$$\end{document}are measurable maps, L2(U,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(U,H)$$\end{document} 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,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(t,\cdot )$$\end{document} are allowed to depend on both the H-norm and V-norm. In the case of classical SPDEs, this means that B(·,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(\cdot ,\cdot )$$\end{document} could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/a72fc4c9188aefa3134926a57ca76363cac88dac