PUB - Publikationen an der Universität Bielefeld

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelf and triple V subset of H subset of V-& lowast;: { dX(t) = A(t, X(t))dt + B(t, X(t))dW (t), t is an element of (0, T ], { X (0) = x is an element of H, where A : [0, T] x V -> V-& lowast;, B:[0, T] x V -> L-2(U, H) 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,& sdot;) are allowed to depend on both the H-norm and V-norm. In the case of classical SPDEs, this means that B(& sdot;,& sdot;) could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.