Shouhong Wang
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SPECIAL GFDI-MATHEMATICS COLLOQUIUM
Speaker: Shouhong Wang Abstract. In the first part of the talk, I shall present a dynamic transition theory, developed by Tian Ma and myself. The key philosophy of the theory is to search for the full set of transition states, attempting to derive a complete characterization on stability and transition. One important part of the theory is to establish a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic, and random. The theory is then applied to study the transitions associated with thermohaline circulations. In the second part of the talk, we shall also introduces a novel approach to deal with the parameterization problem of the "small" spatial scales by the "large" ones for stochastic partial differential equations. This approach relies on stochastic parameterizing manifolds (PMs) which are random manifolds aiming to provide-in a mean square sense-approximate parameterizations of the small scales by the large ones. Backward-forward systems will be introduced to give access to such PMs as pullback limits depending-through the nonlinear terms-on the time-history of the dynamics of the low modes. It will be shown that the corresponding pullback limits can be efficiently determined in practice, leading in turn to an operational procedure for the derivation of non-Markovian reduced equations able to achieve good modeling performances. The talk is based mainly on joint work with Tian Ma, Mickael Chekroun and Honghu Liu. |