Type of Document	Dissertation
Author	Yin, Hong
Author's Email Address	yinhong@math.lsu.edu
URN	etd-04122007-145924
Title	Backward Stochastic Navier-Stokes Equations in Two Dimensions
Degree	Doctor of Philosophy (Ph.D.)
Department	Mathematics
Advisory Committee	Advisor Name Title Padmanabhan Sundar Committee Chair Ambar Sengupta Committee Member Charles Neal Delzell Committee Member Jimmie Lawson Committee Member W. George Cochran Committee Member Anitra WIlson Dean's Representative
Keywords	Backward stochastic Navier-Stokes equations Lorenz system Gronwall inequality Truncated system Galerkin approximation
Date of Defense	2007-04-09
Availability	unrestricted
Abstract There are two parts in this dissertation. The backward stochastic Lorenz system is studied in the first part. Suitable a priori estimates for adapted solutions of the backward stochastic Lorenz system are obtained. The existence and uniqueness of solutions is shown by the use of suitable truncations and approximations. The continuity of the adapted solutions with respect to the terminal data is also established. The backward stochastic Navier-Stokes equations (BSNSEs, for short) corresponding to incompressible fluid flow in a bounded domain $G$ are studied in the second part. Suitable a priori estimates for adapted solutions of the BSNSEs are obtained which reveal a surprising pathwise $L^{infty}(H)$ bound on the solutions. The existence of solutions is shown by using a monotonicity argument. Uniqueness is proved by using a novel method that uses finite-dimensional projections, linearization, and truncations. The continuity of the adapted solutions with respect to the terminal data and the external body force is also established.
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