Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone
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In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Bénard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress-free boundary conditions on the walls, and the fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data are error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity.
KeywordsBénard convection Boussinesq system Continuous data assimilation Signal synchronization Nudging Downscaling
Mathematics Subject Classification35Q30 93C20 37C50 76B75 34D06
The work of A.F. is supported in part by the NSF Grant DMS-1418911. The work of E.L. is supported in part by the ONR Grant N0001416WX01475 and the ONR Grant N0001416WX00796. The work of E.S.T. is supported in part by the ONR Grant N00014-15-1-2333 and the NSF Grants DMS-1109640 and DMS-1109645.
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