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Well -posedness theory of a one parameter family of coupled KdV-type systems and their invariant Gibbs measures
This thesis concentrates on the study of a conserved Hamiltonian system of nonlinear dispersive partial differential equations. The system consists of two equations of Korteweg-de Vries (KdV) type called the Reduced Equations for Equatorial Baroclinic Barotropic Waves. It was proposed by Majda and Biello as a model to study the nonlinear resonant interaction of barotropic Rossby waves and baroclinic Rossby waves in atmospheric sciences. We first discuss the existence of the local in time solutions of the system in both periodic and non-periodic settings with a varying coupling parameter. In the periodic setting, we use number-theoretic idea to characterize our result. Then, by the use of modified functionals that are closely related to the original functionals preserved under the flow, we show how to iterate the local in time results and prove the existence of the solutions for all time. Lastly, we discuss the existence of the Gibbs measures that are invariant under the flow and use it to prove the existence of the global in time solutions on the statistical ensemble of the initial data.
Oh, Choonghong (Tadahiro), "Well -posedness theory of a one parameter family of coupled KdV-type systems and their invariant Gibbs measures" (2007). Doctoral Dissertations Available from Proquest. AAI3289268.