Graduate Studies

 

First Advisor

Adam Larios

Department

Mathematics

Date of this Version

Spring 2024

Document Type

Dissertation

Comments

Copyright 2024, Matthew Enlow. Used by permission

Abstract

We perform an analytical and computational investigation on the effectiveness of a locally bounded truncation function, which we call a calming function, when applied to the nonlinear terms of several dissipative partial differential equations. In particular, the 3D Navier-Stokes equations of incompressible fluid flow, the 2D Kuramoto-Sivashinsky equations of laminar flame fronts, and the 2D MHD-Boussinesq equations of magnetohydrodynamics. Each of these equations have open questions about the global existence and uniqueness of their solutions. These calming functions effectively reduce the algebraic degree of select nonlinear terms, thus one can verify global wellposedness for these "calmed systems." More specifically, in this work we show analytically in this work that the solutions to the calmed systems are globally well-posed, have higher-order regularity, and converge to solutions of the original models on short-time intervals as an introduced parameter in the calmed system tends to 0. We obtain additional results in the case of the 3D Calmed Navier-Stokes equations: when applying calming to the nonlinear term written in its rotational form, we find that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, for calmed Navier-Stokes written either in its advective form or rotational form, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the short-time existence of strong solutions to the 3D Navier-Stokes equations.

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