Electronic Theses and Dissertations

Date

2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Hongqiu Chen

Committee Member

Irena Lasiecka

Committee Member

Mate Wierdl

Committee Member

Roberto Triggiani

Abstract

The purpose of this dissertation is to complete a study of Benjamin Bona Mahony (BBM) type equations. These are non-linear dispersive partial differential equations. The original BBM equation, also known as the regularized long-wave equation, was presented as an alternative to the Korteweg de Vries equation, and it has wide-ranging applications, including water waves and plasma physics. The original BBM equation is well studied. After an introduction of background materials and an explanation of notation, I begin by considering the modified BBM equation; here the nonlinear term is cubic. When the initial data are in the proper Sobolev space, the local well-posedness of the corresponding initial value problem follows from the Banach Fixed Point Theorem. To extend the solution globally, two important invariants of the initial value problem are applied to gain a priori bounds on the norm of the solution. To lower the regularity of the initial data, a new Banach space, with the maximum norm is introduced. This lower regularity case is a new result. Next, I turn my attention to the Regularized Gardner Equation. This equation combines the non-linearity of the original BBM equation with that of its modified form. Similar, but slightly more complicated, computations are performed to show the same results. Then I examine the Extended Benjamin Bona Mahony Equation (EBBM). This equation has two non-linear terms, one of even degree and another of larger and odd degree. To reduce the regularity of the initial data below a certain threshold, the constant coefficients in the equation must be constrained; then to lower it even further, again, a new Banach space endowed with the maximum norm is introduced. The EBBM equation can be considered as a particular case of the generalized BBM (GBBM) type equation. The nonlinearity of the GBBM equation is expressed as the derivative of a polynomial of degree n, greater than or equal to 3. I end my study by proving that the GBBM type equation is globally well-posed, under certain conditions.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

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