Electronic Theses and Dissertations

Date

2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Dr. Irena Lasiecka

Committee Chair

Dr. Hongqiu Chen

Committee Member

Dr. Roberto Triggiani

Committee Member

Pei-Kee Lin

Committee Member

Dr. Irena Lasiecka

Committee Member

Dr. Hongqiu Chen

Abstract

The thesis consists of two Parts. Part I delves into the well-posedness and stability of the Von Karman plate equations with rota-tional inertia, which are critical for understanding the nonlinear oscillations of thin elastic plates. Von Karman nonlinearity, described via Airy’s stress function, provides a good description of large amplitude oscillations. The research develops a rigorous mathematical framework to analyze these equations, proving the existence and uniqueness of weak and strong solutions and exploring their continuous dependence on initial data. The goal undertaken in this thesis is to provide a suit- able framework for stabilization of solutions. This is achieved by introducing damping mechanism which affects the velocity of the displacement as well as the rotational inertia terms. The damping is nonlinear and of monotone type. The study conducts a detailed stability analysis using energy and multipliers methods, compensated compactness, and sharp regularity of Airy’s stress func- tion [55]. The obtained results [estimates] ensure uniform stabilization of the thin-plate subjected to nonlinear dissipation acting on both: the velocity of the displacement and the inertial terms. Stabilization results are quantified by suitable energy decay estimates. The latter are dependent on the ”strength” of the damping and can be exponential, algebraic or logarithmic. It will be shown that the stability results with the corresponding decay rates are uniform with respect to the rotational inertia -hence thickness of the plate. This provides the corresponding result for the limit problem [model without the rotational terms]. Additionally, numerical methods are used to approximate the spectrum of the corresponding linear generators. This provides practical insights into the asymptotic behavior and stability of the system varying with respect to the parameters describing rotational inertia. An interesting observation is the fact that the velocity damping alone would lead to strong [rather than uniform] stability in a rotational model. Thus stabilization of rotational model requires dissipative elements acting also on rotational inertia. If the same rota- tional dissipation is applied to non-rotational model, the dynamics becomes of parabolic type such as seen in structurally damped plates [60]. Part II deals with regularized long wave equation subject to non-homogeneous Robin boundary condition. The main result is local wellposedness of solutions in the spaces C([0, T ], C1 b (R+)) and C([0, T ], H1(R+)). While it has been known that global well posedness does hold for the case of non-homogeneous Dirichlet boundary data, the case of non-homogeneous Robin boundary condi- tions is way more subtle and has been opened in the literature. The thesis provides an affirmative answer in the case of local [short time] well-posedness.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Embargoed until 10-30-2026

Available for download on Friday, October 30, 2026

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