Electronic Theses and Dissertations

Madhumita Roy

Date

2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Irena Lasiecka

Committee Member

Irena Lasiecka

Committee Member

Roberto Triggiani

Committee Member

Mate Wierdl

Committee Member

Bentuo Zheng

Abstract

Long time behavior of a class of nonlinear hyperbolic dynamics governed by Partial Differential Equations [PDEs] is considered in this thesis. The objective is to demonstrate that the long-term behavior can be directed towards coherent structures, such as attractors. Attractors are invariant and compact sets in the phase space that attract all trajectories. The key aspects of the dynamics that allow for these descriptions are: (i) a level of dissipation within the system and (ii) the asymptotic smoothness and regularity [compactness] of the trajectories. In this context, hyperbolic dynamics pose a notable challenge as they inherently lack both of these properties [unlike parabolic dynamics]. The typical behavior of the spectrum of a hyperbolic semigroup is characterized by the presence of infinitely many elements on the imaginary axis, indicating that instability is inherently infinite-dimensional. Regarding regularity, it is well known that singularities propagate over time, making asymptotic smoothness an unnatural phenomenon to expect. In contrast to parabolic dynamics, where solutions smooth out rapidly, here our challenge and goal is to show that the needed properties can be “unlocked” also within hyperbolic framework for some of the fundamental benchmark PDE models such as: semilinear wave equation and nonlinear plate equation [the latter being associated to Schrodinger dynamics]. In particular, we will investigate (1) a model illustrating wave propagation under the influence of external forces with only boundary dissipation, and (2) a fourth-order spatial model representing a nonlinear plate equation. The dynamics of these equations are influenced by significant external disturbances that induce undesirable oscillations, akin to the case of the Tacoma Narrows Bridge. The three fundamental behaviors that a dynamical system exhibits after a long time in presence of some external disturbance are blow-up, global attractor or decay of energy of solutions to some stationary points [stability]. The study focuses on three fundamental behaviors that emerge in dynamical systems over extended periods when subjected to external disturbances and restricted dissipative effects wit the goal to steer the dynamics to a desired coherent structure. In the first part, we study a long-time dynamics for a nonlinearly forced 3-D wave equation subject to a nonlinear boundary dissipation. Nonlinear forces are of critical ex- ponent and supported both in the interior and on the boundary. By critical we mean with respect to critical 3-D Sobolev’s embeddings, so that the forces lead to non-compact trajectories- even with zero initial conditions (at time t = 0). Our aim is to demonstrate that dissipation solely from the boundary can propagate over time throughout the entire domain, leading to the eventual regularization of unstable trajectories as time progresses. We achieve this through the utilization of the “multipliers technique” for propagation, as outlined in the paper [LT93], and compensated compactness methods described in the memoir [CL08], ensuring the ultimate compactness [via smoothness] of trajectories within the attractor. While global attractors for wave equations with geometrically restricted damping and critical interior forcing have been known, a treatment of critical nonlinearity supported on the boundary, within the context of theory of global attractors, was an open question. The method devised for solution to the problem relies on “hidden trace regularity” established for Neumann hyperbolic problems and new tangential boundary trace estimates which do not follow from classical PDE trace theories and as such they are also of independent interest. iii In the second part, we address the long-term evolution of solutions governed by a nonlinear plate model, which captures the dynamics of a suspension bridge under mixed boundary conditions affected by wind forces. The model draws inspiration from the Tacoma Narrows Bridge, famously collapsed in 1940. Here, the dissipation within the dynamics is nonlinear and located within the plate’s interior. The prominent characteristic is instability provoked by external, non-dissipative [non-conservative] sources. Once again, our objective is to demonstrate that the long-term behavior converges to an attracting set, which additionally exhibits finite dimensionality. Initially, we establish the system as a well-posed dynamical system within the framework of nonlinear semigroups. The existence of a global attractor is substantiated by constructing an absorbing set, followed by demonstrating the compensated compactness property of asymptotic and finite energy solutions. In cases where the dissipation is non-degenerate [bounded linearly from below at the origin], we showcase a stronger quasi-stability property of the system, leading to a finite dimensional structure of the attracting set. However, the presence of an un- structured and non-dissipative external force hampers the system’s “gradient” property, challenging the existence of an absorbing set. While in instances of linear damping, this challenge can be managed through the construction of a suitable Lyapunov function, as described in the paper [Bon+22], the nonlinearity of the damping presents a significant obstacle in extending previous arguments. This method, crucial for resolving the current problem, also holds significance independently within the theory of dynamical systems.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Embargoed until 01-24-2025

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Available for download on Friday, January 24, 2025

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