Physical Sciences and Mathematics Commons^™

Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ …

Regularity For Solutions To Parabolic Systems And Nonlocal Minimization Problems, Joe Geisbauer

Department of Mathematics: Dissertations, Theses, and Student Research

The goal of this dissertation is to contribute to both the nonlocal and local settings of regularity within the calculus of variations. We provide analogues of higher differentiability results in the context of Besov spaces for minimizers of nonlocal functionals. We also establish the Holder continuity of solutions to a system of parabolic partial differential equations.

Advisor: Mikil Foss