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Full-Text Articles in Physical Sciences and Mathematics
Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin
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
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
Asymptotic Stability Of A Fluid-Structure Semigroup, George Avalos
Asymptotic Stability Of A Fluid-Structure Semigroup, George Avalos
Department of Mathematics: Faculty Publications
The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. It is now known that this PDE may be modeled by a $C_{0}$-semigroup of contractions on an appropriate Hilbert space. However, because of the nature of the unbounded coupling between fluid and structure, the resolvent of the semigroup generator will \emph{not} be a compact operator. In consequence, the classical solution to the stability problem, by means …