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Mathematical Analysis for Nonlocal Nonlinear Diffusion Models

Hayley Anne Olson, University of Nebraska - Lincoln

Abstract

Nonlinear nonlocal diffusion models arise at the intersection of nonlinear diffusion -- when the diffusivity coefficient is state dependent -- and nonlocal models -- when the operators governing the equations are chosen to have integral, rather than differential, formulations. Nonlinear diffusion models have applications in a wide variety of phenomena, such as temperature or concentration dependent diffusion of materials, liquid movement in porous material, the radiation of heat waves. However, classical diffusion models require a high level of regularity for solutions, namely twice differentiability. Nonlocal models provide the ability to deal with rougher information where solutions are not necessarily continuous nor differentiable. In this dissertation we show that the actions of the nonlocal operators converge to the actions of the classical differential operators, as the horizon $\delta$ of nonlocal interactions (determined by the kernel) shrinks to zero. We also show nonlocal solutions to such systems are close approximations to classical solutions. In fact, two proofs are provided for the convergence of the nonlocal solutions to boundary value problems with Dirichlet-type data: one with lower regularity requirements on the nonlinearity $f$ (allowing a large class of Lipschitz functions) and one which shows a linear rate of convergence in $\delta$, but at a cost of imposing higher regularity assumptions on $f$. To overcome the lack of compactness in the nonlocal framework (given in the classical case by the Rellich-Kondrachov Theorem), the results require bounds on the Lipschitz nonlinearity, as well as on the solutions to the classical system. With these proofs, we provide the first convergence results for nonlocal solutions of nonlinear systems with state-based diffusion coefficients.

Subject Area

Mathematics|Theoretical Mathematics|Applied Mathematics

Recommended Citation

Olson, Hayley Anne, "Mathematical Analysis for Nonlocal Nonlinear Diffusion Models" (2022). ETD collection for University of Nebraska - Lincoln. AAI29168027.
https://digitalcommons.unl.edu/dissertations/AAI29168027

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