Dissertations, Theses, and Capstone Projects

Date of Degree

5-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Jesenko Vukadinovic

Committee Members

Olympia Hadjiliadis

Zeno Huang

Tobias Schafer

Subject Categories

Analysis | Fluid Dynamics | Partial Differential Equations

Keywords

Advection, Diffusion, Dissipation

Abstract

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices representing critical points or connected components of the boundary. A transformation to action-angle coordinates allows for angle-averaging, which in turn allows for quantizing in a similar fashion to the shear flow. The resulting system is an effective diffusion equation (trivial quantum number) paired with a countable family of Schrodinger equations (nontrivial quantum numbers). For the latter, we are able to construct a Lyapunov functional with enhanced characteristic time scales which are much shorter than the inverse of the diffusivity. We apply tools from non-self-adjoint spectral theory to infer enhanced rates of dissipation of the semigroup evolution operator, and we show that the solution of the advection-diffusion equation converges to the solution of the effective diffusion equation as the diffusive parameter becomes vanishingly small.

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