Date of Award

7-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Systems Engineering

First Advisor

Stanley Snelson

Second Advisor

Joo Young Park

Third Advisor

Jian Du

Fourth Advisor

Tariel Kiguradze

Abstract

This dissertation is concerned with the Landau equation, an integro-differential equation that models the particle density of a plasma as it evolves in phase space. The main topic is the (large-data) local existence of classical solutions to the Landau equation in the case of hard potentials (γ ∈ (0, 1]). Solutions have previously been constructed by Chaturvedi [SIAM. J. Math. Anal., 55(5), 5345–5385, 2023] for initial data in an exponentially-weighted Sobolev space of order 10, but it is not a priori clear whether these solutions have more regularity than the initial data. We improve Chaturvedi’s existence result in two ways: our solutions are infinitely differentiable for positive times, and we allow initial data that is more general in terms of regularity and decay, at the cost of requiring a mild positivity condition at time zero. We also prove uniqueness, under the additional assumption that the initial data is H ̈older continuous.
Along the way, we establish some useful results that were previously only known in the case of soft potentials, including spreading of positivity and propagation of H ̈older continuity. Many of the proof strategies from the soft potentials case do not apply here because of the more severe loss of velocity moments.

Download

Share

COinS