Semester

Fall

Date of Graduation

2021

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Adrian Tudorascu

Committee Member

Charis Tsikkou

Committee Member

Harumi Hattori

Committee Member

Jerzy Wojciechowski

Committee Member

Tudor Stanescu

Abstract

In this work we show that the one-dimensional pressureless Euler system admits a Lagragian characterization under fairly general initial conditions, extending recent results by Hynd [7]. Moreover, we show that if the initial velocity is right-continuous and bounded, then we have uniqueness of this Lagrangian solution (called Sticky Particles Flow, or SPF solution), which coincides with the Scalar Conservation Laws (or SCL) solution. An important tool we employed in order to prove existence is a result by Gangbo et al. [5], which establishes a canonical (i.e. the flow is given by the optimal maps pushing the Lebesgue measure restricted to the unit interval forward to the measure-valued solutions) Lagrangian representation of an absolutely continuous ow. Besides the existence result for Lagrangian solutions, which generalizes a recent result by Hynd [7], we obtain uniqueness of said solutions as our main contribution to the field. The uniqueness issue is a long-standing one, with only partial results available. Extra, entropy-like conditions are necessary to single out a solution and such conditions are complicated by the fact that the generic space for existence is the Wasserstein space of probability measures. This means that the Oleinik entropy condition, for example, should naturally be imposed almost everywhere with respect to the measure-valued solution; however, the uniqueness literature uses \everywhere" conditions. These are delicate to obtain because generically the velocity of the flow is a priori well-defined almost everywhere with respect to the measurevalued solution. In this thesis we employ a meticulous extension procedure for the velocity of the flow, which produces the everywhere Oleinik condition as a consequence of the usual, a.e. condition.

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