Semester

Fall

Date of Graduation

2019

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Harumi Hattori

Committee Co-Chair

Adrian Tudorascu

Committee Member

Adam Halas

Committee Member

Casian Pantea

Committee Member

Tudor Stanescu

Abstract

We study global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler-Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical and we also add the friction between two fluids. In addition, we discuss the rates of decay of $L^{p}-L^{q}$ norms for a linear system. Moreover, we use the result for $L^{p}-L^{q}$ estimates to prove the decay rates for the nonlinear systems. In addition, we prove existence of heteroclinic orbits for the nonlinear Vlasov and the one-dimensional Vlasov-Poisson systems. In the nonlinear Vlasov case with sufficiently regular and periodic potential we show that there is at least one orbit emanating from a maximum point of the potential and at least one terminating at it. The Vlasov-Poisson case is more delicate due to the singularity of the potential, and our existence result is limited to one spatial dimension and to weak solutions in the space of probability measures on the torus endowed with the periodic Wasserstein distance.

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