Threshold dynamics in hyperbolic partial differential equations

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Lee, Yongki
Major Professor
Hailiang Liu
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We are interested in the persistence of the $C^1$ solution regularity for time dependent partial differential equations(PDE). As is known that the typical well-posedness result asserts that either a solution of a time-dependent PDE exists for all time or else there is a finite time such that some norm of the solution becomes unbounded as the life span is approached. The natural question is whether there is a critical threshold for the initial data such that the persistence of the $C^1$ solution regularity depends only on crossing such a critical threshold. In this thesis we attempt to study such a critical phenomena in Restricted Euler-Poisson(REP) equations and a class of non-local conservation laws.

In this thesis, we have obtained the following results.

1. For three-dimensional REP equations, we identify both upper thresholds for the finite-time blow up of solutions and sub-thresholds for the global existence of solutions, with the thresholds depending on the relative size of the eigenvalues of the initial velocity gradient matrix and the initial density. For the attractive forcing case, these one-sided threshold conditions of the initial configurations are optimal, and the corresponding results also hold for arbitrary $n$ dimensions ($n\geq 3$).

2. We propose weakly restricted Euler-Poisson(WREP) equations as an effort to gain a better understanding on Euler-Poisson equations in multi-dimensions. The WREP can be viewed as a slight generalization of the REP equations. We then provide upper-thresholds for finite time blow up of solutions for WREP equations with attractive/repulsive forcing. It is shown that the thresholds depend on the relative size of the initial density and each elements of the initial velocity gradient


3. We investigate a class of non-local conservation laws with the nonlinear advection coupling both local and non-local mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics. Our threshold analysis for the traffic flow models is applicable to the class of non-local conservation laws.

4. Lastly, we further study the class of non-local conservation laws. It is well known that the initial value problem for a scalar conservation law may admit more than one weak solution, so we need to find a selection criterion in order to single out the physically relevant solution. We define the Kru\u{z}kov-type entropy solution, and by adapting the doubling of variables method and the method of vanishing viscosity, we obtain a uniqueness and existence of entropy solutions of the non-local conservation laws.

Wed Jan 01 00:00:00 UTC 2014