Thesis Summary:

This thesis has three aims. The first aim of the thesis is to study the well-posedness and to develop numerical methods for scalar conservation laws with nonlocal flux function modeling the phenomenon of aggregation in mathematical biology. The existence of weak solutions to a nonlocal strongly degenerate parabolic aggregation equation is proved using a finite difference method and compactness arguments. For uniqueness, we employ an entropy concept and prove the equivalence between weak and entropy solutions. The finite difference method is utilized to generate numerical examples that illustrate the aggregation process. The second goal of the thesis is to study the well-posedness of a nonlocal conservation but now modeling sedimentation in process industry. We prove existence and uniqueness of entropy solutions for a nonlocal sedimentation equation, again using a finite difference method and standard compactness results. Depending on parameter values, a Lipschitz regularity result or a maximum principle independent by the time variable is found. By the finite difference scheme we obtain numerical examples and compare it with local model results. The layered sedimentation phenomenon is observed. Finally, the Generalized Lagrange Multiplier Finite Volume Method, which was originally developed for the Maxwell equations, is extended to any hyperbolic Friedrichs system of conservation laws with involutions. We prove the convergence of the method. Moreover, the fulfillment of the involution in the weak sense when the mesh parameter goes to zero is shown. Numerical examples illustrate the properties of the method in the Maxwell equations and in the induction equation in the MHD system.

ISI Publications from the Thesis

Fernando BETANCOURT, Raimund BüRGER, Kenneth H. KARLSEN: A strongly degenerate parabolic aggregation equation. Communications in Mathematical Sciences, vol. 9, 3, pp. 711-742, (2011).

Fernando BETANCOURT, Raimund BüRGER, Kenneth H. KARLSEN, Elmer M. TORY: On nonlocal conservation laws modelling sedimentation. Nonlinearity, vol. 24, 3, pp. 855-885, (2011).