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A Geometric Singular Perturbation Theory Approach to Viscous Singular Shocks Profiles for Systems of Conservation Laws
Hsu, Ting-Hao

2015, Doctor of Philosophy, Ohio State University, Mathematics.

The concept of singular shocks was introduced in a series of papers in the 1980s, by Keyfitz and Kranzer, in order to solve Riemann problems for a class of equations which cannot be solved using classical solutions. Classical solutions for Riemann problems are measurable functions composed of regular shocks and rarefactions, and singular shocks are distributions involving delta measures that are weak limits of approximate viscous solutions. During the past decades, many abstract theories of singular shocks were developed, and many examples of this type of solution in problems modeling physical phenomena were discovered.



We study singular shocks as self-similar zero-viscosity limits via the viscous regularization ut+f(u)x=εtuxx for two systems of conservation laws. The first system models incompressible two-phase fluid flow in one space dimension, and the second one is the Keyfitz-Kranzer system. Singular shocks for both systems have been analyzed in the literature, and the results are enhanced in this dissertation.



We improve and apply theorems from Geometric Singular Perturbation Theory, including Fenichel's Theorems, the Exchange Lemma, and the Corner Lemma, to prove existence and convergence of viscous profiles for singular shocks for those two examples. We also derive estimates for the growth rates of the unbounded viscous solutions. In particular, it is demonstrated that, although viscous solutions for these two systems both have shock layers of widths of order ε, they tend to infinity in quantitatively different manners. For the two-phase flow model, the maximum value of the solution is of order log(1/ε), while for the Keyfitz-Kranzer system, the maximum value is of order 1/ε2.

Barbara Keyfitz (Advisor)
Ovidiu Costin (Committee Member)
Saleh Tanveer (Committee Member)
105 p.

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Hsu, T. (2015). A Geometric Singular Perturbation Theory Approach to Viscous Singular Shocks Profiles for Systems of Conservation Laws. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Hsu, Ting-Hao. "A Geometric Singular Perturbation Theory Approach to Viscous Singular Shocks Profiles for Systems of Conservation Laws." Electronic Thesis or Dissertation. Ohio State University, 2015. OhioLINK Electronic Theses and Dissertations Center. 21 Nov 2017.

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Hsu, Ting-Hao "A Geometric Singular Perturbation Theory Approach to Viscous Singular Shocks Profiles for Systems of Conservation Laws." Electronic Thesis or Dissertation. Ohio State University, 2015. https://etd.ohiolink.edu/

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