AIMS Mathematics, 2018, 3(1): 96-130. doi: 10.3934/Math.2018.1.96

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Minimization solutions to conservation laws with non-smooth and non-strictly convex flux

University of Pittsburgh, Mathematics Department, 301 Thackeray Hall, Pittsburgh PA 15260

Conservation laws are usually studied in the context of suffcient regularity conditionsimposed on the flux function, usually C2 and uniform convexity. Some results are proven with theaid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show thatmany of these classical results can be extended to a flux function that is not necessarily smooth oruniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longerhold, by considering the greatest (or supremum, where applicable) of all possible minimizers, wecan successfully extend the results. One specific nonlinear case is that of a piecewise linear fluxfunction, for which we prove existence and uniqueness results. We also approximate it by a smoothed,superlinearized version parameterized by ε and consider the characterization of the minimizers for thesmooth version and limiting behavior as ε ↓ 0 to that of the sharp, polygonal problem. In proving akey result for the solution in terms of the value of the initial condition, we provide a stepping stone toanalyzing the system under stochastic processes, which will be explored further in a future paper.
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