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

Carey Caginalp; Carey Caginalp

doi:10.3934/Math.2018.1.96

AIMS Mathematics

2018, Volume 3, Issue 1: 96-130. doi: 10.3934/Math.2018.1.96

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Research article

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

Carey Caginalp ^,

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

Received: 19 January 2018 Accepted: 06 February 2018 Published: 02 March 2018

Conservation laws are usually studied in the context of suffcient regularity conditions imposed on the flux function, usually C² and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully extend the results. One specific nonlinear case is that of a piecewise linear flux function, 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 the smooth version and limiting behavior as ε ↓ 0 to that of the sharp, polygonal problem. In proving a key result for the solution in terms of the value of the initial condition, we provide a stepping stone to analyzing the system under stochastic processes, which will be explored further in a future paper.
- conservation laws,
- Hopf-Lax-Oleinik,
- shocks,
- variational problems in differential equations,
- minimization
Citation: Carey Caginalp. Minimization solutions to conservation laws with non-smooth and non-strictly convex flux[J]. AIMS Mathematics, 2018, 3(1): 96-130. doi: 10.3934/Math.2018.1.96

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Abstract

Conservation laws are usually studied in the context of suffcient regularity conditions imposed on the flux function, usually C² and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully extend the results. One specific nonlinear case is that of a piecewise linear flux function, 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 the smooth version and limiting behavior as ε ↓ 0 to that of the sharp, polygonal problem. In proving a key result for the solution in terms of the value of the initial condition, we provide a stepping stone to analyzing the system under stochastic processes, which will be explored further in a future paper.

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