Constrained systems of conservation laws: a geometric theory
| Autor: | Moritz Reintjes |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Conservation law
FOS: Physical sciences 35L65 (Primary) Cauchy distribution Mathematical Physics (math-ph) Accumulation function 01 natural sciences 010101 applied mathematics Riemann hypothesis symbols.namesake Mathematics - Analysis of PDEs Riemann problem Flow (mathematics) Geometric group theory FOS: Mathematics symbols Initial value problem Applied mathematics 0101 mathematics Mathematical Physics Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Methods and Applications of Analysis. 24:407-444 |
| ISSN: | 1945-0001 1073-2772 |
| DOI: | 10.4310/maa.2017.v24.n4.a1 |
| Popis: | We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such constrained systems generalize systems of conservation laws in standard form to include various examples of conservation laws in Physics and Engineering beyond gas dynamics, e.g., multi-phase flow in porous media. We prove local well-posedness of the Riemann problem and global existence of the Cauchy problem for initial data with sufficiently small total variation, in one spatial dimension. The key to our existence theory is to generalize the $m\times n$ systems of constrained conservation laws to $n\times n$ systems of conservation laws with states taking values in an $n$-dimensional manifold and to extend Lax's theory for local existence as well as Glimm's random choice method to our geometric framework. Our resulting existence theory allows for the accumulation function to be non-invertible across hypersurfaces. Comment: 48 pages. Version 2: Improvements of wording, specifically in the introduction. Restructuring of Introduction. Version 3: I improved wording; I moved Section 4 to the end; I added Example 2.4 and 2.5; I added references |
| Databáze: | OpenAIRE |
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