TY - JOUR
T1 - Constrained systems of conservation laws
T2 - A geometric theory
AU - REINTJES, Moritz
PY - 2017/12
Y1 - 2017/12
N2 - We address the Riemann and Cauchy problems for systems of n conservation laws in m unknowns which are subject to m − n constraints (m ≥ 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 × n systems of constrained conservation laws to n × 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.
AB - We address the Riemann and Cauchy problems for systems of n conservation laws in m unknowns which are subject to m − n constraints (m ≥ 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 × n systems of constrained conservation laws to n × 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.
KW - Shock waves
KW - hyperbolic conservation laws,
KW - Glimm scheme
KW - Riemann problem
KW - relaxation systems
KW - curved state space
U2 - 10.4310/maa.2017.v24.n4.a1
DO - 10.4310/maa.2017.v24.n4.a1
M3 - RGC 21 - Publication in refereed journal
VL - 24
SP - 407
EP - 444
JO - Methods and Applications of Analysis
JF - Methods and Applications of Analysis
SN - 1073-2772
IS - 4
ER -