Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Holle, Yannick¹ 

[1] Rhein Westfal TH Aachen, Inst Math, Templergraben 55, D-52062 Aachen, Germany

关键词: Hyperbolic conservation laws;    Network;    Coupling condition;    Isentropic gas dynamics;    BGK model;    Relaxation limit;   

DOI  :  10.1016/j.jde.2020.01.005

来源: Elsevier

【 摘 要 】

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free