The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Choi, Young-Pil¹  Kwon, Bongsuk² 

[1] Tech Univ Munich, Fak Math, Boltzmannstr, D-85748 Garching, Germany

[2] Ulsan Natl Inst Sci & Technol, Sch Nat Sci, Dept Math Sci, Ulsan 689798, South Korea

关键词: Pressureless Euler equations;    Compressible Navier-Stokes equations;    Coupled hydrodynamic equations;    Global existence of classical solutions;    Large-time behavior;   

DOI  :  10.1016/j.jde.2016.03.026

来源: Elsevier

【 摘 要 】

We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate L-2 Sobolev space. We also establish the a priori large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free