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

Young Pil Choi; Bongsuk Kwon

doi:10.1016/j.jde.2016.03.026

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

Young Pil Choi, Bongsuk Kwon

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

Abstract

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 L² 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.

Original language	English
Pages (from-to)	654-711
Number of pages	58
Journal	Journal of Differential Equations
Volume	261
Issue number	1
DOIs	https://doi.org/10.1016/j.jde.2016.03.026
Publication status	Published - 2016 Jul 5

Bibliographical note

Funding Information:
Y.-P. Choi was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496) and Engineering and Physical Sciences Research Council (EP/K008404/1). Y.-P. Choi also acknowledges the support of the ERC-Starting Grant (306274) HDSPCONTR “High-Dimensional Sparse Optimal Control”. B. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning(2015R1C1A1A02037662).

Funding Information:
Y.-P. Choi was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2012R1A6A3A03039496 ) and Engineering and Physical Sciences Research Council ( EP/K008404/1 ). Y.-P. Choi also acknowledges the support of the ERC -Starting Grant ( 306274 ) HDSPCONTR “High-Dimensional Sparse Optimal Control”. B. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning ( 2015R1C1A1A02037662 ).

Publisher Copyright:
© 2016 Elsevier Inc.

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1016/j.jde.2016.03.026

Cite this

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title = "The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations",

abstract = "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 L2 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.",

author = "Choi, {Young Pil} and Bongsuk Kwon",

note = "Funding Information: Y.-P. Choi was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496) and Engineering and Physical Sciences Research Council (EP/K008404/1). Y.-P. Choi also acknowledges the support of the ERC-Starting Grant (306274) HDSPCONTR “High-Dimensional Sparse Optimal Control”. B. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning(2015R1C1A1A02037662). Funding Information: Y.-P. Choi was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2012R1A6A3A03039496 ) and Engineering and Physical Sciences Research Council ( EP/K008404/1 ). Y.-P. Choi also acknowledges the support of the ERC -Starting Grant ( 306274 ) HDSPCONTR “High-Dimensional Sparse Optimal Control”. B. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning ( 2015R1C1A1A02037662 ). Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

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N2 - 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 L2 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.

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

Abstract

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