### 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 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.

Original language | English |
---|---|

Pages (from-to) | 654-711 |

Number of pages | 58 |

Journal | Journal of Differential Equations |

Volume | 261 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Jul 5 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*261*(1), 654-711. https://doi.org/10.1016/j.jde.2016.03.026

}

*Journal of Differential Equations*, vol. 261, no. 1, pp. 654-711. https://doi.org/10.1016/j.jde.2016.03.026

**The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations.** / Choi, Young Pil; Kwon, Bongsuk.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

AU - Choi, Young Pil

AU - Kwon, Bongsuk

PY - 2016/7/5

Y1 - 2016/7/5

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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84962092843&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962092843&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2016.03.026

DO - 10.1016/j.jde.2016.03.026

M3 - Article

AN - SCOPUS:84962092843

VL - 261

SP - 654

EP - 711

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -