On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

Young-Pil Choi; Jinwook Jung

doi:10.3934/krm.2022016

Article Contents

2022, Volume 15, Issue 5: 843-891. Doi: 10.3934/krm.2022016

This issue Previous Article Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit Next Article Erratum to: On the entropic property of the ellipsoidal statistical model with the Prandtl number below 2/3

On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

Young-Pil Choi^1, and
Jinwook Jung^2, ,

1.
Department of Mathematics, Yonsei University, Seoul, 03722, Republic of Korea
2.
Research Institute of Basic Sciences, Seoul National University, Seoul, 08826, Republic of Korea

^*Corresponding author: Jinwook Jung
^*Corresponding author: Jinwook Jung

Received: January 2022

Early access: May 20, 2022

Published online: May 20, 2022

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Abstract

We analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.

Keywords:

Vlasov equations,
compressible Navier–Stokes equations with degenerate viscosities,
well-posedness,
finite-time singularity formation.

Mathematics Subject Classification: Primary: 35Q70, 35Q83, 76N10.

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