Global bmo −1 (R N ) radially symmetric solution for compressible Navier-Stokes equations with initial density in L ∞ (R N )

Abstract : In this paper we investigate the question of the existence of global weak solution for the compressible Navier Stokes equations provided that the initial momentum ρ 0 u 0 belongs to bmo −1 (R N) with N = 2, 3 and is radially symmetric. More precisely we deal with the so called viscous shallow water system when the viscosity coefficients verify µ(ρ) = µρ, λ(ρ) = 0 with µ > 0. We prove then a equivalent of the so called Koch-Tataru theorem for the compressible Navier-Stokes equations. In addition we assume that the initial density ρ 0 is only bounded in L ∞ (R N), it allows us in particular to consider initial density admitting shocks. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong, then the initial density which admits initially shocks is instantaneously regularizing inasmuch as the density becomes Lipschitz. This coupling is expressed via the regularity of the so called effective velocity v = u + 2µ ln ρ. In our case v 0 belongs to L 2 (R N) ∩ L ∞ (R N), it is important to point out that this choice on the initial data implies that we work in a setting of infinite energy on the initial data (ρ 0 , u 0), it extends in particular the results of [48]. In a similar way, we consider also the case of the dimension N = 1 where the momentum ρ 0 u 0 belongs to bmo −1 (R) without any geometric restriction. To finish we prove the global existence of strong solution for large initial data provided that the initial data are radially symmetric and sufficiently regular in dimension N = 2, 3 for γ law pressure.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [49 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01976953
Contributor : Boris Haspot <>
Submitted on : Thursday, January 10, 2019 - 2:04:35 PM
Last modification on : Wednesday, May 15, 2019 - 3:39:39 AM
Long-term archiving on : Thursday, April 11, 2019 - 3:40:28 PM

File

Radial.Symétrique.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01976953, version 1

Citation

Boris Haspot. Global bmo −1 (R N ) radially symmetric solution for compressible Navier-Stokes equations with initial density in L ∞ (R N ). 2019. ⟨hal-01976953⟩

Share

Metrics

Record views

128

Files downloads

103