Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density

Boris Haspot (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 ANGE - Numerical Analysis, Geophysics and Ecology

We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum ρ 0 u 0 belongs to the set of the finite measure M(R) and when the initial density ρ 0 is in the set of bounded variation functions BV (R). In particular it includes at the same time the case of initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. 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 and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity v = u + µ(ρ) ρ 2 ∂ x ρ (see [20, 18, 3]) with µ(ρ) the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is to weak (it means ρ 0 v 0 ∈ M(R)) then we prove the existence of weak energy in finite time but the density remains a prirori discontinuous on the time interval of existence.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density. 2018. ⟨hal-01716150v3⟩