New effective pressure and existence of global strong solution for compressible Navier–Stokes equations with general viscosity coefficient in one dimension

Abstract

In this paper we prove the existence of global strong solution for the Navier-Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the technics developed by Hoff for the constant viscosity case.

Vanishing Capillarity Limit of the Navier--Stokes--Korteweg System in One Dimension with Degenerate Viscosity Coefficient and Discontinuous Initial Density

“…First, we recall the finite-time existence results of strong solutions and some explosion criterion stating that the only way in which a classical solution might blow-up is because of the appearance of vacuum regions or because the density does not remain bounded, more precisely, the L ∞ -norm of 1 ρ or of ρ blows up. These results were stated and proved in [BH20] and [BH21] (see Theorem 3.1 from these papers).…”

“…The main difficulty is to obtain a priori estimates assuring that the density is bounded and bounded by below. Once this is achieved, one may follow the approach of D. Hoff in order to obtain the estimates necessary to prove existence and uniqueness, see [Hof87], [Hof98] for the original approach or our more recent contributions [BH20], [BH21]. Thus, we will not insist on these, by now well-understood points.…”

“…Again, the fact that the solution itself verifies the estimates announced in Theorem 1.2 is a consequence of the Fatou lemma. One can consult in particular [BH20].…”

“…This is sufficient to control the L ∞ norm of 1 ρ . In [BH20] we proved that if α > 1 2 , γ ≥ max{1, α} and if the so-called effective velocity satisfies initially an Oleinik type inequality (see [Ole59]) then we have the existence of global strong solution provided that (ρ 0 , 1 ρ0 ) are bounded. In particular there is no restriction on the sign of the effective flux.…”

“…ρ > 0) all along the time. Indeed, explosion criteria shows that this is the possible cause for the breakdown of strong solutions, see for instance Theorem 1.1. from [CDNP20] and Theorem 3.3. from [BH20,BH21]. Even, for less regular solutions in the spirit of Hoff-Serre [Hof87,Ser86a,Ser86b], the former qualitative information for the densities ensure that one can carry out estimation in the Hoff-class of regularity (referred to as intermediate-regularity solutions).…”