Compressible Navier–Stokes system: Large solutions and incompressible limit

Abstract

Abstract. Here we prove the existence of global in time regular solutions to the twodimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity v0 and almost constant density ̺0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier-Stokes equations supplemented with the divergence-free projection of v0, is global. The systems are examined in R d with d ≥ 2 , in the criticalḂ s 2,1 Besov spaces framework. IntroductionWe are concerned with the following compressible Navier-Stokes equations in the whole space R d :supplemented with initial data: ̺| t=0 = ̺ 0 and v| t=0 = v 0 .The pressure function P is given and assumed to be strictly increasing. The shear and volume viscosity coefficients λ and µ are constant (just for simplicity) and fulfill the standard strong parabolicity assumption:(1.2) µ > 0 and ν := λ + 2µ > 0.Starting with the pioneering work by Matsumura and Nishida [18,19] in the beginning of the eighties, a number of papers have been dedicated to the challenging issue of proving the global existence of strong solutions for (1.1) in different contexts (whole space or domains, dimension d = 2 or d ≥ 3, and so on). One may mention in particular the works by Zajaczkowski [26], Shibata [13], Danchin [4], Mucha [21,23,24] and, more recently, by Kotschote [14,15]. The common point between all those papers is that the initial velocity is assumed to be small, and that the initial density is close to a stable constant steady state.Our main goal is to prove the global existence of strong solutions to (1.1) for a class of large initial data. In the two-dimensional case, we establish that, indeed, for fixed shear viscosity µ and any initial velocity-field v 0 (with critical regularity), the solution to (1.1) is global if λ is sufficiently large, and ̺ 0 sufficiently close (in terms of λ) to some positive constant (say 1 for notational simplicity). This result will strongly rely on the fact that, at least formally, the limit velocity for λ → +∞ satisfies the incompressible Navier-Stokes equations:with V 0 being the Leray-Helmholtz projection of v 0 on divergence-free vector-fields. Then the initial data V 0 of (1.3) is inḂ 0 2,1 (R 2 ). Therefore, in light of the well-known embeddingḂ 0 2,1 (R 2 ) ⊂ L 2 (R 2 ), we are guaranteed that it generates a unique global solution V in the energy classthat satisfies the energy identity:Based on that fact, one may prove that the additional regularity of V 0 is preserved through the time evolution (see Theorem 4.1 in the Appendix), that is). Let us now state our main existence result for (1.1) in the two-dimensional setting.There exists a large constant C such that for V 0 = Pv 0 , the divergence-free part of the initial velocity, we setand if ν satisfieswhere Q stands for the projection operator on potential vector-fields, then there exists a unique global in time regular solution (̺, v) to (1.1) such tha...

Global Large Solutions to the Three Dimensional Compressible Navier--Stokes Equations

“…The aim of this paper is to construct global solutions to the compressible Navier-Stokes equations in R 2 with the large component of the incompressible velocity without restriction condition on the volume viscosity. This is significant different to [4] and [18] assuming the volume viscosity being sufficiently large. For simplicity of notation, we use the viscosity coefficient values µ = 1 and λ = 0 throughout the paper.…”

Compression effects in heterogeneous media

“…But the incompressible equations can be also obtain from a large bulk viscosity limit: if in the bulk viscosity term ∇(λ 0 div u) one lets λ 0 go to +∞ then, formally, div u should tend to 0. This result has been recently proved by Danchin and Mucha in [10]. In our paper, the main novelty is to consider both singular pressure and singular bulk viscosity depending on the density which will encode incompressibility of the material at the maximal packing value ρ * = 1, assuming the shear viscosity to be constant.…”

Global Large Solutions and Incompressible Limit for the Compressible Navier–Stokes Equations

“…More precisely, the present study is motivated by the following result in R 2 : Theorem 1.1. (Danchin and Mucha [12]) Let ν ≥ µ, v 0 ∈Ḃ 0 2,1 (R 2 ) and a 0 :…”

“…To get rid of this difficulty, we shall derive the smoothing effect of density in the L 2 setting in low frequency and the damping effect of density in the L p setting in the high frequency, respectively. In the low frequency, one can follow the method used in [12] to derive the desired estimates, whereas in the high frequency, we follow an elementary energy approach in terms of effective velocity developed by Haspot [20], using Hoff's viscous effective flux in [24].…”

“…We follow from [12] to bound each term (∆ j a, Q∆ j u). More precisely, testing (3.34) and (3.35) by∆ j a and Q∆ j u, respectively, yields…”