Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

Valli, Alberto; Zaja̧czkowski, Wojciech M.

doi:10.1007/bf01206939

Cited by 265 publications

(144 citation statements)

References 21 publications

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“…To the best of our knowledge, there is no existence results for the stochastic compressible Navier-Stokes system in the class of strong solutions. The regularity hypotheses imposed in Definition 4.1 are inspired by the deterministic case studied by Valli [22] and Valli, Zajaczkowski [23].…”

Section: Weak-strong Uniquenessmentioning

confidence: 99%

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

Breit

Feireisl

Hofmanová³

2017

Commun. Math. Phys.

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We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak-strong uniqueness implies weak-strong uniqueness in law.

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Section: Weak-strong Uniquenessmentioning

confidence: 99%

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

Breit

Feireisl

Hofmanová³

2017

Commun. Math. Phys.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Compressible Navier–Stokes system: Large solutions and incompressible limit

Danchin

Mucha

2017

Advances in Mathematics

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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...

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“…Then using delicate energy methods in Sobolev spaces, Matsumura and Nishida showed in their pioneering papers [18,19] that the classical solutions exist globally in time provided that the data are small in some sense. See also the papers [6,12,23,27,28,29] for some further local or global results in case of positive densities.…”

Section: Introductionmentioning

confidence: 99%

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

2006

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Abstract. We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3 . We first prove the local existence of solutions (ρ, u) ) under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t > 0, we conclude that (ρ, u) is a classical solution in (0, T * * ) × Ω for some T * * ∈ (0, T * ]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.

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Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

Cited by 265 publications

References 21 publications

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

Compressible Navier–Stokes system: Large solutions and incompressible limit

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

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