Vanishing Viscosity Limit of the Compressible Navier--Stokes Equations with General Pressure Law

Schrecker, Matthew R. I.; Schulz, Simon

doi:10.1137/18m1224362

Cited by 9 publications

(21 citation statements)

References 27 publications

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“…Note that the class of gases considered in the previous definition is significantly wider than the one introduced in [15]. Indeed, therein the authors considered the vanishing viscosity limit of solutions of (1.2) under the assumption of a pressure law satisfying the first two hypotheses of Definition 1.1, but where p(ρ) = c * ρ for ρ ≥ R instead of (1.5).…”

Section: Introductionmentioning

confidence: 99%

“…This assumption for large density allowed the authors to provide an explicit representation for entropies in this regime (cf. [15,Theorem 2.6]). Such a representation is no longer possible without the explicit form of the pressure.…”

Section: Introductionmentioning

confidence: 99%

“…The convergence of the vanishing physical viscosity limit from the Navier-Stokes equations to the Euler equations is a difficult problem with a rich history. Prior to the contribution of the authors in [15], Chen and Perepelitsa proved in [5] that, for a γ-law gas of index γ ∈ (1, ∞) and given any initial data of relative finite-energy, a solution of (1.1) could be obtained as the inviscid limit of solutions of (1.2). Earlier than that, the existence of L ∞ entropy solutions of the isentropic Euler equations had been obtained by means of vanishing artificial viscosities and from finite-difference schemes by DiPerna [8], Chen [2], Ding, Chen, and Luo [7], Lions, Perthame, and Tadmor [13], and Lions, Perthame, and Souganidis [12] for polytropic gases, by Chen and LeFloch [3,4] for general pressure laws, and by Huang and Wang [10] for isothermal gases.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of this theorem relies on the techniques of compensated compactness, initiated in this context by DiPerna [8], and the finite-energy framework begun by LeFloch-Westdickenberg [11] and extended by the authors in [15]. Our approach is the following.…”

Section: Introductionmentioning

confidence: 99%

“…In order to do this, we split the entropy kernel into two distinct quantities: a re-scaled version of the kernel obtained in [15], which we call χ * , and a perturbation, called χ. To this end, we make the following definitions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Inviscid limit of the compressible Navier-Stokes equations for asymptotically isothermal gases

Schrecker¹,

Schulz²

2019

Preprint

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We prove the existence of relative finite-energy vanishing viscosity solutions of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p(ρ)/ρ = O(1) in the limit ρ → ∞. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier-Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier-Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Inviscid limit of the compressible Navier-Stokes equations for asymptotically isothermal gases

Schrecker¹,

Schulz²

2019

Preprint

Self Cite

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Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

Wang

2022

Arch Rational Mech Anal

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We are concerned with global finite-energy solutions of the three-dimensional compressible Euler-Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions with spherical symmetry of the Cauchy problem for the Euler-Poisson equations as the inviscid limit of the corresponding compressible Navier-Stokes-Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in L p , and a more general compensated compactness framework via several new main ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum (ρ = 0) and far-field (ρ = ∞) are carefully analyzed. Owing to the generality of the pressure law, only the W −1,p loc -compactness of weak entropy dissipation measures with p ∈ [1, 2) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the L p compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

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Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry

Chen,

Huang,

et al. 2024

Commun. Math. Phys.

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We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in $$L^p$$ L p , and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum ($$\rho =0$$ ρ = 0 ) and far-field ($$\rho =\infty $$ ρ = ∞ ) are carefully analyzed. Owing to the generality of the pressure law, only the $$W^{-1,p}_{\textrm{loc}}$$ W loc - 1 , p -compactness of weak entropy dissipation measures with $$p\in [1,2)$$ p ∈ [ 1 , 2 ) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the $$L^p$$ L p compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

show abstract

Vanishing Viscosity Limit of the Compressible Navier--Stokes Equations with General Pressure Law

Cited by 9 publications

References 27 publications

Inviscid limit of the compressible Navier-Stokes equations for asymptotically isothermal gases

Inviscid limit of the compressible Navier-Stokes equations for asymptotically isothermal gases

Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry

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