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

Schrecker, Matthew R. I.; Schulz, Simon

doi:10.1016/j.jde.2020.06.018

Cited by 3 publications

(7 citation statements)

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“…Singularities of the entropy kernel and the entropy flux kernel. As indicated in [10,61,62], understanding the singularities of the entropy kernel and the entropy flux kernel is essential for the reduction of the Young measure. Thus, it requires some detailed estimates of the singularities of the entropy kernel and the entropy flux kernel.…”

Section: 3mentioning

confidence: 99%

“…However, it is not clear how the H −1 loc -compactness for the general pressure satisfying (1.4)-(1.6) can be shown by using the expansions of the weak entropy kernel established in [10,11]. Motivated by [62], we instead show the W −1,p loc -compactness for 1 ≤ p < 2, so that an improved div-curl lemma (cf. [18]) applies, which leads to the commutation identity for the entropy pairs.…”

mentioning

confidence: 91%

“…An L p compensated compactness framework was first studied by LeFloch-Westdickenberg [41] for the isentropic Euler equations for the case γ ∈ (1, 5 3 ) in (1.2), and was further developed by Chen-Perepelitsa [12] to all γ > 1 for (1.2) with a simplified proof; see also [16] for spherically symmetric solutions of the M-D isentropic Euler equations. We also refer to [61,62] for the 1-D case of asymptotically isothermal gas, i.e., γ 2 = 1 in (1.6). More recently, Chen-He-Wang-Yuan [9] established both the strong inviscid limit of CNSPEs (1.10) and the global existence of spherically symmetric solutions of CEPEs (1.1) with large data for polytropic gases (1.2).…”

mentioning

confidence: 99%

“…For the general pressure law satisfying (1.4)-(1.6), the above method does not apply directly, since it is difficult to show that every connected subset of the support of the Young measure is a bounded interval. Motivated by [10,11,46,61,62], we carefully analyze the singularities of ∂ λ 1 +1 χ with λ 1 = 3−γ 1 2(γ 1 −1) for large ρ and fully exploit the property: (ρ γ 2 +1 , ρ|u| 3 ) ∈ L 1 (dν (t,r) ) so that the ∂ λ 1 +1 −derivatives can be operated in the commutation relation; see Lemmas 4.11-4.14 for details. Then we prove that the Young measure is either a delta measure or supported on the vacuum line by similar arguments as in [10,11,46,62].…”

mentioning

confidence: 99%

“…Motivated by [10,11,46,61,62], we carefully analyze the singularities of ∂ λ 1 +1 χ with λ 1 = 3−γ 1 2(γ 1 −1) for large ρ and fully exploit the property: (ρ γ 2 +1 , ρ|u| 3 ) ∈ L 1 (dν (t,r) ) so that the ∂ λ 1 +1 −derivatives can be operated in the commutation relation; see Lemmas 4.11-4.14 for details. Then we prove that the Young measure is either a delta measure or supported on the vacuum line by similar arguments as in [10,11,46,62]. 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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confidence: 99%

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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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Section: 3mentioning

confidence: 99%

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confidence: 91%

mentioning

confidence: 99%

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confidence: 99%

mentioning

confidence: 99%

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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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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.

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Corrigendum to “Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases” [J. Differ. Equ. 269 (10) (2020) 8640–8685]

Schrecker

Schulz

2021

Journal of Differential Equations

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

Cited by 3 publications

References 13 publications

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

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

Corrigendum to “Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases” [J. Differ. Equ. 269 (10) (2020) 8640–8685]

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