Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

Huang, Xiangdi; Li, Jing; Xin, Zhouping

doi:10.1002/cpa.21382

Cited by 452 publications

(423 citation statements)

References 32 publications

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“…Under this initial layer compatibility condition, a local theory was established successfully; see also [28]. Such a local solution was further extended globally by Huang-Li-Xin [16], when initial energy is small. Some similar results can also be found in [17] and [18].…”

Section: 3)mentioning

confidence: 88%

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

Pan

Zhu

2016

J. Math. Fluid Mech.

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Abstract. 2D shallow water equations have degenerate viscosities proportional to surface height, which vanishes in many physical considerations, say, when the initial total mass, or energy are finite. Such a degeneracy is a highly challenging obstacle for development of well-posedness theory, even local-in-time theory remains open for long time. In this paper, we will address this open problem with some new perspectives, independent of the celebrated BD-entropy [2,3]. After exploring some interesting structures of most models of 2D shallow water equations, we introduced a proper notion of solution class, called regular solutions, and identified a class of initial data with finite total mass and energy, and established the local-in-time well-posedness of this class of smooth solutions. The theory is applicable to most relatively physical shallow water models, broader than those with BD-entropy structures. Later, a Beale-Kato-Majda type blow-up criterion is also established. This paper is mainly based on our early preprint [22].

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Section: 3)mentioning

confidence: 88%

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

Pan

Zhu

2016

J. Math. Fluid Mech.

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“…(D 2 ) : Zlotnik inequality (see Appendix A in Section 7) which was used in [16] for isentropic flow to get the upper bound of the density does not work here. In [16], g(ρ) is defined as g(ρ) = − ρP (ρ) 2µ+λ for the case thatρ = 0, where P (ρ) = aρ γ for a > 0 and γ > 1.…”

Section: Introductionmentioning

confidence: 99%

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

Wen¹,

Zhu²

2017

SIAM J. Math. Anal.

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We consider the Cauchy problem for the full compressible Navier-Stokes equations with vanishing of density at infinity in R 3 . Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data is more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of R 3 and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first result so far for the global existence of solutions to the full compressible NavierStokes equations when density vanishes at infinity (in space). In addition, the exponential decay rate of the strong solution is of independent interest.

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“…Because of its physical importance and mathematical challenging, the well-posed theory has been widely studied for the system (1.1), (1.2) combined with Fourier's law (1.6), see [1,2,3,4,6,8,11,12,14,15,16,17,18,21,23]. In particular, the local existence and uniqueness of smooth solutions was established by Serrin [21] and Nash [18] for initial data far away from vacuum.…”

Section: Introductionmentioning

confidence: 99%

Compressible Navier–Stokes Equations with hyperbolic heat conduction

Racke

2016

J. Hyper. Differential Equations

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Abstract. In this paper, we investigate the system of compressible Navier-Stokes equations with hyperbolic heat conduction, i.e., replacing the Fourier's law by Cattaneo's law. First, by using Kawashima's condition on general hyperbolic parabolic systems, we show that for small relaxation time τ , global smooth solution exists for small initial data. Moreover, as τ goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier-Stokes equations.

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Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations

Cited by 452 publications

References 32 publications

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

Compressible Navier–Stokes Equations with hyperbolic heat conduction

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