Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier–Stokes equations

Rozanova, Olga

doi:10.1016/j.jde.2008.07.007

Cited by 109 publications

(50 citation statements)

References 13 publications

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“…Remark 2.2.5 For κ > 0, with small initial mass, one can not expect generally that the global solutions as in Theorems 2.2.1 and 2.4.2 are highly decreasing at infinity (in space) due to [30] even if they are initially, or that the entropy S has better regularity due to [31].…”

Section: Remark 224mentioning

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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Section: Remark 224mentioning

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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“…Lemma 5.5 implies that 35) for any t ∈ [0, T ) with C > 0 a finite number. Noting that (5.2) is essentially a parabolichyperbolic system, it is then standard to derive other higher order estimates for the regularity of the regular solutions.…”

Section: Blow-up Criterionmentioning

confidence: 99%

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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“…On the other hand, lots of finite time blowup results were established, see [6,21,24,26], for various solution classes or conditions. However, it is not clear yet if such solutions exist locally in time.…”

Section: Formation Of Singularitiesmentioning

confidence: 99%

Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum

Pan

Zhu

2016

Bull Braz Math Soc, New Series

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The purpose of this paper is to present our recent progress on the local theory of classical solutions for compressible isentropic Navier-Stokes equations with density dependent viscosities in a power law and vacuum, which is a long-standing open problem due to the very high degeneracy caused by vacuum for this system. We will also mention some related open problems of high mathematical interest for this system.

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Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier–Stokes equations

Cited by 109 publications

References 13 publications

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

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

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

Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum

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