Existence results for viscous polytropic fluids with vacuum

Cho, Yonggeun; Kim, Hyunseok

doi:10.1016/j.jde.2006.05.001

Cited by 223 publications

(284 citation statements)

References 24 publications

Supporting

Mentioning

275

Contrasting

Order By: Relevance

“…Hence the existence of a unique local solution (ρ, u) with the regularity (1.8) was already proved in [2,3] and our new theorem shows that (ρ, u) has some additional regularity if the data satisfy a stronger compatibility condition (1.10). It is easy to show that (1.10) is also necessary for the existence of solutions with the regularity (1.11).…”

Section: Remark 12 From the Continuity Equation (11) It Follows Imentioning

confidence: 58%

“…To estimate each term in the right hand side of (3.10), we follow the arguments in [2,3,4]; we first apply the standard inequalities such as Hölder, Sobolev and Young's inequalities and then use Lemma 3.2.…”

Section: A Priori Estimates For the Linearized Problemmentioning

confidence: 99%

“…Then Lions' theory has been improved by several authors to deduce more general results; see [7,8,9,10,14,15] for details. The very recent papers [2,3,4] by Choe and the authors are devoted to establishing some local existence results on strong solutions. Among other things, we showed in [2,3] (see also the paper [21] by Salvi and Straškraba) that if the initial data ρ 0 , u 0 satisfy the regularity condition…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2006

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Remark 12 From the Continuity Equation (11) It Follows Imentioning

confidence: 58%

Section: A Priori Estimates For the Linearized Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

2006

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the sake of generality, we will study the blow-up criterion for local strong solutions with initial vacuum, the existence of which is essentially obtained in [2]. The case that the initial density has a positive lower bound can be dealt with in the same manner (in fact, simpler) and the same result holds.…”

Section: Introductionmentioning

confidence: 99%

“…Before giving our main result, we state the following local existence of the strong solutions with initial vacuum, the proof of which can be found in [2]. 10) and the compatibility conditions …”

Section: Introductionmentioning

confidence: 99%

A blow-up criterion for compressible viscous heat-conductive flows

Jiang

2010

Acta Mathematica Scientia

View full text Add to dashboard Cite

We study an initial boundary value problem for the three-dimensional Navier-Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale-Kato-Majda criterion for ideal incompressible flows. © 2009 Elsevier Masson SAS. All rights reserved. RésuméNous étudions un problème de valeur limite initiale pour les équations de Navier-Stokes tridimensionnelles des fluides visqueux conducteurs de chaleur dans un domaine délimité lisse. Nous établissons un critère d'explosion pour les solutions fortes en termes de température et de gradient de vitesse seulement, semblable au critère de Beale-Kato-Majda pour les écoulements incompressibles idéaux.

show abstract

Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum

Xin

2021

Comm Pure Appl Math

View full text Add to dashboard Cite

In the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one-dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than O. 1x 2 /. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi-type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

show abstract

Existence results for viscous polytropic fluids with vacuum

Cited by 223 publications

References 24 publications

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

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

A blow-up criterion for compressible viscous heat-conductive flows

Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum

Contact Info

Product

Resources

About