2015
DOI: 10.1016/j.jde.2015.04.007
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Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities

Abstract: In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial da… Show more

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Cited by 34 publications
(26 citation statements)
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“…For example, the aforementioned works by Xin and Yan [50,51] point out that the appearance of vacuum areas of the initial density profile will cause finite time blow-up of the classical solutions. See also [27,3]. Recently, Li, Wang and Xin [28] show that there are no classical solutions with bounded entropy for the compressible Navier-Stokes equations in some Sobolev spaces in any time interval (0, T ) if the initial data contains vacuum.…”
Section: Introductionmentioning
confidence: 99%
“…For example, the aforementioned works by Xin and Yan [50,51] point out that the appearance of vacuum areas of the initial density profile will cause finite time blow-up of the classical solutions. See also [27,3]. Recently, Li, Wang and Xin [28] show that there are no classical solutions with bounded entropy for the compressible Navier-Stokes equations in some Sobolev spaces in any time interval (0, T ) if the initial data contains vacuum.…”
Section: Introductionmentioning
confidence: 99%
“…It should be remarked that the conditions (13) guarantee that integration by parts in our calculations makes sense (see also Rozanova and Jiu et al 14,19 ).…”
Section: Introduction and Main Resultsmentioning
confidence: 79%
“…In this paper, we are interested in the blowup of classical solutions to the Cauchy problem (1-5). There are huge literatures on the blow-up results for the compressible Euler equations 3-10 and compressible Navier-Stokes equations, [11][12][13][14][15][16][17]19 but the blow-up results for the coupled kinetic-fluid equations are very few because the kinetic and fluid equations have different characteristic curves; see Choi. 18 Very recently, Choi 18 deals with the finite-time blowup phenomena of classical solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the initial configurations.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…Proof The proof is the same as Lemma . Indeed, for CNS, in view of Lemma , if 1<γ1+2n, we have rightd2dt2G(t)left=H(t)=2E(t)+(n(γ1)2)Ei(t)rightrightleft2E(t)=2E(0). Integrating over [0, t ], we get .…”
Section: Blow‐up Of the Smooth Solution In Whole Spacementioning
confidence: 92%
“…Because in the authors got the blow‐up of the smooth solution under the restriction 1<γ1+2n, we will firstly extend the results about the blow‐up of the smooth solution of CNS or ICNS without any restriction about γ . Then, we use the same method to research the blow‐up of the smooth solution of CNS or ICNS in half space with Naiver‐slip boundary condition.…”
Section: Introductionmentioning
confidence: 92%