Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

40. Finite Time Blow-Up of Regular Solutions for Compressible Flows

verfasst von : Xiangdi Huang, Zhou Ping Xin

Erschienen in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The development of finite time singularity of smooth solutions to the compressible Navier-Stokes system as well as its blowup mechanism is discussed in the presence of vacuum. It is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support or isolated mass group. Besides, unified Serrin-type regularity criteria are established for the barotropic and full compressible Navier-Stokes equations with or without heat conduction. As an immediate corollary, it gives an affirmative answer to a problem proposed by J. Nash in the 1950s which asserts that the finite time blowup must be due to the concentration of either the density or the temperature.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups
Nächstes Kapitel Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations
Literatur
1.
J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)MathSciNetCrossRef
2.
L.C. Berselli, G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)MathSciNetCrossRef
3.
4.
Y. Cho, B. Jin, Blow-up of viscous heat-conducting compressible flows. J. Math. Anal. Appl. 320, 819–826 (2006)MathSciNetCrossRef
5.
Y. Cho, H.J. Choe, H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83, 243–275 (2004)MathSciNetCrossRef
6.
Y. Cho, H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscr. Math. 120, 91–129 (2006)MathSciNetCrossRef
7.
Y. Cho, H. Kim, Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)MathSciNetCrossRef
8.
H.J. Choe, J. Bum, Regularity of weak solutions of the compressible Navier-Stokes equations. J. Korean Math. Soc. 40, 1031–1050 (2003)MathSciNetCrossRef
9.
H.J. Choe, H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Eqs. 190, 504–523 (2003)MathSciNetCrossRef
10.
D.P. Du, J.Y. Li, K.J. Zhang, Blowup of smooth solutions to the Navier-Stokes equations for compressible isothermal fluids (2011). arXiv:1108.1613v1
11.
E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford University Press, Oxford/New York, 2004)MATH
12.
E. Feireisl, E.A. Novotny, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetCrossRef
13.
D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Am. Math. Soc. 303, 169–181 (1987)MATH
14.
D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)MathSciNetCrossRef
15.
D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multi-dimensional heat-conducting fluids. Arch. Rat. Mech. Anal. 193, 303–354 (1997)CrossRef
16.
D. Hoff, D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991)MathSciNetCrossRef
17.
D. Hoff, J.A. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations. Commun. Math. Phys. 216(2), 255–276 (2001)MathSciNetCrossRef
18.
D. Hoff, K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44(2), 603–676 (1995)MathSciNetCrossRef
19.
X.D. Huang, Some results on blowup of solutions to the compressible Navier-Stokes equations. Ph.D. thesis, The Chinese University of Hong Kong, 2009
20.
X.D. Huang, J. Li, On breakdown of solutions to the full compressible Navier-Stokes equations. Methods Appl. Anal. 16(4), 479–490 (2009)MathSciNetMATH
21.
X.D. Huang, J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations (2011). arXiv:1107.4655v3
22.
X.D. Huang, J. Li, On breakdown of solutions to the full compressible Navier-Stokes equations. Methods Appl. Anal. 16, 479–490 (2009)MathSciNetMATH
23.
X.D. Huang, J. Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows. Commun. Math. Phys. 324, 147–171 (2013)MathSciNetCrossRef
24.
X.D. Huang, J. Li, Z.P. Xin, Blowup criterion for viscous barotropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011)CrossRef
25.
X.D. Huang, J. Li, Z.P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J. Math. Anal. 43, 1872–1886 (2011)MathSciNetCrossRef
26.
X.D. Huang, J. Li, Z.P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)MathSciNetCrossRef
27.
X.D. Huang, Z.P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stkoes equations. Sci. China 53(3), 671–686 (2010)MathSciNetCrossRef
28.
X.D. Huang, Z.P. Xin, On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discret. Contin. Dyn. Syst. 36(8), 4477–4493 (2016).MathSciNetCrossRef
29.
J.S. Fan, S. Jiang, Blow-Up criteria for the Navier-Stokes equations of compressible fluids. J. Hyper. Diff. Equ. 5, 167–185 (2008)MathSciNetCrossRef
30.
J.S. Fan, S. Jiang, Y. Ou, A blow-up criterion for compressible viscous heat-conductive flows. Annales de l’Institut Henri Poincare (C) Analyse non lineaire 27, 337–350 (2010)MathSciNetCrossRef
31.
S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral dissertation, Kyoto University, 1983
32.
S. Kawashima, T. Nishida, The initial-value problems for the equations of viscous compressible and perfect compressible fluids, RIMS, Kokyuroku, 428, Kyoto University, Nonlinear functional analysis, 34–59 June 1981
33.
A.V. Kazhikhov, V.V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh 41, 282–291 (1977)MathSciNet
34.
J.I. Kanel, A model system of equations for the one-dimensional motion of a gas. Differ. Uravn. 4, 721–734 (1968) (in Russian)MathSciNet
35.
H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. Siam J. Math. Anal. 37, 1417–1434 (2006)MathSciNetCrossRef
36.
O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type (American Mathematical Society, Providence, 1968)CrossRef
37.
L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959)MATH
38.
39.
P.L. Lions, Mathematical topics in fluid mechanics, vol. 2 (The Clarendon Press, Oxford; Oxford University Press, New York, 1998)
40.
T.P. Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations. J. Differ. Equ. 33, 92–111 (1979)MathSciNetCrossRef
41.
Z. Luo, Global existence of classical solutions to two-dimensional Navier-Stokes equations with Cauchy data containing vacuum. Math. Methods Appl. Sci. 37(9), 1333–1352 (2014)MathSciNetMATH
42.
T. Makino, S. Ukai, S. Kawashima, Sur solutions a support compact de l’equation d’ Euler compressible. Jpn. J. Appl. Math. 3, 249–257 (1986)CrossRef
43.
A. Matsumura, T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)MathSciNetCrossRef
44.
A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)MathSciNetCrossRef
45.
J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France. 90, 487–497 (1962)MathSciNetCrossRef
46.
47.
L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)MathSciNetMATH
48.
O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity. J. Differ. Eqs. 245, 1762–1774 (2008)MathSciNetCrossRef
49.
R. Salvi, I. Straskraba, Global existence for viscous compressible fluids and their behavior as t → ∞. J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 40, 17–51 (1993)
50.
D. Serre, Variations de grande amplitude pour la densite d’un fluide visqueux compressible. Phys. D. 48(1), 113–128 (1991)MathSciNetCrossRef
51.
D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér, C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986)
52.
D. Serre, Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math. 303, 703–706 (1986)MathSciNetMATH
53.
J. Serrin, Mathematical principles of classical fluid-mechanics. Handbuch der Physik, vol. 811 (Springer, Berlin/Heidelberg, 1959), pp 125–265
54.
55.
J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–95 (1962)MathSciNetCrossRef
56.
T. Sideris, Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–487 (1985)MathSciNetCrossRef
57.
J.A. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edn. (Springer, New York, 1994)CrossRef
58.
V.A. Vaigant, A.V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid. Sib. Math. J. 36(6), 1283–1316 (1995)MathSciNetCrossRef
59.
Y.Z. Sun, C. Wang, Z.F. Zhang, A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pures Appl. 95, 36–47 (2011)MathSciNetCrossRef
60.
Y.Z. Sun, C. Wang, Z.F. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Ration. Mech. Anal. 201, 727–742 (2011)MathSciNetCrossRef
61.
Z.P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)MathSciNetCrossRef
62.
Z.P. Xin, W. Yan, On Blowup of classical solutions to the compressible Navier-Stokes equations. Commun. Math. Phys. 321, 529–541 (1995)MathSciNetCrossRef
63.
Y. Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. Math. Ann. 328, 173–192 (2004)MathSciNetCrossRef
Metadaten
Titel
Finite Time Blow-Up of Regular Solutions for Compressible Flows
verfasst von
Xiangdi Huang
Zhou Ping Xin
Copyright-Jahr
2018
Buch
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-13344-7_57