Abstract
The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.
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References
Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)
Cho Y., Kim H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)
Fan J., Jiang S., Ou Y.: A blow-up criterion for compressible viscous heat-conductive flows. Annales de l’Institut Henri Poincare (C) Analyse non lineaire 27, 337–350 (2010)
Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford Science Publication, Oxford (2004)
Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)
Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)
Hoff D.: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Rational Mech. Anal. 139, 303–354 (1997)
Huang, X.D.: Some results on blowup of solutions to the compressible Navier–Stokes equations. PhD thesis, The Chinese University of Hong Kong (2009)
Huang X.D., Li J.: On breakdown of solutions to the full compressible Navier–Stokes equations. Methods Appl. Anal. 16, 479–490 (2009)
Huang, X.D., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations. http://arxiv.org/abs/1107.4655
Huang X.D., Li J., Xin Z.P.: Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J. Math. Anal. 43, 1872–1886 (2011)
Huang X.D., Li J., Xin Z.P.: Blowup criterion for viscous barotropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011)
Huang X.D., Li J., Xin Z.P.: 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)
Huang X.D., Xin Z.P.: A blow-up criterion for classical solutions to the compressible Navier–Stokes equations. Sci. China 53, 671–686 (2010)
Kazhikhov A.V.: Cauchy problem for viscous gas equations. Sib. Math. J. 23, 44–49 (1982)
Kazhikhov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)
Lions P.L.: Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models. Oxford University Press, New York (1998)
Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)
Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)
Rozanova O.: Blow up of smooth solutions to the compressible Navier–Stokes equations with the data highly decreasing at infinity. J. Differ. Equ. 245, 1762–1774 (2008)
Serrin J.: On the uniqueness of compressible fluid motion. Arch. Rational Mech. Anal. 3, 271–288 (1959)
Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)
Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier–Stokes equations. J. Math. Pures Appl. 95, 36–47 (2011)
Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Rational Mech. Anal. 201, 727–742 (2011)
Xin Z.P.: Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)
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Huang, X., Li, J. & Wang, Y. Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System. Arch Rational Mech Anal 207, 303–316 (2013). https://doi.org/10.1007/s00205-012-0577-5
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DOI: https://doi.org/10.1007/s00205-012-0577-5