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Blowup criterion for viscous baratropic flows with vacuum states. (English) Zbl 1213.35135
The paper is concerning barotropic and not baratropic flow of a viscous compressible Navier-Stokes fluid. In some previous papers, the authors proved a blow-up criteria in terms of the norm of the velocity gradient of the velocity, but with a strong restriction involving shear and bulk viscosity coefficients. In the present paper this result is improved, by removing this restriction. A new blow-up criterion is given in terms of the deformation tensor. The key steps are some new estimates in $$L_2$$ of both density and velocity gradients. The result is obtained by a new energy estimate which uses the effective stress tensor. The case of zero density (vacuum state) is also studied. A very interesting estimate of the velocity gradient, in terms of divergence and vorticity, is given in Lemma 2.3, by using Poincaré and Ehrling type inequalities.

MSC:
 35B44 Blow-up in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35Q30 Navier-Stokes equations
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 [1] Beal 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) · Zbl 0573.76029 [2] Bendali A., Domíguez J.M., Gallic S.: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107(2), 537–560 (1985) · Zbl 0591.35053 [3] Bourguignon J.P., Brezis H.: Remarks on the Euler equation. J. Funct. Analysis 15, 341–363 (1974) · Zbl 0279.58005 [4] Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluid. J.Math. Pures Appl. 83, 243–275 (2004) · Zbl 1080.35066 [5] Cho Y., Kim H.: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscript Math. 120, 91–129 (2006) · Zbl 1091.35056 [6] Cho Y., Kim H.: Existence results for viscous polytropic fluids with vacuum. J. Diff. Eqs. 228, 377–411 (2006) · Zbl 1139.35384 [7] Choe H.J., Kim H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Diff. Eqs. 190, 504–523 (2003) · Zbl 1022.35037 [8] Choe H.J., Bum J.: Regularity of weak solutions of the compressible Navier-Stokes equations. J. Korean Math. Soc. 40(6), 1031–1050 (2003) · Zbl 1034.76049 [9] Constantin P.: Nonlinear inviscid incompressible dynamics. Phys. D. 86, 212–219 (1995) · Zbl 0899.76103 [10] Desjardins B.: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Comm. Part. Diff. Eqs. 22(5-6), 977–1008 (1997) · Zbl 0885.35089 [11] Fan J.S., Jiang S.: Blow-Up criteria for the navier-stokes equations of compressible fluids. J.Hyper. Diff. Eqs. 5(1), 167–185 (2008) · Zbl 1142.76049 [12] Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001) · Zbl 0997.35043 [13] Feireisl E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53(6), 1705–1738 (2004) · Zbl 1087.35078 [14] Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004 · Zbl 1080.76001 [15] Hoff D.: Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc. 303(1), 169–181 (1987) · Zbl 0656.76064 [16] Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995) · Zbl 0836.76082 [17] Hoff D., Serre D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991) · Zbl 0741.35057 [18] Huang, X.D.: Some results on blowup of solutions to the compressible Navier-Stokes equations. PhD Thesis, Chinese University of Hong Kong, 2009 [19] Huang, X.D., Xin, Z.P.: A Blow-up criterion for the compressible Navier-Stokes equations. http://arxiv.org/abs/0902.2606v1 [math-ph], 2009 [20] Huang X.D., Xin Z.P.: A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci. in China 53(3), 671–686 (2010) · Zbl 1256.35059 [21] 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. Prikl. Mat. Meh. 41, 282–291 (1977) [22] Lions, P.L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. New York: Oxford University Press, 1998 · Zbl 0908.76004 [23] Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980) · Zbl 0429.76040 [24] Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962) · Zbl 0113.19405 [25] Ponce G.: Remarks on a paper: ”Remarks on the breakdown of smooth solutions for the 3-D Euler equations”. Commun. Math. Phys. 98(3), 349–353 (1985) · Zbl 0589.76040 [26] Rozanova O.: Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity. J. Diff. Eqs. 245, 1762–1774 (2008) · Zbl 1154.35070 [27] Salvi R., Straskraba I.: Global existence for viscous compressible fluids and their behavior as t J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 40, 17–51 (1993) · Zbl 0785.35074 [28] Serre D.: Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986) · Zbl 0597.76067 [29] Serre D.: 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) · Zbl 0611.35070 [30] Serrin J.: On the uniqueness of compressible fluid motion. Arch. Rat. Mech. Anal. 3, 271–288 (1959) · Zbl 0089.19103 [31] Vaigant V.A., Kazhikhov A.V.: On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid. Siberian Math. J. 36(6), 1108–1141 (1995) · Zbl 0860.35098 [32] Xin Z.P.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998) · Zbl 0937.35134
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