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A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. (English) Zbl 1256.35059

Summary: In this paper, we obtain a blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations just in terms of the gradient of the velocity, analogous to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, the initial vacuum is allowed in our case.

MSC:

35Q30 Navier-Stokes equations
35B44 Blow-up in context of PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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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, 1984, 94: 61–66 · Zbl 0573.76029
[2] Chemin J Y, Masmoudi N. About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J Math Anal, 2001, 33: 84–112 · Zbl 1007.76003
[3] Cho Y G, Choe H J, Kim H S. Unique solvablity of the initial boundary value problems for compressible viscous fluid. J Math Pure Appl, 2004, 83: 243–275 · Zbl 1080.35066
[4] Cho Y G, Kim H S. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscript Math, 2006, 120: 91–129 · Zbl 1091.35056
[5] Cho Y G, Kim H S. Existence results for viscous polytropic fluids with vacuum. J Differential Equations, 2006, 228: 377–411 · Zbl 1139.35384
[6] Choe H J, Jin B J. Regularity of weak solutions of the compressible Navier-Stokes equations. J Korean Math Soc, 2003, 40: 1031–1050 · Zbl 1034.76049
[7] Choe H J, Kim H S. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differential Equations, 2003, 190: 504–523 · Zbl 1022.35037
[8] Constantin P. Nonlinear inviscid incompressible dynamics. Phys D, 1995, 86: 212–219 · Zbl 0899.76103
[9] Constantin P, Fefferman C. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ Math J, 1993, 42: 775–789 · Zbl 0837.35113
[10] Constantin P, Fefferman C, Majda A J. Geometric constraints on potentially singular solutions for the 3-D Euler equations. Comm Partial Differential Equations, 1996, 21: 559–571 · Zbl 0853.35091
[11] Constantin P, Fefferman C, Titi E S, et al. Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Commun Math Phys, 2007, 270: 789–811 · Zbl 1123.35043
[12] Desjardins B. Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Comm Partial Differential Equations, 1997, 22: 977–1008 · Zbl 0885.35089
[13] DiPerna R J, Lions P L. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math, 1989, 98: 511–547 · Zbl 0696.34049
[14] Fan J S, Jiang S. Blow-Up criteria for the Navier-Stokes equations of compressible fluids. J Hyperbolic Differ Equa, 2008, 5: 167–185 · Zbl 1142.76049
[15] Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004 · Zbl 1080.76001
[16] Feireisl E. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ Math J, 2004, 53: 1705–1738 · Zbl 1087.35078
[17] Hoff D. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303: 169–181 · Zbl 0656.76064
[18] Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch Rational Mech Anal, 1995, 132: 1–14 · Zbl 0836.76082
[19] Hoff D. Compressible flow in a half-space with Navier boundary conditions. J Math Fluid Mech, 2005, 7: 315–338 · Zbl 1095.35025
[20] Hoff D, Serre D. The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J Appl Math, 1991, 51: 887–898 · Zbl 0741.35057
[21] Huang X D, Xin Z P. A blow-up criterion for the compressible Navier-Stokes equations. Comm Math Phys, submitted · Zbl 1256.35059
[22] Jiang S, Zhang P. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Comm Math Phys, 2001, 215: 559–581 · Zbl 0980.35126
[23] 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, 1977, 41: 282–291
[24] Leray J. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math, 1934, 63: 193–248 · JFM 60.0726.05
[25] Lions P L. Mathematical Topics in Fluid Mechanics, Vol. 1. Oxford: Oxford University Press, 1996 · Zbl 0866.76002
[26] Lions P L. Mathematical Topics in Fluid Mechanics, Vol. 2. Oxford: Oxford University Press, 1998 · Zbl 0908.76004
[27] Liu T P, Xin Z P, Yang T. Vacuum states for compressible flow. Discrete Contin Dyn Syst, 1998, 4: 1–32 · Zbl 0970.76084
[28] Matsumura A, Nishida T. Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm Math Phys, 1983, 89: 445–464 · Zbl 0543.76099
[29] Salvi R, Straskraba I. Global existence for viscous compressible fluids and their behavior as t J Fac Sci Univ Tokyo Sect IA Math, 1993, 40: 17–51 · Zbl 0785.35074
[30] Serre D. Solutions faibles globales deséquations de Navier-Stokes pour un fluide compressible. C R Acad Sci Paris Sér I Math, 1986, 303: 639–642 · Zbl 0597.76067
[31] Serre D. Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C R Acad Sci Paris Sér I Math, 1986, 303: 703–706 · Zbl 0611.35070
[32] Solonnikov V A. Solvability of the initial boundary value problem for the equation a viscous compressible fluid. J Sov Math, 1980, 14: 1120–1133 · Zbl 0451.35092
[33] Sun WJ, Jiang S, Guo Z H. Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids. J Differential Equations, 2006, 222: 263–296 · Zbl 1091.35063
[34] Xin Z P. Blow-up of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51: 229–240 · Zbl 0937.35134
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