Global Well-Posedness of Compressible Navier–Stokes Equations for Some Classes of Large Initial Data

We prove the global well-posedness of three dimensional compressible Navier–Stokes equations for some classes of large initial data, which may have large oscillation for the density and large energy for the velocity. The proof uses the special structure of the system (especially the effective viscous flux).

[1]  Ting Zhang.  Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space , 2007, 0712.2652.

[2]  Takaaki Nishida,et al.  Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids , 1983 .

[3]  Ting Zhang.  Erratum to: Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space , 2009 .

[4]  Zhouping Xin,et al.  Blowup of smooth solutions to the compressible Navier‐Stokes equation with compact density , 1998 .

[5]  Song Jiang,et al.  On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations , 2001 .

[6]  Ping Zhang,et al.  Sums of large global solutions to the incompressible Navier-Stokes equations , 2010, 1002.4736.

[7]  Chao Wang,et al.  A Beale–Kato–Majda Criterion for Three Dimensional Compressible Viscous Heat-Conductive Flows , 2011 .

[8]  David Hoff,et al.  Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids , 1997 .

[9]  Takaaki Nishida,et al.  The Initial Value Problem for the Equations of Motion of compressible Viscous and Heat-conductive Fluids. , 1979 .

[10]  J. Nash,et al.  Le problème de Cauchy pour les équations différentielles d'un fluide général , 1962 .

[11]  N. Lerner,et al.  Flow of Non-Lipschitz Vector-Fields and Navier-Stokes Equations , 1995 .

[12]  Qionglei Chen,et al.  Global well‐posedness for compressible Navier‐Stokes equations with highly oscillating initial velocity , 2009, 0907.4540.

[13]  David Hoff,et al.  Compressible Flow in a Half-Space with Navier Boundary Conditions , 2005 .

[14]  E. Feireisl,et al.  On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations , 2001 .

[15]  On the global wellposedness of the 3-D Navier–Stokes equations with large initial data , 2005, math/0508374.

[16]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[17]  Hiroshi Fujita,et al.  On the Navier-Stokes initial value problem. I , 1964 .

[18]  R. Danchin.  Global Existence in Critical Spaces¶for Flows of Compressible¶Viscous and Heat-Conductive Gases , 2001 .

[19]  Raphaël Danchin,et al.  A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical Lp Framework , 2010 .

[20]  Zhifei Zhang,et al.  Global regularity for the Navier–Stokes equations with some classes of large initial data , 2011 .

[21]  Chao Wang,et al.  A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations , 2010, 1001.1247.

[22]  Gui-Qiang G. Chen,et al.  Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow , 2009, 0910.2360.

[23]  David Hoff,et al.  Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions , 2002 .

[24]  Ping Zhang,et al.  Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids , 2003 .

[25]  Zhouping Xin,et al.  Global well‐posedness of classical solutions with large oscillations and vacuum to the three‐dimensional isentropic compressible Navier‐Stokes equations , 2010, 1004.4749.

[26]  Ting Zhang.  Global solutions of compressible Navier-Stokes equations with a density-dependent viscosity coefficient , 2009, 0904.1659.

[27]  Marco Cannone,et al.  A generalization of a theorem by Kato on Navier-Stokes equations , 1997 .

[28]  David Hoff,et al.  Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data , 1995 .

[29]  Raphaël Danchin,et al.  Global existence in critical spaces for compressible Navier-Stokes equations , 2000 .

[30]  Y. Meyer,et al.  Solutions auto-similaires des équations de Navier-Stokes , 1994 .

[31]  R. Danchin.  Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density , 2007 .

[32]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[33]  B. Haspot.  Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces , 2011 .

[34]  Raphaël Danchin,et al.  On the uniqueness in critical spaces for compressible Navier-Stokes equations , 2005 .

[35]  Isabelle Gallagher,et al.  Large, global solutions to the Navier-Stokes equations, slowly varying in one direction , 2007, 0710.5408.

[36]  Zhifei Zhang,et al.  ON THE ILL-POSEDNESS OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS , 2011, 1109.6092.

[37]  Isabelle Gallagher,et al.  Global regularity for some classes of large solutions to the Navier-Stokes equations , 2008, 0807.1265.

[38]  R. Danchin.  LOCAL THEORY IN CRITICAL SPACES FOR COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE GASES , 2001 .

[39]  Ping Zhang,et al.  On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations , 2007 .

[40]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[41]  Zhifei Zhang,et al.  Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities , 2008, 0811.4215.

[42]  Boris Haspot,et al.  Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids , 2010, 1005.0706.