On the Global Well-Posedness of the 3D Compressible Navier-Stokes-Fourier Equations with General Pressure Law

Abstract

For the general compressible 3D compressible Navier-Stokes-Fourier (NSF) equations, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state. The main novelties of this paper are: (1) The NSF equations we study are fully general in the sense that we only assume the most basic thermodynamic conditions, i.e., the pressure and internal energy are monotone with respect to the density and temperature, respectively. (2) We fully employ the entropy structure and illustrate its role played in the analytical study of the compressible NSF.

Keywords

Compressible Navier-Stokes system / Littlewood-Paley theory / Critical spaces / 35Q30 / 76D03

References

[1] Alinhac S. Paracomposition et opérateurs paradifférentiels. Comm. Partial Diff. Equ.. 1986, 11: 87-121.

[2] Bahouri H, Chemin JY, Danchin R. Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematischen Wissenschaften 343. 2011, Berlin Heidelberg, Springer-Verlag.

[3] Bony JM. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup.. 1981, 14(4): 209-246.

[4] Chemin JY. Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. Anal. Math.. 1999, 77: 27-50.

[5] Chemin, J.Y., Lerner, N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Diff. Equ. 121, 314–328 (1995)

[6] Danchin R.: Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae, 141 (2000), 579–614

[7] Danchin R.: Global existence in critical spaces for flows of compressible viscous and heat-conductive gase, Arch. Rational Mech. Anal., 160 (2001), 1–39

[8] Danchin R. Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A. 2003, 133: 1311-1334.

[9] Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math.. 2000, 141: 579-614.

[10] Danchin R. Global existence in critical spaces for flows of compressible viscous and heat-conductive gase. Arch. Rational Mech. Anal.. 2001, 160: 1-39.

[11] Danchin R. Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math.. 2002, 124(6): 1153-1219.

[12] Danchin R. Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A. 2003, 133: 1311-1334.

[13] Danchin R. Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Diff. Equ.. 2004, 9: 353-386

[14] Danchin R. Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoam.. 2005, 21(3): 861-886.

[15] Danchin R. The inviscid limit for density-dependent incompressible fluids. Anneles de la Faculté de Sciences des Toulouse Sér.. 2006, 6(15): 637-688

[16] Fujita H., Kato T.: On the Navier-Stokes initial value problem I, Archiv for Rational Mechanic Analysis, 16 (1964), 269–315

[17] Fujita H, Kato T. On the Navier-Stokes initial value problem I, Archiv for Rational Mechanic. Analysis. 1964, 16: 269-315

[18] Jiang N, Levermore CD. Weakly nonlinear-dissipative approximations of hyperbolic-parabolic systems with entropy. Arch. Ration. Mech. Anal.. 2011, 201(2): 377-412.

[19] Jiang N, Levermore CD. Global existence and regularity of the weakly compressible Navier-Stokes system. Asymptot. Anal.. 2012, 76(2): 61-86

[20] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.. 1980, 20: 67-104

[21] Nash J. Le problème de Cauchy pour les équations différentielles d’un fluide général. Bulletin de la Soc. Math. de France. 1962, 90: 487-497.

[22] Serrin, J.: On the uniqueness of compressible fluid motions. Arch. Rational Mech. Anal. 3, 271–288 (1959)

[23] Triebel H.: Theory of Function Spaces. Monograph in mathematics, Vol.78 (1983), Birkhauser Verlag, Basel

[24] Zhang P.: Global smooth solutions to the 2-D nonhomogeneous Navier-Stokes equations, Int. Math. Res. Not. IMRN , 2008, Art. ID rnn 098, 26 pp

Funding

National Natural Science Foundation of China(12371211)

National Natural Science Foundation of China(11731008)

RIGHTS & PERMISSIONS

The Author(s)