Acta mathematica scientia, Series B >
GLOBAL CLASSICAL SOLUTIONS TO THE 3-D ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL INITIAL ENERGY
Received date: 2011-07-04
Online published: 2012-11-20
Supported by
This work was partially supported by National Natural Science Foundation of China (11001090) and the Fundamental Research Funds for the Central Universities (11QZR16).
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ||ρ0||L∞ is appropriate small and 1 << 6/ 5 . Here the initial density could have vacuum and we do not require that the initial energy is small.
ZHANG Pei-Xin , DENG Xue-Mei , ZHAO Jun-Ning . GLOBAL CLASSICAL SOLUTIONS TO THE 3-D ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL INITIAL ENERGY[J]. Acta mathematica scientia, Series B, 2012 , 32(6) : 2141 -2160 . DOI: 10.1016/S0252-9602(12)60166-0
[1] Hoff D. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303(1): 169–181
[2] Kazhikov 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
[3] Serre D. Solutions faibles globales des ´equations de Navier-Stokes pour un fluide compressible. C R Acad Sci Paris S´er I Math, 1986, 303: 639–642
[4] Serre D. 1´équation monodimensionnelle d´un fluide visqueux, compressible et conducteur de chaleur. C R Acad Sci Paris S´er I Math, 1986, 303: 703–706
[5] Nash J. Le problème de Cauchy pour les équations diff´erentielles d´un fluide g´en´eral. Bull Soc Math France, 1962, 90: 487–497
[6] Serrin J. On the uniqueness of compressible fluid motion. Arch Rational Mech Anal, 1959, 3: 271–288
[7] Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluid. J Math Pures Appl, 2004, 83: 243–275
[8] Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative intial densities. Manuscript Math, 2006, 120: 91–129
[9] Cho Y, Kim H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differ Eqs, 2003, 190: 504–523
[10] Salvi R, Straskraba I. Global existence for viscous compressible fluids and their behavior as t ! 1. J Fac Sci Univ Tokyo Sect IA Math, 1993, 40: 17–51
[11] 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(1): 67–104
[12] Hoff D. Global solutions of the Navier-Stokes equations for multidimendional compressible flow with dis-constinuous initial data. J Differ Eqs, 1995, 120(1): 215–254
[13] Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations fo state and discontinuous initial data. Arch Rational Mech Anal, 1995, 132: 1–14
[14] Lions P L. Mathematical Topics in Fluid Mechanics. Vol 2. Compressible Models. New York: Oxford University Press, 1998
[15] Feiresl E, Novotny A, Petzeltov´a H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3(4): 358–392
[16] Zhang T, Fang D Y. Compressible flows with a density-dependent viscosity coefficient. SIAM J Math Anal, 2010, 41(6): 2453–2488
[17] Zhang T. Global solutions of compressible barotropic Navier-Stokes equations with a density-dependent viscosity coefficient. J Math Phys, 2011, 52(4): 043510
[18] 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. Comm Pure Appl Math, 2012, 65(4): 549–585
[19] Deng X M, Zhang P X, Zhao J N. Global Well-posedness of Classical Solutions with Large Initial data and Vacuum to the Three-dimensional isentropic compressible Navier-Stokes equations. Preprint
[20] 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
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