Abstract
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in the three space dimensions with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient µ is large enough.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluid. J Math Pure Appl, 2004, 83: 243–275
Cho Y, Kim H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J Differential Equations, 2003, 190: 504–523
Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscript Math, 2006, 120: 91–129
Hoff D. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303: 169–181
Hoff D. Global solutions of the Navier-Stokes equations for multidimendional compressible flow with discontinuous initial data. J Differential Equations, 1995, 120: 215–254
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
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: 549–585
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
Kong D X, Liu K F, Wang Y Z. Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases. Sci China Math, 2010, 53: 719–738
Ladyzenskaja O A, Solonnikov V V, Ural’ceva N N. Linear and Quasilinear Equatons of Parabolic Type. Providence, RI: Amer Math Soc, 1968
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
Nash J. Le problème de Cauchy pour leséquations différentielles d’un fluide général. Bull Soc Math France, 1962, 90: 487–497
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
Serre D. 1’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C R Acad Sci Paris Sér I Math, 1986, 303: 703–706
Serrin J. On the uniqueness of compressible fluid motion. Arch Rational Mech Anal, 1959, 3: 271–288
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
Sun Y Z, Wang C, Zhang Z F. A Beale-Kato-Majda blow-up criterion criterion for 3-D compressible Navier-Stokes equations. J Math Pure Appl, 2011, 95: 36–47
Zhao W J, Du D P. Local well-posedness of lower regularity solutions for the incompressible viscoelastic fluid system. Sci China Math, 2010, 53: 1521–1530
Zlotnik A A. Uniform estimates and stabilization of symmetric solutions of a system of quaslinear equations. Differential Equations, 2000, 36: 701–716
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Deng, X., Zhang, P. & Zhao, J. Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data. Sci. China Math. 55, 2457–2468 (2012). https://doi.org/10.1007/s11425-012-4481-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4481-0