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The Compressible Navier-Stokes Equations with Weak Viscosity and Heat Conductivity
Department of Mathematics, Jinan University, Guangzhou, China
Department of Mathematics, Jinan University, Guangzhou, China
Department of Mathematics, Jinan University, Guangzhou, China
- 1 Department of Mathematics, Jinan University, Guangzhou, China
- 2 Department of Mathematics, Jinan University, Guangzhou, China
- 3 Department of Mathematics, Jinan University, Guangzhou, China
American Journal of Computational Mathematics·Volume 09 (2019)·Pages 32–47·Published 8 April 2019·DOI10.4236/ajcm.2019.92003
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Abstract
It is well known that the full compressible Navier-Stokes equations with viscosity and heat conductivity coefficients of order of the Knudsen number ò>0 can be deduced from the Boltzmann equation via the Chapman-Enskog expansion. In this paper, we carry out the rigorous mathematical study of the compressible Navier-Stokes equation with the initial-boundary value problems. We construct the existence and most importantly obtain the higher regularities of the solutions of the full compressible Navier-Stokes system with weak viscosity and heat conductivity in a general bounded domain.
KeywordsCompressible Navier-Stokes SystemEnergy Estimatethe Helmholtz DecompositionElliptic Estimatesthe Galerkin Method
- Galdi, G.P. (2012) Navier-Stokes Equations: A Mathematical Analysis. In: Meyers, R., Eds., Mathematics of Complexity and Dynamical Systems, Spring, New York, NY.
- Bardos, C., Golse, F. and Levermore, D.C. (1991) Fluid Dynamic Limits of the Kinetic Equation. I. Formal Derivation. Journal of Statistical Physics, 63, 323-344. https://doi.org/10.1007/BF01026608
- Liu, S.-Q., Yang, T. and Zhao, H.-J. (2014) Compressible Navier-Stokes Approximation to the Boltzmann Equation. Journal of Differential Equations, 256, 3770-3816. https://doi.org/10.1016/j.jde.2014.02.020
- Duan, R.-J. and Liu, S.-Q. Compressible Navier-Stokes Approximation for the Boltzmann Equation in Bounded Domains. arXiv:1806.09796
- Wang, Y. (2016) Uniform Regularity and Vanishing Dissipation Limit for the Full Compressible Navier-Stokes System in Three Dimensional Bounded Domain. Archive for Rational Mechanics and Analysis, 221, 1345-1415. https://doi.org/10.1007/s00205-016-0989-8
- Wang, Y., Xin, Z.-P. and Yong, Y. (2015) Uniform Regularity and Vanishing Viscosity Limit for the Compressible Navier-Stokes with General Navier-Slip Boundary Conditions in Three-Dimensional Domains. SIAM Journal on Mathematical Analysis, 47, 4123-4191. https://doi.org/10.1137/151003520
- Masmoudi, N. and Rousset, F. (2017) Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier-Stokes Equations. Archive for Rational Mechanics and Analysis, 223, 301-417. https://doi.org/10.1007/s00205-016-1036-5
- Matsumura, A. and Nishida, T. (1980) The Initial Value Problem for the Equations of Motion of Viscous and Heat-Conductive Gases. Journal of Mathematics of Kyoto University, 26, 67-104. https://doi.org/10.1215/kjm/1250522322
- Matsumura, A. and Nishida, T. (1983) Initial-Boundary Value Problems for the Equations of Motion of Compressible Viscous and Heat-Conductive Fluids. Communications in Mathematical Physics, 89, 445-464. https://doi.org/10.1007/BF01214738
- Huang, X.-D., Li, J. and Xin, Z.-P. (2012) Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations. Communications on Pure and Applied Mathematics, 65, 549-585. https://doi.org/10.1002/cpa.21382
- Ukai, S., Yang, T. and Zhao, H.-J. (2006) Convergence Rate for the Compressible Navier-Stokes Equations with External Force. Journal of Hyperbolic Differential Equations, 3, 561-574. https://doi.org/10.1142/S0219891606000902