RESEARCH ARTICLE

A combination of energy method and spectral analysis for study of equations of gas motion

  • Renjun DUAN , 1 ,
  • Seiji UKAI 2 ,
  • Tong YANG 3
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  • 1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria
  • 2. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, China
  • 3. Department of Mathematics, City University of Hong Kong, Hong Kong, China

Received date: 25 Jul 2008

Accepted date: 10 Mar 2009

Published date: 05 Jun 2009

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier- Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.

Cite this article

Renjun DUAN , Seiji UKAI , Tong YANG . A combination of energy method and spectral analysis for study of equations of gas motion[J]. Frontiers of Mathematics in China, 2009 , 4(2) : 253 -282 . DOI: 10.1007/s11464-009-0020-x

1
Deckelnick K. Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math Z, 1992, 209: 115-130

DOI

2
Deckelnick K. L2-decay for the compressible Navier-Stokes equations in unbounded domains. Comm Partial Differential Equations, 1993, 18: 1445-1476

DOI

3
Desvillettes L, Villani C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent Math, 2005, 159(2):245-316

DOI

4
Duan R -J. The Boltzmann equation near equilibrium states in ℝn. Methods and Applications of Analysis, 2007, 14(3): 227-250

5
Duan R -J. On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in Lξ2(HxN). Journal of Differential Equations, 2008, 244: 3204-3234

DOI

6
Duan R -J. Some Mathematical Theories on the Gas Motion under the Influence of External Forcing. Ph D Thesis. Hong Kong: City University of Hong Kong, 2008

7
Duan R -J, Liu H -X, Ukai S, Yang T. Optimal Lp-Lq convergence rates for the Navier-Stokes equations with potential force. Journal of Differential Equations, 2007, 238: 220-233

DOI

8
Duan R -J, Ukai S, Yang T, Zhao H -J. Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. Comm Math Phys, 2008, 277: 189-236

DOI

9
Duan R -J, Yang T, Zhu C -J. Navier-Stokes equations with degenerate viscosity, vacuum and gravitational force. Mathematical Methods in the Applied Sciences, 2007, 30: 347-374

DOI

10
Guo Y. The Boltzmann equation in the whole space. Indiana Univ Math J, 2004, 53: 1081-1094

DOI

11
Guo Y. Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm Pure Appl Math, 2006, 59: 626-687

DOI

12
Hoff D, Zumbrun K. Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves. Z angew Math Phys, 1997, 48: 597-614

DOI

13
Kawashima S. Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis. Kyoto: Kyoto University, 1983

14
Kobayashi T. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in ℝ3. J Differential Equations, 2002, 184:587-619

DOI

15
Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3. Commun Math Phys, 1999, 200: 621-659

DOI

16
Ladyzhenskaya O A. TheMathematical Theory of Viscous Incompressible Flow. 2nd English Ed (revised and enlarged). New York-London-Paris: Science Publishers, 1969, 1-224

17
Liu T -P, Wang W -K. The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions. Commun Math Phys, 1998, 196: 145-173

DOI

18
Liu T -P, Yang T, Yu S -H. Energy method for the Boltzmann equation. Physica D, 2004, 188(3-4): 178-192

DOI

19
Liu T -P, Yu S -H. Boltzmann equation: Micro-macro decompositions and positivity of shock profiles. Commun Math Phys, 2004, 246(1): 133-179

DOI

20
Liu T -P, Yu S -H. Diffusion under gravitational and boundary effects. Bull Inst Math Acad Sin (N S), 2008, 3: 167-210

21
Matsumura A, Nishida T. The initial value problems for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67-104

22
Matsumura A, Yamagata N. Global weak solutions of the Navier-Stokes equations for multidimensional compressible flow subject to large external potential forces. Osaka J Math, 2001, 38(2): 399-418

23
Nishida T, Imai K. Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ Res Inst Math Sci, 1976/77, 12: 229-239

DOI

24
Ponce G, Global existence of small solutions to a class of nonlinear evolution equations. Nonlinear Anal, 1985, 9: 339-418

DOI

25
Shibata Y, Tanaka K. Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid. Comput Math Appl, 2007, 53: 605-623

DOI

26
Shizuta Y. On the classical solutions of the Boltzmann equation. Commun Pure Appl Math, 1983, 36: 705-754

DOI

27
Strain R M. The Vlasov-Maxwell-Boltzmann system in the whole space. Commun Math Phys, 2006, 268(2): 543-567

DOI

28
Strain R M, Guo Y. Almost exponential decay near Maxwellian. Communications in Partial Differential Equations, 2006, 31: 417-429

DOI

29
Ukai S. On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proceedings of the Japan Academy, 1974, 50: 179-184

DOI

30
Ukai S. Les solutions globales de l’´equation de Boltzmann dans l’espace tout entier et dans le demi-espace. C R Acad Sci Paris, 1976, 282A(6): 317-320

31
Ukai S, Yang T. The Boltzmann equation in the space L2∩Lβ∞: Global and timeperiodic solutions. Analysis and Applications, 2006, 4: 263-310

DOI

32
Ukai S, Yang T, Zhao H -J. Global solutions to the Boltzmann equation with external forces. Analysis and Applications, 2005, 3(2): 157-193

DOI

33
Ukai S, Yang T, Zhao H -J. Convergence rate for the compressible Navier-Stokes equations with external force. J Hyperbolic Diff Equations, 2006, 3: 561-574

DOI

34
Ukai S, Yang T, Zhao H -J. Convergence rate to stationary solutions for Boltzmann equation with external force. Chinese Ann Math, Ser B, 2006, 27: 363-378

35
Yang T, Zhao H -J. A new energy method for the Boltzmann equation. Journal of Mathematical Physics, 2006, 47: 053301

DOI

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