Duan, Renjun; Ukai, Seiji; Yang, Tong A combination of energy method and spectral analysis for study of equations of gas motion. (English) Zbl 1180.35103 Front. Math. China 4, No. 2, 253-282 (2009). Summary: 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 first illustrate this method by using a simple model and then 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 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. Cited in 13 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35Q30 Navier-Stokes equations 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 35B35 Stability in context of PDEs 35Q20 Boltzmann equations Keywords:linearization; asymptotic stability; optimal rate; non-trivial steady state; time-space dependent forces; Vlasov-Poisson-Boltzmann system PDFBibTeX XMLCite \textit{R. Duan} et al., Front. Math. China 4, No. 2, 253--282 (2009; Zbl 1180.35103) Full Text: DOI References: [1] Deckelnick K. Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math Z, 1992, 209:115–130 · Zbl 0752.35048 [2] Deckelnick K. L 2-decay for the compressible Navier-Stokes equations in unbounded domains. Comm Partial Differential Equations, 1993, 18:1445–1476 · Zbl 0798.35124 [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 · Zbl 1162.82316 [4] Duan R-J. The Boltzmann equation near equilibrium states in \(\mathbb{R}\)n. 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 L {\(\xi\)} 2 (H x N ). Journal of Differential Equations, 2008, 244:3204–3234 · Zbl 1150.82019 [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 L p-L q convergence rates for the Navier-Stokes equations with potential force. Journal of Differential Equations, 2007, 238:220–233 · Zbl 1121.35096 [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 · Zbl 1175.82047 [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 · Zbl 1128.35082 [10] Guo Y. The Boltzmann equation in the whole space. Indiana Univ Math J, 2004, 53:1081–1094 · Zbl 1065.35090 [11] Guo Y. Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm Pure Appl Math, 2006, 59:626–687 · Zbl 1089.76052 [12] Hoff D, Zumbrun K. Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves. Z angew Math Phys, 1997, 48:597–614 · Zbl 0882.76074 [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 \(\mathbb{R}\)3. J Differential Equations, 2002, 184:587–619 · Zbl 1069.35051 [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 \(\mathbb{R}\)3. Commun Math Phys, 1999, 200:621–659 · Zbl 0921.35092 [16] Ladyzhenskaya O A. The Mathematical 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 · Zbl 0912.35122 [18] Liu T-P, Yang T, Yu S-H. Energy method for the Boltzmann equation. Physica D, 2004, 188(3–4):178–192 · Zbl 1098.82618 [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 · Zbl 1092.82034 [20] Liu T-P, Yu S-H. Diffusion under gravitational and boundary effects. Bull Inst Math Acad Sin (N S), 2008, 3:167–210 · Zbl 1388.76337 [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 · Zbl 0429.76040 [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 · Zbl 0980.35128 [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 · Zbl 0344.35003 [24] Ponce G. Global existence of small solutions to a class of nonlinear evolution equations. Nonlinear Anal, 1985, 9:339–418 · Zbl 0576.35023 [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 · Zbl 1122.76077 [26] Shizuta Y. On the classical solutions of the Boltzmann equation. Commun Pure Appl Math, 1983, 36:705–754 · Zbl 0522.35004 [27] Strain R M. The Vlasov-Maxwell-Boltzmann system in the whole space. Commun Math Phys, 2006, 268(2):543–567 · Zbl 1129.35022 [28] Strain R M, Guo Y. Almost exponential decay near Maxwellian. Communications in Partial Differential Equations, 2006, 31:417–429 · Zbl 1096.82010 [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 · Zbl 0312.35061 [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 · Zbl 0345.45012 [31] Ukai S, Yang T. The Boltzmann equation in the space L 2 L {\(\beta\)} : Global and timeperiodic solutions. Analysis and Applications, 2006, 4: 263–310 · Zbl 1096.35012 [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 · Zbl 1152.76464 [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 · Zbl 1184.35251 [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 · Zbl 1151.76571 [35] Yang T, Zhao H-J. A new energy method for the Boltzmann equation. Journal of Mathematical Physics, 2006, 47:053301 · Zbl 1111.82048 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.