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姓名	張育晟(Yo-Cheng Zhang)  查詢紙本館藏	畢業系所	數學系
論文名稱	Navier-Stokes 方程组弱解的存在性 (On The Existence Of Solutions For Navier-Stokes Equations)
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摘要(中)	在這篇研究報告中，我們仔細檢視了 P.L. Lions [Lio98] 與 E. Feireisl [FNP01] 對可壓縮等熵 Navier-Stokes 方程組弱解整體存在性的證明。Lions [Lio98] 證明了在三維空間中，當熱容比大於 9/5，而且初值所對應的動能與位能是有限的，則方程組的弱解在任意時間內都存在。之後，Feireisl把 Lions 的結果推廣到熱容比大於 3/2 的情形。本文按 E. Feireisl [FNP01] 與 A. Novotny [NS04] 的證明方式，試著給予一個比較詳細的過程。
摘要(英)	Abstract In this survey artical, we scrutinize the paper by P.L. Lions [Lio98] and E. Feireisl [FNP01] that contribute to the global in time existence of solutions for the compressible isentropic Navier-Stokes equations. If initial data has finite energy, Lions obtained global existence of weak solutions when the adiabatic constant gamma>9/5 . The result was later improved by Feireisl for gamma>3/2. This artical is intended to give some more details about the proofs of the global in time existence by E. Feireisl [FNP01] and A. Novotny [NS04].
關鍵字(中)	★ 可壓縮 ★ 整體存在性	關鍵字(英)	★ compressible ★ global in time existence ★ Navier-Stokes
論文目次	中文摘要.........................................................................................................i 英文摘要........................................................................................................ii Contents..........................................................................................................iii 1. Introduction..............................................................................................p.2 2. Main results............................................................................................p.11 3. Galerkin method.....................................................................................p.16 4. Compactness of weak solutions..............................................................p.23 5. Renormalized continuity equation..........................................................p.34 6. Vanishing articial pressure limit.............................................................p.39 References..................................................................................................p.57
參考文獻	[Eva98] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19, AMS, Providence, 1998. [FNP01] E. Feireisl, A.Novotny and H. Petzeltova, On the Existence of Globally Defined Weak Solutions to the Navier-Stokes Equations. J. Math. Fluid Mech. 3 (2001) 358–392. [Fei04a] E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J., 53(6):1705-1738, 2004. [Fei04b] E. Feireisl, Dynamics of viscous compressible fluids, Oxford University Press, Oxford, 2004. [FN09] E. Feireisl and A. Novotny, Singular Limits in Thermodynamics of Viscous Flu- ids, Birkhauser Verlag, Basel, 2009. [Gal94] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, I. Springer-Verlag, New York, 1994. [Hof87] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equa- tions with large initial data. Trans. Amer. Math. Soc., 303(1):169-181, 1987. [Hof95] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differential Equations, 120(1):215-254, 1995. [Hof98] D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states. Z. Angew. Math. Phys., 49(5):774-785, 1998. [HS01] D. Hoff and J. Smoller, Non-formation of vacuum states for com- pressible Navier-Stokes equations. Comm. Math. Phys., 216(2):255–276, 2001. [KS77] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh., 41(2):282-291, 1977. [KJF77] A. Kufner, O. John, and S. Fucik, Function Spaces, Academia, Prague., 1977. [Lio96] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible models, Oxford University Press, Oxford, 1996. [Lio98] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2, Compressible models, Oxford University Press, Oxford, 1998. [Lun95] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Prob- lems. Birkhauser, Basel, 1995. [MN79] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979. [Mur81] F. Murat, Compacite par compensation: condition necessaire et suffsante de continuite faible sous une hypothese de rang constant, Ann. Sc. Norm Super. Pisa, Cl. Sci., IV. Ser. 8 (1981), 69–102. [NS04] A. Novotny and I. Straskraba, Introduction to the mathematical theory of com- pressible flow. Oxford University Press, Oxford, 2004. [Rud91] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill Inc., New York, 1991. [Ser86a] D. Serre, Solutions faibles globales des equations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Ser. I Math., 303(13):639-642, 1986. [Ser86b] D. Serre, Sur l’equation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Ser. I Math., 303(14):703-706, 1986. [She82] V. V. Shelukhin, Motion with a contact discontinuity in a viscous heat conduct- ing gas. Dinamika Sploshn. Sredy, (57):131-152, 1982. [She83] V. V. Shelukhin, Evolution of a contact discontinuity in the barotropic flow of a viscous gas. Prikl. Mat. Mekh., 47(5):870-872, 1983. [She84] V. V. Shelukhin, On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas. Prikl. Mat. Mekh., 48(6):912-920, 1984. [She86] V. V. Shelukhin, Boundary value problems for equations of a barotropic viscous gas with nonnegative initial density. Dinamika Sploshn. Sredy, (74):108-125, 162-163, 1986. [Sim86] J. Simon, Compact sets in the space Lp (0; T;B). Ann. Mat. Pura Appl. 146 (1986), 65-96. [Sol76] V. A. Solonnikov, The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56:128-142, 197, 1976. Investigations on linear operators and theory of functions, VI. [Ste70] E.M. Stein, Singular integrals and differential properties of functions. Princeton University Press, Princeton, 1970. [Tem01] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Ameri- can Mathematical Society, 2001. [Zie89] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, 1989.
指導教授	鄭經斅(Ching-hsiao Cheng)	審核日期	2011-7-28
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