Boltzmann方程的Tjon-Wu模型
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摘要
Boltzmann方程是从统计层次描述稀薄气体中粒子的位置和速度的分布函数时空演化的数学模型，具有很强的实用性。由于碰撞算子的复杂性，此类方程的研究相当困难。通过对空间齐次的Boltzmann方程进行Abel变换，人们得到了形式简单的Tjon-Wu方程([1]-[4])或者说是Boltzmann方程的Tjon-Wu模型。人们已对这类方程做了大量的研究(例如：[5]-[11])。本文研究由A．Lasola[11]引入的一类广义的Tjon-Wu方程。
首先，我们考虑方程的Cauchy问题：1．利用半群知识，我们建立了Cauchy问题的解的存在唯一性。2．我们利用不动点的性质和基本的矩不等式，运用逐次逼近法得到了在一定条件下Cauchy问题的解的严格正性及高阶矩估计。
其次，我们讨论了方程的定态性质：1．我们通过验证Schauder不动点定理的条件，证明了稳态解的存在性。2．我们利用算子ρ的正性及不动点的性质证明了稳态解的严格正性。3．我们证明了此类方程的稳态解的唯一性及渐近稳定性(A．Lasola[11]已证明了这类广义的Tjon-Wu方程的稳态解在弱拓扑意义下是指数稳定的)。在这里，稳态解的正性对渐近稳定性的证明起了重要作用，它使我们可以利用Kantorovich-Rubinstein最大值原理来说明稳态解在强拓扑意义下是全局渐近稳定的。4．我们利用对偶及嵌入的方法得到在特殊情况下稳态解是C~∞的。
Boltzmann equation is the practically mathematical model to describe the time and space evolution, at a statistical level, of the position and velocity distribution function of particles in a dilute gas. However in view of the complication of the collision operator the study of the equation is very difficult. Tjon—Wu equation([l]—[4]) or Tjon—Wu model of the Boltzmann equation with easier form was derived from the spatially homogeneous Boltzmann equation by use of the Abel transformation. Several authors investigated the equation(see, e.g.:[5]—[11]).This paper studies a generalized Tjon—Wu equation proposed by A.Lasola[11].
First,the Cauchy problem for the equation is considered. 1. We obtain the existence and uniqueness for the solution of the Cauchy problem by means of semigroup theory. 2. We show positivity and estimates of moments of senior order for the solution of the Cauchy problem in some special conditions by using the properties of fixed point ,the basic moment inequality and the method of successive approximations.
Second, we discuss the stationary problem for the equation. 1. We establish the existence of the stationary solution through verifying the condition of the Schauder fixed point theorem. 2. We show that the stationary solution is positive by means of positivity of the operator P and the properties of fixed point. 3. We obtain the uniqueness and the asymptotic stability of the stationary solution (A.Lasola [11] has proved that the stationary solution for the equation is exponentially stable in weak topology ). The positivity of the stationary solution plays an important role in the proof of stability. It allows us to apply the Kantorovich—Rubinstein maximum principle to prove that the stationary solution is globally asymptotically stable in strong topology. 4. We show by dulity and embedding method that the stationary solution is C~∞in some special conditions.
首先，我们考虑方程的Cauchy问题：1．利用半群知识，我们建立了Cauchy问题的解的存在唯一性。2．我们利用不动点的性质和基本的矩不等式，运用逐次逼近法得到了在一定条件下Cauchy问题的解的严格正性及高阶矩估计。
其次，我们讨论了方程的定态性质：1．我们通过验证Schauder不动点定理的条件，证明了稳态解的存在性。2．我们利用算子ρ的正性及不动点的性质证明了稳态解的严格正性。3．我们证明了此类方程的稳态解的唯一性及渐近稳定性(A．Lasola[11]已证明了这类广义的Tjon-Wu方程的稳态解在弱拓扑意义下是指数稳定的)。在这里，稳态解的正性对渐近稳定性的证明起了重要作用，它使我们可以利用Kantorovich-Rubinstein最大值原理来说明稳态解在强拓扑意义下是全局渐近稳定的。4．我们利用对偶及嵌入的方法得到在特殊情况下稳态解是C~∞的。
Boltzmann equation is the practically mathematical model to describe the time and space evolution, at a statistical level, of the position and velocity distribution function of particles in a dilute gas. However in view of the complication of the collision operator the study of the equation is very difficult. Tjon—Wu equation([l]—[4]) or Tjon—Wu model of the Boltzmann equation with easier form was derived from the spatially homogeneous Boltzmann equation by use of the Abel transformation. Several authors investigated the equation(see, e.g.:[5]—[11]).This paper studies a generalized Tjon—Wu equation proposed by A.Lasola[11].
First,the Cauchy problem for the equation is considered. 1. We obtain the existence and uniqueness for the solution of the Cauchy problem by means of semigroup theory. 2. We show positivity and estimates of moments of senior order for the solution of the Cauchy problem in some special conditions by using the properties of fixed point ,the basic moment inequality and the method of successive approximations.
Second, we discuss the stationary problem for the equation. 1. We establish the existence of the stationary solution through verifying the condition of the Schauder fixed point theorem. 2. We show that the stationary solution is positive by means of positivity of the operator P and the properties of fixed point. 3. We obtain the uniqueness and the asymptotic stability of the stationary solution (A.Lasola [11] has proved that the stationary solution for the equation is exponentially stable in weak topology ). The positivity of the stationary solution plays an important role in the proof of stability. It allows us to apply the Kantorovich—Rubinstein maximum principle to prove that the stationary solution is globally asymptotically stable in strong topology. 4. We show by dulity and embedding method that the stationary solution is C~∞in some special conditions.
引文
[1] Bobylev A. V. . Exact solutions of Boltzmann equation. Soviet Phys. Dokl. , 1976, 20: 822-824
[2] Krook M. , Wu T. T. . Exact solutions of the Tjon-Wu equation. Phys. Fluids , 1977, 20: 1589-1595
[3] Tjon J. A. , Wu T. T. . Numerical aspects of the approach to a Maxwellian distribution. Phys. Rev. A. , 1979, 19: 883-888
[4] Barnsley M. F. , Cornille H. . General solution of a Boltzmann equation and the formation of Maxwellian tails. Proc. Roy. Soc. London Ser. A , 1981, 374: 371—400
[5] Krook M. , Wu T. T. . Formation of Maxwellian Tails. Phys. Rev. Lett. , 1976, 36: 1107-1109
[6] Barnsley M. F. , Turchetti G. . New Results on the Nonlinear Boltzmann Equations (C. Bardos and D. Bessis, Eds. ), Bifurcation Phenomena in Mathematical Physics and Related Topics. Boston: Reidel, 1980.
[7] Dlotko T. , Lasota A. . On the Tjon—Wu representation of the Boltzmann equation. Ann. Polon. Math. , 1983, 421: 73-82
[8] Kielek Z. . Asymptotic behaviour of solutions to the Tjon—Wu equation. Ann. Polon. Math. , 1990, 52: 109-118
[9] Lasota A. , Traple J. . An application of the Kantorovich—Rubinstein maximum principle in the theory of the Tjon—Wu equation. J. Differential Equs. , 1999, 159: 578-596
[10] Carraffini G. L. , Iori M. , Spiga G. . The Tjon—Wu model in extended kinetic theory: Exact stationary solutions and trend to equilibrium. Riv. Mat. Univ. Parma, V. Ser. , 1993, 2: 83-92
[11] Lasota A. . Asymptotic stability of some nonlinear Boltzmann—type equation. J. Math. Anal. Appl. , 2002, 268: 291-309
[12] Boltzmann L. . Weitere studien uber das Warmegleichgewicht unter Gasmolekulen.Sitzungsberichta Akad. Wiss. Vienner, part II, 1872. 275—370
[13] Cercignani C. . Mathematical Methods in Kinetic Theory. New York: Plenum Press, 1969.
[14] Truesdell C. , Muncaster R. G. . Fundaments of Maxwell's Kinetic Theory of a Simple Monoatomic Gas. New York: Academic Press, 1980.
[15] Cercignani C. . The Boltzmann equation and Its Applications. New York: Springer,1988.
[16] Cercignani C. , Illner R. , Pulvirenti M. . The Mathematical Theory of Dilute Gases. New York: Springer, 1994.
[17] Barnsley M. F. , Turchetti G. . On the Abel Transformation and the nonlinear Boltzmann equestion. Phys. Lett. A , 1979, 72: 417—419
[18] Krook M. , Wu T. T. . Exact solution of Boltzmann Equations for Multicomponent Systems. Phys. Rev. Lett. , 1977, 38: 991-993
[19] Barnsley M. F. , Cornille H. . On a class of solutions of the Krook—Tjon—Wu model of the Boltzmann equation. J. Math. Phys. , 1980, 21: 1176-1193
[20] Barnsley M. F. , Herod J. V. , Jory V. V. , et al.The Tjon-Wu equation in Banach space settings. J. Funct. Anal. , 1981, 43: 32-51
[21] Turchetti G. . On the structure of the generralized Tjon—Wu equations. J. Math. Phys. , 1982, 23: 639-645
[22] Barnsley M. F. , Herod J. V. , Mosher D. L. , et al. Analysis of Boltzmann equationsin Hilbert space by means of a non—linear eigenvalue property. Spectral theory of differential operators, Proc. Conf., Birmingham/ USA 1981, North-Holland Math. Stud. , 1981, 55: 45-52
[23] Barnsley M. F. , Herod J. V. , Mosher D. L. , et al. Solutions for a model Boltzmann equation by monotonicity methods. Houston J. Math. , 1983, 9: 345—355
[24] Malczak J. . An application of Markov operators in differential and integral equations. Rend. Semin. Mat. Univ. Padova, 1992, 87: 281-297
[25] Grad H. . Asymptomic theory of the Boltzmann equation II; in Rarefied Gas Dynamics, Vol.1, Ed. J. Laurmann. New York: Academic Press, 1963. 26—59
[26] Arkeryd L. , Espositio R. , Pulvirenti M. . The Boltzmann equation for weakly inho- mogeneous data. Comm. Math. Phys. , 1987, 111: 393-407
[27] Arkeryd L. . Stability in L~1 for the spatially homogeneous Boltzmann equation. Arch Rational Mech. Anal. , 1988, 103: 151-167
[28]Diperna R. , Lions P. -L. .On the Cauchy problem for the Boltzmann equation: Globalexistence and weak stability results. Ann. Math. , 1989, 130: 321—366
[29]Wennberg B..Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonl. Analysis T. 54. A. , 1993, 20:935～964
[30]Wennberg B.. Stability and exponential convergence for the Boltzmann equation,thesis. Chalmers University of Technology, 1993.
[31]Wennberg B.. Stability and exponential convergence for Boltzmann equation. Ration Mech.Anal.,1995, 130:103～144
[32]Prohorov Yu. V.. Convergence of random processes and limit theorems in probability theory. Theory Probab.Appl.,1956,1:157～214
[33]Crandall 54. G.. Differential equations on convex sets. J. Math. Soc. Japan , 1970,22:443～455
[34]Dudley R. 54.. Probbabilities and Metrics. Lecture Notes Aarhus Universitet, Vol.45, 1976.
[35]Zolotarev V. M.. Ideal metrics in the problems of approximating the distributions of sums of independent random variables. Theory Probab. Appl.,1977, 22:433～449
[36]Zolotarev V. M..Ideal metrics in the problems of probability theory.Austral.J.Statist.,1979, 21:193～208
[37]Ethier S. N.,Kurtz T. G..Markov Processes,Characterization and Convergence.New York: Wiley,1986.
[38]Lasota A.,Traple J..Asymptotic stability of differential equations on convex sets. J.Dyn. Differ. Equations,2003, 15:335～355
[39]Barnsley 54. F..Fractals Everywhere. Boston: Academic Press,1993.
[40]Dunford N.,Schwartz J. T.. Linear Operators, I. General Theory. New York: Interscience, 1958.
[41]Kuczma M.,Choczewski B.,Ger R..Iterative Functional Equetions.Cambridge:Cambridge Univ. Press,1990.
[42]Rachev S.T..Probability Metrics and Stability of Stochastic Models. New York:Wiley, 1991.
[43]Adams R．A．．索伯列夫空间．叶其孝等译．北京：人民教育出版社，1981．106，114～115
[44]Barros-Neto J．．广义函数引论．欧阳光中等译．上海：上海科学技术出版社，1981．32
[2] Krook M. , Wu T. T. . Exact solutions of the Tjon-Wu equation. Phys. Fluids , 1977, 20: 1589-1595
[3] Tjon J. A. , Wu T. T. . Numerical aspects of the approach to a Maxwellian distribution. Phys. Rev. A. , 1979, 19: 883-888
[4] Barnsley M. F. , Cornille H. . General solution of a Boltzmann equation and the formation of Maxwellian tails. Proc. Roy. Soc. London Ser. A , 1981, 374: 371—400
[5] Krook M. , Wu T. T. . Formation of Maxwellian Tails. Phys. Rev. Lett. , 1976, 36: 1107-1109
[6] Barnsley M. F. , Turchetti G. . New Results on the Nonlinear Boltzmann Equations (C. Bardos and D. Bessis, Eds. ), Bifurcation Phenomena in Mathematical Physics and Related Topics. Boston: Reidel, 1980.
[7] Dlotko T. , Lasota A. . On the Tjon—Wu representation of the Boltzmann equation. Ann. Polon. Math. , 1983, 421: 73-82
[8] Kielek Z. . Asymptotic behaviour of solutions to the Tjon—Wu equation. Ann. Polon. Math. , 1990, 52: 109-118
[9] Lasota A. , Traple J. . An application of the Kantorovich—Rubinstein maximum principle in the theory of the Tjon—Wu equation. J. Differential Equs. , 1999, 159: 578-596
[10] Carraffini G. L. , Iori M. , Spiga G. . The Tjon—Wu model in extended kinetic theory: Exact stationary solutions and trend to equilibrium. Riv. Mat. Univ. Parma, V. Ser. , 1993, 2: 83-92
[11] Lasota A. . Asymptotic stability of some nonlinear Boltzmann—type equation. J. Math. Anal. Appl. , 2002, 268: 291-309
[12] Boltzmann L. . Weitere studien uber das Warmegleichgewicht unter Gasmolekulen.Sitzungsberichta Akad. Wiss. Vienner, part II, 1872. 275—370
[13] Cercignani C. . Mathematical Methods in Kinetic Theory. New York: Plenum Press, 1969.
[14] Truesdell C. , Muncaster R. G. . Fundaments of Maxwell's Kinetic Theory of a Simple Monoatomic Gas. New York: Academic Press, 1980.
[15] Cercignani C. . The Boltzmann equation and Its Applications. New York: Springer,1988.
[16] Cercignani C. , Illner R. , Pulvirenti M. . The Mathematical Theory of Dilute Gases. New York: Springer, 1994.
[17] Barnsley M. F. , Turchetti G. . On the Abel Transformation and the nonlinear Boltzmann equestion. Phys. Lett. A , 1979, 72: 417—419
[18] Krook M. , Wu T. T. . Exact solution of Boltzmann Equations for Multicomponent Systems. Phys. Rev. Lett. , 1977, 38: 991-993
[19] Barnsley M. F. , Cornille H. . On a class of solutions of the Krook—Tjon—Wu model of the Boltzmann equation. J. Math. Phys. , 1980, 21: 1176-1193
[20] Barnsley M. F. , Herod J. V. , Jory V. V. , et al.The Tjon-Wu equation in Banach space settings. J. Funct. Anal. , 1981, 43: 32-51
[21] Turchetti G. . On the structure of the generralized Tjon—Wu equations. J. Math. Phys. , 1982, 23: 639-645
[22] Barnsley M. F. , Herod J. V. , Mosher D. L. , et al. Analysis of Boltzmann equationsin Hilbert space by means of a non—linear eigenvalue property. Spectral theory of differential operators, Proc. Conf., Birmingham/ USA 1981, North-Holland Math. Stud. , 1981, 55: 45-52
[23] Barnsley M. F. , Herod J. V. , Mosher D. L. , et al. Solutions for a model Boltzmann equation by monotonicity methods. Houston J. Math. , 1983, 9: 345—355
[24] Malczak J. . An application of Markov operators in differential and integral equations. Rend. Semin. Mat. Univ. Padova, 1992, 87: 281-297
[25] Grad H. . Asymptomic theory of the Boltzmann equation II; in Rarefied Gas Dynamics, Vol.1, Ed. J. Laurmann. New York: Academic Press, 1963. 26—59
[26] Arkeryd L. , Espositio R. , Pulvirenti M. . The Boltzmann equation for weakly inho- mogeneous data. Comm. Math. Phys. , 1987, 111: 393-407
[27] Arkeryd L. . Stability in L~1 for the spatially homogeneous Boltzmann equation. Arch Rational Mech. Anal. , 1988, 103: 151-167
[28]Diperna R. , Lions P. -L. .On the Cauchy problem for the Boltzmann equation: Globalexistence and weak stability results. Ann. Math. , 1989, 130: 321—366
[29]Wennberg B..Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonl. Analysis T. 54. A. , 1993, 20:935～964
[30]Wennberg B.. Stability and exponential convergence for the Boltzmann equation,thesis. Chalmers University of Technology, 1993.
[31]Wennberg B.. Stability and exponential convergence for Boltzmann equation. Ration Mech.Anal.,1995, 130:103～144
[32]Prohorov Yu. V.. Convergence of random processes and limit theorems in probability theory. Theory Probab.Appl.,1956,1:157～214
[33]Crandall 54. G.. Differential equations on convex sets. J. Math. Soc. Japan , 1970,22:443～455
[34]Dudley R. 54.. Probbabilities and Metrics. Lecture Notes Aarhus Universitet, Vol.45, 1976.
[35]Zolotarev V. M.. Ideal metrics in the problems of approximating the distributions of sums of independent random variables. Theory Probab. Appl.,1977, 22:433～449
[36]Zolotarev V. M..Ideal metrics in the problems of probability theory.Austral.J.Statist.,1979, 21:193～208
[37]Ethier S. N.,Kurtz T. G..Markov Processes,Characterization and Convergence.New York: Wiley,1986.
[38]Lasota A.,Traple J..Asymptotic stability of differential equations on convex sets. J.Dyn. Differ. Equations,2003, 15:335～355
[39]Barnsley 54. F..Fractals Everywhere. Boston: Academic Press,1993.
[40]Dunford N.,Schwartz J. T.. Linear Operators, I. General Theory. New York: Interscience, 1958.
[41]Kuczma M.,Choczewski B.,Ger R..Iterative Functional Equetions.Cambridge:Cambridge Univ. Press,1990.
[42]Rachev S.T..Probability Metrics and Stability of Stochastic Models. New York:Wiley, 1991.
[43]Adams R．A．．索伯列夫空间．叶其孝等译．北京：人民教育出版社，1981．106，114～115
[44]Barros-Neto J．．广义函数引论．欧阳光中等译．上海：上海科学技术出版社，1981．32
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