一类带有非牛顿位势的正则化Vlasov方程初边值问题
摘要
本文主要研究一类带有非牛顿位势的正则化Vlasov方程探讨了这类方程满足下列条件的初边值问题其中f是在时刻t,在(x,v)处单位体积及单位速度变化范围中的粒子分布函数.φ为非牛顿重力位势.ε>0,(?)I×(-L,L),I=(0,1),x∈I,v∈(-L,L).
本文重点研究上述问题弱解的存在性.
The kinetic theory of the gas is also known as compressible fluid dynamics. It is research the fluid dynamics of variable density or fluid played a significant compression.
In the macroscopic scales where the gas and fluid are regarded as a continuum, their motion is described by the macroscopic quantities such as macroscopic mass density, bulk velocity, temperature, pressure, stresses, heat flux and so on. The Euler and Navier-Stokes equations, compressible or incompressible, are the most famous equations among governing equations proposed so far in fluid dynamics.
The extreme contrary is the microscopic scale where the gas, fluid, and hence any matter, are looked at as a many-body system of microscopic particles. Thus, the motion of the system is governed by the coupled Newton equations, within the framework of the classical mechanics. Although the Newton equation is the first principle of the classical mechanics, it is not of practical use because the number of the equations is so enormous.
From a cognitive perspective, we also hope that a way can be reflected macroscopic scales and microscopic scale at the same time. The Boltzmann Equation is able to embody the characteristics. So we can say that the Boltzmann equation, which is the subject of these notes, is the most classical but fundamental equation in the mesoscopic kinetic theory. In fact, the Boltzmann equation is a convergence between the Euler and Navier-Stokes equations.
The first existence theorem of the solutions to the Boltzmann equation goes back to 1932 when Carleman proved the existence of global (in time) solutions to the Cauchy problem for the spatially homogeneous case. It should be stressed that this is two years before the incompressible Navier-Stokes equation was solved by Leray on the existence of global weak solutions. On the other hand, the research on the spatially inhomogeneous Boltzmann equation started much later. It is only in 1963 when Grad constructed the first local solutions near the Maxwellian, and it is in 1974 when the first author of these notes constructed global solutions that are also near the Maxwellian, extending Grad's mathematical framework.
The Vlasov equation of this paper is the Boltzmann equation in a special case of collision-free items.
In this paper, we delicate to the model of the fluid dynamics, that is, the initial-boundary value problem to a class of the regularization for the Vlasov equations with non-Newtonian potential. That is, with initial and boundary condition:where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,ε> 0. (x, v)∈(?)= I×(-L, L), I = (0,1), (?), T>0.x∈I,v∈(-L,L). Physically, this system describes the motion of compressible viscous isentropic gas flow under the gravitational force.
The difficult of this type model is mainly that the equations are coupled with elliptic and parabolic. Also the degenerate of the elliptic equation, and so on.
For the case of the parameter p∈(1,2), we consider its solution defined by:
Definition 1 The (f,φ) is called a solution to the initial boundary value problem(1)-(2), if the following conditions are satisfied:
(i)
(ii) For all (?), for a.e.t∈(0,T), we have:
(iii) For allφ∈L~∞(0,T;H_0~1(I)), for a.e.t∈(0,T), we have:We get the main result as following.
定理1 Assume f_0, f_0≥0, f_0∈L~∞(?), for any fix T > 0. Then there exist aweak solution of the problem (1)-(2) in (?) [0, T], such that:
定理2 Assume f_0, f_0≥0, (?), for any fix T > 0. 1 < p < 2.Assume(?) is a weak solution of the problem (1)-(2) in (?) [0, T], and satisfies (?).If (f,φ) is a weak solution of the problem (1)-(2)in (?)[0, T], then (?)=(f,φ) .
In this chapter, we use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 1 and Theorem 2.
So we first consider the following: (?), (3) with initial boundary value:(?), (4)whereδ> 0,ε> 0, 1 < p < 2, (?)(0,T), T > 0, (?),I=(0,1), x∈I, v∈(-L,L).
For the singularity, we need to regularize it: Letφ~(δ,0) = 0(?), (5) (?), (6)(?), (7)wherewe get the uniform estimates of solution for (5)-(7).ess For the uniform estimate, we can take limits with respect to k,δand get the Theorem 1. Finally, we obtain the proving of the Theorem 2.
For the case of the parameter p∈(2, +∞), we consider the following initial boundary problem:(?). (8)with(?), (9)where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,μ> 0 and (?) =
we consider its solution defined by:
Definition 2 The (f,φ) is called a solution to the initial boundary value problem(8)-(9), if the following conditions are satisfied:
(i)
(ii) For all (?), for a.e. t∈(0, T), we have:
(iii) For allφ∈L~∞(0,T; H_0~1(I)), for a.e.t∈(0,T), we have:The main result as following:
定理3 Assume f_0, f_0≥0, f_0∈L~∞(?), for any fix T > 0.p > 2,μ> 0. Then there exist a weak solution of the problem (8)-(9) in (?)[0, T], such that:
定理4 Assume f_0, f_0≥0, (?), for any fix T > 0. p > 2,μ> 0. Assume(?) is a weak solution of the problem (8)-(9) in (?) [0. T], and satisfies (?). If (f,φ) is a weak solution of the problem (8)-(9) in (?)[0, T], then (?) =(f,φ).
In this chapter, the proof by the same technique as in the case of the p∈(1.2). We use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 3 and Theorem 4.
本文重点研究上述问题弱解的存在性.
The kinetic theory of the gas is also known as compressible fluid dynamics. It is research the fluid dynamics of variable density or fluid played a significant compression.
In the macroscopic scales where the gas and fluid are regarded as a continuum, their motion is described by the macroscopic quantities such as macroscopic mass density, bulk velocity, temperature, pressure, stresses, heat flux and so on. The Euler and Navier-Stokes equations, compressible or incompressible, are the most famous equations among governing equations proposed so far in fluid dynamics.
The extreme contrary is the microscopic scale where the gas, fluid, and hence any matter, are looked at as a many-body system of microscopic particles. Thus, the motion of the system is governed by the coupled Newton equations, within the framework of the classical mechanics. Although the Newton equation is the first principle of the classical mechanics, it is not of practical use because the number of the equations is so enormous.
From a cognitive perspective, we also hope that a way can be reflected macroscopic scales and microscopic scale at the same time. The Boltzmann Equation is able to embody the characteristics. So we can say that the Boltzmann equation, which is the subject of these notes, is the most classical but fundamental equation in the mesoscopic kinetic theory. In fact, the Boltzmann equation is a convergence between the Euler and Navier-Stokes equations.
The first existence theorem of the solutions to the Boltzmann equation goes back to 1932 when Carleman proved the existence of global (in time) solutions to the Cauchy problem for the spatially homogeneous case. It should be stressed that this is two years before the incompressible Navier-Stokes equation was solved by Leray on the existence of global weak solutions. On the other hand, the research on the spatially inhomogeneous Boltzmann equation started much later. It is only in 1963 when Grad constructed the first local solutions near the Maxwellian, and it is in 1974 when the first author of these notes constructed global solutions that are also near the Maxwellian, extending Grad's mathematical framework.
The Vlasov equation of this paper is the Boltzmann equation in a special case of collision-free items.
In this paper, we delicate to the model of the fluid dynamics, that is, the initial-boundary value problem to a class of the regularization for the Vlasov equations with non-Newtonian potential. That is, with initial and boundary condition:where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,ε> 0. (x, v)∈(?)= I×(-L, L), I = (0,1), (?), T>0.x∈I,v∈(-L,L). Physically, this system describes the motion of compressible viscous isentropic gas flow under the gravitational force.
The difficult of this type model is mainly that the equations are coupled with elliptic and parabolic. Also the degenerate of the elliptic equation, and so on.
For the case of the parameter p∈(1,2), we consider its solution defined by:
Definition 1 The (f,φ) is called a solution to the initial boundary value problem(1)-(2), if the following conditions are satisfied:
(i)
(ii) For all (?), for a.e.t∈(0,T), we have:
(iii) For allφ∈L~∞(0,T;H_0~1(I)), for a.e.t∈(0,T), we have:We get the main result as following.
定理1 Assume f_0, f_0≥0, f_0∈L~∞(?), for any fix T > 0. Then there exist aweak solution of the problem (1)-(2) in (?) [0, T], such that:
定理2 Assume f_0, f_0≥0, (?), for any fix T > 0. 1 < p < 2.Assume(?) is a weak solution of the problem (1)-(2) in (?) [0, T], and satisfies (?).If (f,φ) is a weak solution of the problem (1)-(2)in (?)[0, T], then (?)=(f,φ) .
In this chapter, we use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 1 and Theorem 2.
So we first consider the following: (?), (3) with initial boundary value:(?), (4)whereδ> 0,ε> 0, 1 < p < 2, (?)(0,T), T > 0, (?),I=(0,1), x∈I, v∈(-L,L).
For the singularity, we need to regularize it: Letφ~(δ,0) = 0(?), (5) (?), (6)(?), (7)wherewe get the uniform estimates of solution for (5)-(7).ess For the uniform estimate, we can take limits with respect to k,δand get the Theorem 1. Finally, we obtain the proving of the Theorem 2.
For the case of the parameter p∈(2, +∞), we consider the following initial boundary problem:(?). (8)with(?), (9)where f is the mass density function of gas particles having position x and velocity v at time t.Φ=Φ(x, t) the non-Newton gravitational potential,μ> 0 and (?) =
we consider its solution defined by:
Definition 2 The (f,φ) is called a solution to the initial boundary value problem(8)-(9), if the following conditions are satisfied:
(i)
(ii) For all (?), for a.e. t∈(0, T), we have:
(iii) For allφ∈L~∞(0,T; H_0~1(I)), for a.e.t∈(0,T), we have:The main result as following:
定理3 Assume f_0, f_0≥0, f_0∈L~∞(?), for any fix T > 0.p > 2,μ> 0. Then there exist a weak solution of the problem (8)-(9) in (?)[0, T], such that:
定理4 Assume f_0, f_0≥0, (?), for any fix T > 0. p > 2,μ> 0. Assume(?) is a weak solution of the problem (8)-(9) in (?) [0. T], and satisfies (?). If (f,φ) is a weak solution of the problem (8)-(9) in (?)[0, T], then (?) =(f,φ).
In this chapter, the proof by the same technique as in the case of the p∈(1.2). We use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the Theorem 3 and Theorem 4.
引文
[1] Abdallah N B. Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system[J]. Math. Methods Appl. Sci. 17 (1994), no. 6, 451-476.
[2] Adams R A. Sobolev Space[M]. Academic Press, New York, 1975.
[3] Arsenev A A. Global existence of a weak solution of Vlasov's system of equations[J]. U.S.S.R. Comput. Math. Phys. Vol. 15 (1975), 131-143.
[4] Alexandre R. Weak solutions of the Vlasov-Poisson initial-boundary value problem [J]. Math. Methods Appl. Sci. 16 (1993), no. 8, 587-607.
[5] Andreasson H. Global existence of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzmann equation[J]. Nonlinear Anal. 28 (1997), no. 7, 1193-1211.
[6] Bae H. Existence, regularity, and decay rate of solutions of non-Nwetonian flow[J]. J. Math. Anal. Appl. 231(1999), 467-491.
[7] Bae H, Choe H J. Existence of weak solutions to a class of non-Newtonian flows[J]. Houston Journal of Mathematics. 26(2000), no.2, 387-408.
[8] Bardos C, Degond P. Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[J]. Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), no. 2, 101-118.
[9] Batt J, Rein G. Global classical solutions of the periodic Vlasov-Poisson system in three dimensions [J]. C. R. Acad. Sci. Paris. I Math. 313 (1991), no. 6, 411-416.
[10] Batt J, Philip J, Rein G. Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions [J]. Arch. Rational Mech. Anal. 130 (1995), no. 2, 163-182.
[11] Bellout H, Bloom F, Necas J, Young measure-valued solutions for non-Newtonian incompressible fluids [J]. Comm. Partial Diffrential Equations. 19(1994), 1763-1803.
[12] Bostan M. Permanent regimes for the 1D Vlasov-Poisson system with boundary conditions [J]. SIAM J. Math. Anal. 35 (2003), no. 4, 922-948 (electronic).
[13] Bostan M. Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem[J]. SIAM J. Math. Anal. 37 (2005), no. 1, 156-188 (electronic).
[14] Bostan M. Boundary value problem for the TV-dimensional time periodic Vlasov-Poisson system[J]. Math. Methods Appl. Sci. 29(2006), no.15, 1801-1848.
[15] Bostan M. Weak solutions for the Vlasov-Poisson initial-boundary value problem with bounded electric field[J]. Chin. Ann. Math. Ser. B 28 (2007), no. 4, 389-420.
[16] Carmack L, Majda G. Concentrations in the one-dimensional Vlasov-Poisson equations: additional regularizations[J]. Phys. D 121 (1998), no. 1-2, 127-162.
[17] Castella F. Propagation of space moments in the Vlasov-Poisson equation and further resuits[J]. Ann. Inst. H. Poincare Anal. Non Lineaire 16 (1999), no. 4, 503-533.
[18] Diperna R J, Lions P L. On the Cauchy problem for Boltzmann equations: global existence and weak stability [J]. Ann. of Math. (2) 130 (1989), no. 2, 321-366.
[19] Dogbe C. Spherical harmonics expansion of the Vlasov-Poisson initial boundary value problem[J]. ZAMM Z. Angew. Math. Mech. 86 (2006), no. 5, 339-357.
[20] Dolbeault J. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation[C]. Current developments in partial differential equations (Temuco, 1999). Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 361-380.
[21] Dolbeault J, Fernandez J, Sanchez O. Stability for the gravitational Vlasov-Poisson system in dimension two[J]. Comm. Partial Differential Equations 31 (2006), no. 10-12, 1425-1449.
[22] Duan R, Yang T, Zhu C J. Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system[J]. J. Math. Anal. Appl. 327 (2007), no. 1, 425-434.
[23] Gajewski H, Groger K, Zacharias K. Nichtlineare operatorgleichungen und operatordifferen- tialgleichungen[C]. Akademie-Verlag, Berlin, 1974.
[24] Glassey R T, Schaeffer J. On symmetric solutions of the relativistic Vlasov-Poisson system[J]. Comm. Math. Phys. 101 (1985), no. 4, 459-473.
[25] Goudon T, Jabin P E, Vasseur A. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime[J]. Indiana Univ. Math. J. 53 (2004), no. 6, 1495-1515.
[26] Goudon T, Jabin P E, Vasseur A. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime[J]. Indiana Univ. Math. J. 53 (2004), no. 6, 1517-1536.
[27] Guo Y. The Vlasov-Poisson-Boltzmann system near vacuum[J]. Comm. Math. Phys. 218 (2001), no. 2, 293-313.
[28] Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians[J]. Comm. Pure Appl. Math. 55 (2002), no. 9, 1104-1135.
[29] Hadzie M, Rein G, Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case[J]. Indiana Univ. Math. J. 56 (2007), no. 5, 2453-2488.
[30] Horst E. New results and open problems in the theory of the Vlasov equation. Progress in Nuclear Energy[J], Volume 8, Issues 2-3, 1981, Pages 185-189.
[31] Horst E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory [J]. Math. Methods Appl. Sci. 3 (1981), no. 2, 229-248.
[32] Horst E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases [J]. Math. Methods Appl. Sci. 4 (1982), no. 1, 19-32.
[33] Horst E, Hunze R. Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation[J]. Math. Methods Appl. Sci. 6 (1984), no. 2, 262-279.
[34] Horst E. Global strong solutions of Vlasov's equation-necessary and sufficient conditions for their existence [J]. Partial differential equations (Warsaw, 1984), 143-153, Banach Center Publ., 19, PWN, Warsaw, 1987.
[35] Hwang H J, Schaeffer J. Uniqueness for weak solutions of a one-dimensional boundary value problem for the Vlasov-Poisson system[J]. J. Differential Equations 244 (2008), no. 10, 2665-2691.
[36] Jabin P E. The Vlasov-Poisson system with infinite mass and energy [J]. J. Statist. Phys. 103 (2001), no. 5-6, 1107-1123.
[37] Karimov A R, Lewis H. Ralph Nonlinear solutions of the Vlasov-Poisson equations [J]. Phys. Plasmas 6 (1999), no. 3, 759-761.
[38] Kufner A, John O, Fucik S. Function Space[M], Academia, Prague, 1977.
[39] Ladyzhenskaya O A. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them[J], in "Boundary Value Problems of Mathematical Physics V,", Amer. Math. Soc, Providence, RL, 1970.
[40] Ladyzhenskaya O A. Boundary Value Problems of Mathematical Physics[M], Spring-Verlag, New York, 1985.
[41] Letavin M I. A difference scheme for solving the initial-boundary value problem for one-dimensional Vlasov equations[J]. (Russian) Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), no. 2, 418-423, 494.
[42] Li L. On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation [J]. J. Differential Equations 244 (2008), no. 6, 1467-1501.
[43] Lions P L, Perthame B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system[J]. Invent. Math. 105 (1991), no. 2, 415-430.
[44] Lions P L. Global existence of solutions for isentropic compressible Navier-Stokes equations[J]. C. R. Acad. Sci. Paris Ser.I Math., 1993, Vol.316, 1335-1340.
[45] Lions P L. Mathematical topics in fluid dynamics. Vol. 1, Incompressible models[M]. Oxford Science Publication, Oxford, 1996.
[46] Lions P L. Mathematical topics in fluid dynamics. Vol. 2, Compressible models[M]. Oxford Science Publication, Oxford, 1998.
[47] Malek J, Necas J, Rokyta M, Ruzcka M. Weak and Measure-valued Solutions to Evolutionary PDEs[M]. Chapman and Hall, New York 1996.
[48] Mellet A, Vasseur A. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations[J]. Math. Models Methods Appl. Sci. 17 (2007), no. 7, 1039-1063.
[49] Mellet A, Vasseur A. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations [J]. Comm. Math. Phys. 281 (2008), no. 3, 573-596.
[50] Mingalev I V. Global solvability of the Cauchy problem for a system of Vlasov-Poisson equations with some collision integrals[J]. (Russian) Dokl. Akad. Nauk 351 (1996), no. 4, 459-461.
[51] Mischler S. On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[J]. Comm. Math. Phys. 210 (2000), no. 2, 447-466.
[52] Neshchadim M V. Differential identities and uniqueness theorem in inverse problem for the Boltzmann-Vlasov equation[J]. J. Inverse Ill-Posed Probl. 16 (2008), no. 3, 283-290.
[53] Pankavich S. Global existence for the Vlasov-Poisson system with steady spatial asymp-totics[J]. Comm. Partial Differential Equations 31 (2006), no. 1-3, 349-370.
[54] Pankavich S. Local existence for the one-dimensional Vlasov-Poisson system with infinite mass[J]. Math. Methods Appl. Sci. 30 (2007), no. 5, 529-548.
[55] Pfaffelmoser K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[J]. J. Differential Equations 95 (1992), no. 2, 281-303.
[56] Phillips T N. On the potential of spectral methods to solve problems in non-Newtonian fluid mechanics[J]. Numer. Methods Partial Differential Equations 5 (1989), no. 1, 35-43. 76-08
[57] Ramkissoon H. Lame-type potentials in a non-Newtonian fluid theory[J]. Acta Mech. 60 (1986), no. 3-4, 135-141.
[58] Rein G, Rendall A D. Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting [J]. Arch. Rational Mech. Anal. 126 (1994), no. 2, 183-201.
[59] Schaeffer J. The Vlasov-Poisson system with steady spatial asymptotics[J]. Comm. Partial Differential Equations 28 (2003), no. 5-6, 1057-1084.
[60] Schaeffer J. Steady spatial asymptotics for the Vlasov-Poisson system[J]. Math. Methods Appl. Sci. 26 (2003), no. 4, 273-296.
[61] Schaeffer J. Global existence for the Vlasov-Poisson system with nearly symmetric data[J]. J. Differential Equations 69 (1987), no. 1, 111-148.
[62] Schaeffer J. Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions[J]. Comm. Partial Differential Equations 16 (1991), no. 8-9, 1313-1335.
[63] Tran V C. A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions [J]. Stochastic Process. Appl. 118 (2008), no. 2, 284-318.
[64] Weckler J. On the initial-boundary-value problem for the Vlasov-Poisson system: existence of weak solutions and stability[J]. Arch. Rational Mech. Anal. 130 (1995), no. 2, 145-161.
[65] Wennberg B. Stability and exponential convergence for the Boltzmann equation[J]. Arch. Rational Mech. Anal. 130 (1995), no. 2. 103-144.
[66] Wollman S. The spherically symmetric Vlasov-Poisson system[J]. J. Differential Equations Volume 35, Issue 1, (1980), 30-35.
[67] Wollman S. Existence and uniqueness theory of the Vlasov-Poisson system with application to the problem with cylindrical symmetry[J]. J. Math. Anal. Appl. 90 (1982), no. 1, 138-170.
[68] Wollman S. A refinement of the existence and uniqueness theorem for the Viasov- Poisson system [J]. J. Math. Anal. Appl. 126 (1987), no. 2, 574-578.
[69] Wollman S. Convergence of a numerical approximation to the one-dimensional Vlasov-Poisson system[J]. Transport Theory Statist. Phys. 19 (1990), no. 6, 545-562.
[70] Wollman S. Global-in-time solutions to the three-dimensional Vlasov-Poisson system [J]. J. Math. Anal. Appl. 176 (1993), no. 1, 76-91.
[71] Yang T, Yu H, Zhao H. Cauchy problem for the Vlasov-Poisson-Boltzmann system[J]. Arch. Ration. Mech. Anal. 182 (2006), no. 3, 415-470.
[72] Yang T, Zhao H. Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system [J]. Comm. Math. Phys. 268 (2006), no. 3, 569-605.
[73] Zeidler E. Nonlinear fnctional analysis Ⅱ/B-Nonlinera Monotone operators[M]. Springer-Verlag, Berlin-heidelberg-New York, 1990.
[74] Zhang P, Qiu Q. On the existence of almost global weak solution to multidimensional Vlasov-Poisson equation[J]. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998). no. 4, 534. Chinese Ann. Math. Ser. B 19 (1998), no. 3, 381-390.
[75] Zhang X, Wei J. The Vlasov-Poisson system with infinite kinetic energy and initial data in L~p(R~6)[J]. J. Math. Anal. Appl. 341 (2008), no. 1. 548-558.
[76] Zhidkov P. On global solutions for the Vlasov-Poisson system. Electron [J]. J. Differential Equations 2004, No. 58, 11 pp. (electronic).[77]郭柏灵,林国广,尚亚东,非牛顿流动力系统[M].北京:国防工业出版社,2006.
[78]李大潜,秦铁虎,物理学与偏微分方程(第2版)(上、下册)[M].北京:高等教育出版社, 2005.
[79]许孝精,一类具有真空的可压缩非牛顿流[D].长春:吉林大学数学学院,2007.
[80]应隆安,滕振寰,双曲型守恒律方程及其差分方法[M].北京:科学出版社,1991.
[2] Adams R A. Sobolev Space[M]. Academic Press, New York, 1975.
[3] Arsenev A A. Global existence of a weak solution of Vlasov's system of equations[J]. U.S.S.R. Comput. Math. Phys. Vol. 15 (1975), 131-143.
[4] Alexandre R. Weak solutions of the Vlasov-Poisson initial-boundary value problem [J]. Math. Methods Appl. Sci. 16 (1993), no. 8, 587-607.
[5] Andreasson H. Global existence of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzmann equation[J]. Nonlinear Anal. 28 (1997), no. 7, 1193-1211.
[6] Bae H. Existence, regularity, and decay rate of solutions of non-Nwetonian flow[J]. J. Math. Anal. Appl. 231(1999), 467-491.
[7] Bae H, Choe H J. Existence of weak solutions to a class of non-Newtonian flows[J]. Houston Journal of Mathematics. 26(2000), no.2, 387-408.
[8] Bardos C, Degond P. Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data[J]. Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), no. 2, 101-118.
[9] Batt J, Rein G. Global classical solutions of the periodic Vlasov-Poisson system in three dimensions [J]. C. R. Acad. Sci. Paris. I Math. 313 (1991), no. 6, 411-416.
[10] Batt J, Philip J, Rein G. Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions [J]. Arch. Rational Mech. Anal. 130 (1995), no. 2, 163-182.
[11] Bellout H, Bloom F, Necas J, Young measure-valued solutions for non-Newtonian incompressible fluids [J]. Comm. Partial Diffrential Equations. 19(1994), 1763-1803.
[12] Bostan M. Permanent regimes for the 1D Vlasov-Poisson system with boundary conditions [J]. SIAM J. Math. Anal. 35 (2003), no. 4, 922-948 (electronic).
[13] Bostan M. Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem[J]. SIAM J. Math. Anal. 37 (2005), no. 1, 156-188 (electronic).
[14] Bostan M. Boundary value problem for the TV-dimensional time periodic Vlasov-Poisson system[J]. Math. Methods Appl. Sci. 29(2006), no.15, 1801-1848.
[15] Bostan M. Weak solutions for the Vlasov-Poisson initial-boundary value problem with bounded electric field[J]. Chin. Ann. Math. Ser. B 28 (2007), no. 4, 389-420.
[16] Carmack L, Majda G. Concentrations in the one-dimensional Vlasov-Poisson equations: additional regularizations[J]. Phys. D 121 (1998), no. 1-2, 127-162.
[17] Castella F. Propagation of space moments in the Vlasov-Poisson equation and further resuits[J]. Ann. Inst. H. Poincare Anal. Non Lineaire 16 (1999), no. 4, 503-533.
[18] Diperna R J, Lions P L. On the Cauchy problem for Boltzmann equations: global existence and weak stability [J]. Ann. of Math. (2) 130 (1989), no. 2, 321-366.
[19] Dogbe C. Spherical harmonics expansion of the Vlasov-Poisson initial boundary value problem[J]. ZAMM Z. Angew. Math. Mech. 86 (2006), no. 5, 339-357.
[20] Dolbeault J. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation[C]. Current developments in partial differential equations (Temuco, 1999). Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 361-380.
[21] Dolbeault J, Fernandez J, Sanchez O. Stability for the gravitational Vlasov-Poisson system in dimension two[J]. Comm. Partial Differential Equations 31 (2006), no. 10-12, 1425-1449.
[22] Duan R, Yang T, Zhu C J. Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system[J]. J. Math. Anal. Appl. 327 (2007), no. 1, 425-434.
[23] Gajewski H, Groger K, Zacharias K. Nichtlineare operatorgleichungen und operatordifferen- tialgleichungen[C]. Akademie-Verlag, Berlin, 1974.
[24] Glassey R T, Schaeffer J. On symmetric solutions of the relativistic Vlasov-Poisson system[J]. Comm. Math. Phys. 101 (1985), no. 4, 459-473.
[25] Goudon T, Jabin P E, Vasseur A. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime[J]. Indiana Univ. Math. J. 53 (2004), no. 6, 1495-1515.
[26] Goudon T, Jabin P E, Vasseur A. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime[J]. Indiana Univ. Math. J. 53 (2004), no. 6, 1517-1536.
[27] Guo Y. The Vlasov-Poisson-Boltzmann system near vacuum[J]. Comm. Math. Phys. 218 (2001), no. 2, 293-313.
[28] Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians[J]. Comm. Pure Appl. Math. 55 (2002), no. 9, 1104-1135.
[29] Hadzie M, Rein G, Global existence and nonlinear stability for the relativistic Vlasov-Poisson system in the gravitational case[J]. Indiana Univ. Math. J. 56 (2007), no. 5, 2453-2488.
[30] Horst E. New results and open problems in the theory of the Vlasov equation. Progress in Nuclear Energy[J], Volume 8, Issues 2-3, 1981, Pages 185-189.
[31] Horst E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory [J]. Math. Methods Appl. Sci. 3 (1981), no. 2, 229-248.
[32] Horst E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases [J]. Math. Methods Appl. Sci. 4 (1982), no. 1, 19-32.
[33] Horst E, Hunze R. Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation[J]. Math. Methods Appl. Sci. 6 (1984), no. 2, 262-279.
[34] Horst E. Global strong solutions of Vlasov's equation-necessary and sufficient conditions for their existence [J]. Partial differential equations (Warsaw, 1984), 143-153, Banach Center Publ., 19, PWN, Warsaw, 1987.
[35] Hwang H J, Schaeffer J. Uniqueness for weak solutions of a one-dimensional boundary value problem for the Vlasov-Poisson system[J]. J. Differential Equations 244 (2008), no. 10, 2665-2691.
[36] Jabin P E. The Vlasov-Poisson system with infinite mass and energy [J]. J. Statist. Phys. 103 (2001), no. 5-6, 1107-1123.
[37] Karimov A R, Lewis H. Ralph Nonlinear solutions of the Vlasov-Poisson equations [J]. Phys. Plasmas 6 (1999), no. 3, 759-761.
[38] Kufner A, John O, Fucik S. Function Space[M], Academia, Prague, 1977.
[39] Ladyzhenskaya O A. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them[J], in "Boundary Value Problems of Mathematical Physics V,", Amer. Math. Soc, Providence, RL, 1970.
[40] Ladyzhenskaya O A. Boundary Value Problems of Mathematical Physics[M], Spring-Verlag, New York, 1985.
[41] Letavin M I. A difference scheme for solving the initial-boundary value problem for one-dimensional Vlasov equations[J]. (Russian) Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), no. 2, 418-423, 494.
[42] Li L. On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation [J]. J. Differential Equations 244 (2008), no. 6, 1467-1501.
[43] Lions P L, Perthame B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system[J]. Invent. Math. 105 (1991), no. 2, 415-430.
[44] Lions P L. Global existence of solutions for isentropic compressible Navier-Stokes equations[J]. C. R. Acad. Sci. Paris Ser.I Math., 1993, Vol.316, 1335-1340.
[45] Lions P L. Mathematical topics in fluid dynamics. Vol. 1, Incompressible models[M]. Oxford Science Publication, Oxford, 1996.
[46] Lions P L. Mathematical topics in fluid dynamics. Vol. 2, Compressible models[M]. Oxford Science Publication, Oxford, 1998.
[47] Malek J, Necas J, Rokyta M, Ruzcka M. Weak and Measure-valued Solutions to Evolutionary PDEs[M]. Chapman and Hall, New York 1996.
[48] Mellet A, Vasseur A. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations[J]. Math. Models Methods Appl. Sci. 17 (2007), no. 7, 1039-1063.
[49] Mellet A, Vasseur A. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations [J]. Comm. Math. Phys. 281 (2008), no. 3, 573-596.
[50] Mingalev I V. Global solvability of the Cauchy problem for a system of Vlasov-Poisson equations with some collision integrals[J]. (Russian) Dokl. Akad. Nauk 351 (1996), no. 4, 459-461.
[51] Mischler S. On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[J]. Comm. Math. Phys. 210 (2000), no. 2, 447-466.
[52] Neshchadim M V. Differential identities and uniqueness theorem in inverse problem for the Boltzmann-Vlasov equation[J]. J. Inverse Ill-Posed Probl. 16 (2008), no. 3, 283-290.
[53] Pankavich S. Global existence for the Vlasov-Poisson system with steady spatial asymp-totics[J]. Comm. Partial Differential Equations 31 (2006), no. 1-3, 349-370.
[54] Pankavich S. Local existence for the one-dimensional Vlasov-Poisson system with infinite mass[J]. Math. Methods Appl. Sci. 30 (2007), no. 5, 529-548.
[55] Pfaffelmoser K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[J]. J. Differential Equations 95 (1992), no. 2, 281-303.
[56] Phillips T N. On the potential of spectral methods to solve problems in non-Newtonian fluid mechanics[J]. Numer. Methods Partial Differential Equations 5 (1989), no. 1, 35-43. 76-08
[57] Ramkissoon H. Lame-type potentials in a non-Newtonian fluid theory[J]. Acta Mech. 60 (1986), no. 3-4, 135-141.
[58] Rein G, Rendall A D. Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting [J]. Arch. Rational Mech. Anal. 126 (1994), no. 2, 183-201.
[59] Schaeffer J. The Vlasov-Poisson system with steady spatial asymptotics[J]. Comm. Partial Differential Equations 28 (2003), no. 5-6, 1057-1084.
[60] Schaeffer J. Steady spatial asymptotics for the Vlasov-Poisson system[J]. Math. Methods Appl. Sci. 26 (2003), no. 4, 273-296.
[61] Schaeffer J. Global existence for the Vlasov-Poisson system with nearly symmetric data[J]. J. Differential Equations 69 (1987), no. 1, 111-148.
[62] Schaeffer J. Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions[J]. Comm. Partial Differential Equations 16 (1991), no. 8-9, 1313-1335.
[63] Tran V C. A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions [J]. Stochastic Process. Appl. 118 (2008), no. 2, 284-318.
[64] Weckler J. On the initial-boundary-value problem for the Vlasov-Poisson system: existence of weak solutions and stability[J]. Arch. Rational Mech. Anal. 130 (1995), no. 2, 145-161.
[65] Wennberg B. Stability and exponential convergence for the Boltzmann equation[J]. Arch. Rational Mech. Anal. 130 (1995), no. 2. 103-144.
[66] Wollman S. The spherically symmetric Vlasov-Poisson system[J]. J. Differential Equations Volume 35, Issue 1, (1980), 30-35.
[67] Wollman S. Existence and uniqueness theory of the Vlasov-Poisson system with application to the problem with cylindrical symmetry[J]. J. Math. Anal. Appl. 90 (1982), no. 1, 138-170.
[68] Wollman S. A refinement of the existence and uniqueness theorem for the Viasov- Poisson system [J]. J. Math. Anal. Appl. 126 (1987), no. 2, 574-578.
[69] Wollman S. Convergence of a numerical approximation to the one-dimensional Vlasov-Poisson system[J]. Transport Theory Statist. Phys. 19 (1990), no. 6, 545-562.
[70] Wollman S. Global-in-time solutions to the three-dimensional Vlasov-Poisson system [J]. J. Math. Anal. Appl. 176 (1993), no. 1, 76-91.
[71] Yang T, Yu H, Zhao H. Cauchy problem for the Vlasov-Poisson-Boltzmann system[J]. Arch. Ration. Mech. Anal. 182 (2006), no. 3, 415-470.
[72] Yang T, Zhao H. Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system [J]. Comm. Math. Phys. 268 (2006), no. 3, 569-605.
[73] Zeidler E. Nonlinear fnctional analysis Ⅱ/B-Nonlinera Monotone operators[M]. Springer-Verlag, Berlin-heidelberg-New York, 1990.
[74] Zhang P, Qiu Q. On the existence of almost global weak solution to multidimensional Vlasov-Poisson equation[J]. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998). no. 4, 534. Chinese Ann. Math. Ser. B 19 (1998), no. 3, 381-390.
[75] Zhang X, Wei J. The Vlasov-Poisson system with infinite kinetic energy and initial data in L~p(R~6)[J]. J. Math. Anal. Appl. 341 (2008), no. 1. 548-558.
[76] Zhidkov P. On global solutions for the Vlasov-Poisson system. Electron [J]. J. Differential Equations 2004, No. 58, 11 pp. (electronic).[77]郭柏灵,林国广,尚亚东,非牛顿流动力系统[M].北京:国防工业出版社,2006.
[78]李大潜,秦铁虎,物理学与偏微分方程(第2版)(上、下册)[M].北京:高等教育出版社, 2005.
[79]许孝精,一类具有真空的可压缩非牛顿流[D].长春:吉林大学数学学院,2007.
[80]应隆安,滕振寰,双曲型守恒律方程及其差分方法[M].北京:科学出版社,1991.