补偿列紧方法在一类非线性双曲系统中的应用
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摘要
本文介绍了补偿列紧方法在单个守恒律方程和一些重要的双曲守恒律系统中的应用.借助于著名的Bernstein-Weierstrass定理,我们用Lax熵导出了不带凸性的单个方程在L~∞或L_(loc)~p中一致有界的逼近解序列的强收敛性,并据此研究了一个对称双曲系统的松弛极限;用补偿列紧方法和动力学公式相结合的思想,我们极大地简化了测度约化这一最为关键、困难的步骤,对二次流系统和Le Roux系统的L~∞熵解的存在性给了凝练的证明,并对ρ—u方程组1＜γ＜3的情形建立一个紧性框架.
本文共分五章,具体安排如下:
在第一章,我们介绍了一些基本概念和补偿列紧理论中的几个重要定理,并证明了一个抛物型系统解的存在性定理.
在第二章,我们用Lax熵导出了不带凸性的单个方程在L~∞或L_(loc)~p解一致有界的逼近解序列的强收敛性,并据此研究了一个对称双曲系统的松弛极限.
在第三、四章,我们用补偿列紧方法和动力学公式相结合的思想简化证明了二次流系统和Le Roux系统的L~∞熵解的存在性,得到了一些更一般的结果.我们还讨论了这两个系统的零松弛现象.
在第五章,我们对一维可压缩流体流的Euler方程组(ρ—u方程组)1＜γ＜3的情形建立一个紧性框架,以及在远离真空的假设下,对1＜γ＜3的一维等熵气体动力学系统(ρ—m方程组)的熵解的存在性给了一个非常简洁的证明.我们还得到了一些重要的带源项的ρ—u方程组和ρ—m方程组的解的存在性.
In this paper,we introduce some applications of the compensated compactness method to scalar conservation law as well as some important hyperbolic conservation laws.With the aid of the well-known Bernstein-Weierstrass theorem, we derive the strong convergence of a sequence of uniform L~∞or L_(loc)~p bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit for a symmetrically hyperbolic system based on the results;by using the compensated compactness method coupled with some basic ideas of the kinetic formulation,we considerably simplify the most crucial and difficult step,reduction of Young measures,to give refined proofs for the existence of L~∞entropy solutions to a system of quadratic flux and the Le Roux system,and establish a compactness framework of theρ- u equations for the case of 1＜γ＜3.
This paper consists of five chapters,the arrangement of it is as follows: In Chapter 1,we introduce some basic definitions and important theorems of the theory of compensated compactness,and prove the existence of smooth solutions for the parabolic system.
In Chapter 2,we derive the strong convergence of a sequence of uniform L~∞or L_(loc)~P bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit lot a symmetrically hyperbolic system based on the results.
In Chapter 3,4,we use the compensated compactness method coupled with some basic ideas of the kinetic formulation to simplify the proofs for the existence of globally bounded entropy solutions to a system of quadratic flux and the Le Roux system,and then obtain some more general results.The zero relaxation limits for the two systems are also discussed.
In Chapter 5,we establish a compactness framework of the Euler equations of one dimensional,compressible fluid flow(ρ- u equations)for the case of 1＜γ＜3,and give a much concise proof for the existence of entropy solutions to one dimensional isentropic gas dynamics(ρ- m equations)with 1＜γ＜3,provided that the initial data are away from vacuum state.We also obtain global existence for some importantρ- u equations with sources as well asρ- m equations with sources.
本文共分五章,具体安排如下:
在第一章,我们介绍了一些基本概念和补偿列紧理论中的几个重要定理,并证明了一个抛物型系统解的存在性定理.
在第二章,我们用Lax熵导出了不带凸性的单个方程在L~∞或L_(loc)~p解一致有界的逼近解序列的强收敛性,并据此研究了一个对称双曲系统的松弛极限.
在第三、四章,我们用补偿列紧方法和动力学公式相结合的思想简化证明了二次流系统和Le Roux系统的L~∞熵解的存在性,得到了一些更一般的结果.我们还讨论了这两个系统的零松弛现象.
在第五章,我们对一维可压缩流体流的Euler方程组(ρ—u方程组)1＜γ＜3的情形建立一个紧性框架,以及在远离真空的假设下,对1＜γ＜3的一维等熵气体动力学系统(ρ—m方程组)的熵解的存在性给了一个非常简洁的证明.我们还得到了一些重要的带源项的ρ—u方程组和ρ—m方程组的解的存在性.
In this paper,we introduce some applications of the compensated compactness method to scalar conservation law as well as some important hyperbolic conservation laws.With the aid of the well-known Bernstein-Weierstrass theorem, we derive the strong convergence of a sequence of uniform L~∞or L_(loc)~p bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit for a symmetrically hyperbolic system based on the results;by using the compensated compactness method coupled with some basic ideas of the kinetic formulation,we considerably simplify the most crucial and difficult step,reduction of Young measures,to give refined proofs for the existence of L~∞entropy solutions to a system of quadratic flux and the Le Roux system,and establish a compactness framework of theρ- u equations for the case of 1＜γ＜3.
This paper consists of five chapters,the arrangement of it is as follows: In Chapter 1,we introduce some basic definitions and important theorems of the theory of compensated compactness,and prove the existence of smooth solutions for the parabolic system.
In Chapter 2,we derive the strong convergence of a sequence of uniform L~∞or L_(loc)~P bounded approximate solutions for scalar equations without convexity by constructing four families of Lax entropies,and study the relaxation limit lot a symmetrically hyperbolic system based on the results.
In Chapter 3,4,we use the compensated compactness method coupled with some basic ideas of the kinetic formulation to simplify the proofs for the existence of globally bounded entropy solutions to a system of quadratic flux and the Le Roux system,and then obtain some more general results.The zero relaxation limits for the two systems are also discussed.
In Chapter 5,we establish a compactness framework of the Euler equations of one dimensional,compressible fluid flow(ρ- u equations)for the case of 1＜γ＜3,and give a much concise proof for the existence of entropy solutions to one dimensional isentropic gas dynamics(ρ- m equations)with 1＜γ＜3,provided that the initial data are away from vacuum state.We also obtain global existence for some importantρ- u equations with sources as well asρ- m equations with sources.
引文
[1]T.Aubin,Some nonlinear problems in Riemannian geometry,Springer-Verlag,Berlin-Heidelberg-New York,1998.
[2]G.Q.Chen,Hyperbolic system of conservation laws with a symmetry,Comm.Partial Diff.Eqs.,16(1991),1461-1487.
[3]G.Q.Chen and J.Glimm,Global solutions to the compressible Euler equations with geometric structure,Comm.Math.Phys.,180(1996),153-193.
[4]G.Q.Chen and P.T.Kan,Hyperbolic conservation laws with umbilic degeneracy I,Arch.Rat.Mech.Anal.,130(1995),231-276.
[5]G.Q.Chen and Y.G.Lu,A study of approaches to applying the theory of compensated compactness,Chinese Science Bulletin,34(1989),15-19.
[6]G.Q.Chen and Y.G.Lu,Convergence of the approximation solutions to isentropic dynamics,Acta Math.Sci.,10(1990),39-46.
[7]Z.Cheng,Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity,J.Math.Anal.Appl.,340(2008),558-568.
[8]Z.Cheng,Global entropy solutions to a variant of the compressible Euler equations,Appl.Math.Lett.,21(2008),410-415.
[9]Z.Cheng,The application of the kinetic formulation to a system of quadratic flux,accepted by Acta Math.Sci..
[10]Z.Cheng,On application of kinetic formulation of the Le Roux system,accepted by Proc.Edinburgh Math.Soc..
[11]Z.Cheng,Relaxation limit for a symmetrically hyperbolic system,submitted.
[12]K.N.Chueh,C.C.Conley and J.A.Smoller,Positive invariant regions for systems of nonlinear equations,Indiana.Univ.Math.J.,26(1977),373-392.
[13]X.Q.Ding,G.Q.Chen and P.Z.Luo,Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics Ⅰ,Acta Math.Sci.,4(1985),483-500.
[14]R.J.DiPerna,Global solutions to a class of nonlinear hyperbolic systems of equations,Comm.Pure Appl.Math.,26(1973),1-28.
[15]R.J.DiPerna,Convergence of approximate solutions to conservation laws,Arch. Rat.Mech.Anal.,82(1983),27-70.
[16]S.Earnshaw,On the mathematical theory of sound,Philos.Trans.,150(1858),1150-1154.
[17]J.Glimm,Solutions in the large for nonlinear hyperbolic systems of equations,Comm.Pure Appl.Math.,18(1965),697-715.
[18]F.J.James,Y.J.Pang and B.Perthame,Kinetic formulation for chromatography and some other hyperbolic systems,J.Math.Pure.Appl.,74(1995),367-385.
[19]P.T.Kan,On the Cauchy problem of a 2×2 system of nonstrictly hyperbolic conservation laws[Ph D thesis],New York Univ.,1989.
[20]A.Y.LeRoux,Numerical stability for some equations of gas dynamics,Mathematics of Computation,37(1981),435-446.
[21]P.X.Lin,Young measuses and an application of compensated compactness to onedimensional nonlinear elastodynamics,Trans.Am.Math.Soc.,329(1992),377-413.
[22]P.L.Lions,B.Perthame and P.E.Souganidis,Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,Comm.Pure Appl.Math.,49(1996),599-638.
[23]P.L.Lions,B.Perthame and E.Tadmor,Kinetic formulation of the isentropic gas dynamics and p-system,Commun.Math.Phys.,163(1994),415-431.
[24]M.Liu and Z.Chang,Conservation laws Ⅲ:relaxation limit,Revista Colom.Mater.,41(2007),107-115.
[25]Y.G.Lu,Convergence of solutions to nonlinear dispersive equations without convexity conditions,Appl.Anal.,31(1989),239-246.
[26]Y.G.Lu,Convergence of the viscosity method for a nonstrictly hyperbolic system,Acta Math.Sci.,12(1992),349-360.
[27]Y.G.Lu,Singular limits of stiff relaxation and dominant diffusion for nonlinear systems,J.Diff.Eqs.,179(2002),687-713.
[28]Y.G.Lu,Hyperbolic conservation laws and the compensated compactess method,128,Chapman and Hall,CRC Press,New York,2002.
[29]Y.G.Lu,Existence of global entropy solutions to a nonstrictly hyperbolic system,Arch.Rat.Mech.Anal.,178(2005),287-299.
[30]Y.G.Lu,Global weak solution for a symmetrically hyperbolic system,Appl.Math.Lett.,19(2006),522-526.
[31]Y.G.Lu,I.Mantilla and L.Rendon,Convergence of approximated solutions to a nonstrictly hyperbolic system,Advanced Nonlinear Studies,1(2001),65-79.
[32]F.Murat,Compacite par compensation,Ann.Scuola Norm.Sup.Pisa,5(1978),489-507.
[33]B.Perthame,Kinetic formulations,Oxford Univ.Press,2002.
[34]M.E.Schonbek,Convergence of solutions to nonlinear dispersive equations,Comm.Partial Diff.Eqs.,7(1982),959-1000.
[35]D.Serre,La Compacite par compensation pour les systemes hyperboliques non lineaires de deux equations a une dimension d'space,J.Math.Pures Appl.,65(1986),423-468.
[36]D.Serre,Systems of conservation laws,Cambridge Univ.Press,2000.
[37]J.Smoller,Shock waves and reaction-diffusion equations,Springer-Verlag,BerlinHeidelberg-New York,1983.
[38]M.Tao,Z.Cheng and J.Yan,Conservation laws Ⅱ:relaxation limit,Revista Colom.Mater.,41(2007),91-106.
[39]T.Tartar,Compensated compactness and applications to partial differential equations,Nonlinear Analysis and Mechanics,Heriot-Watt symposium,Ⅳ(R.J.Knops ed.),Pitman Res.Notes Math.Ser.,Longman Sci.Tech.,Harlow,1979.
[40]T.Tartar,The compensated compactness method applied to systems of conservation laws,Systems of Nonlinear Partial Differential Equations(J.M.Ball,ed.),NATO ASI Series,Reidel,1983.
[41]J.Yan,Z.Cheng and M.Tao,Conservation laws Ⅰ:viscosity solutions,Revista Colom.Mater.,41(2007),81-90.
[42]T.Yang,C.Z.Zhu and H.J.Zhao,Compactness framework of L~p approximate solutions for scalar conservation laws,J.Math.Anal.Appl.,220(1998),164-186.
[43]应隆安,滕振寰,双曲型守恒律方程及其差分方法,北京:科学出版社,1991.
[44]K.Yosida,Functional analysis,Springer,New York,1968.
[2]G.Q.Chen,Hyperbolic system of conservation laws with a symmetry,Comm.Partial Diff.Eqs.,16(1991),1461-1487.
[3]G.Q.Chen and J.Glimm,Global solutions to the compressible Euler equations with geometric structure,Comm.Math.Phys.,180(1996),153-193.
[4]G.Q.Chen and P.T.Kan,Hyperbolic conservation laws with umbilic degeneracy I,Arch.Rat.Mech.Anal.,130(1995),231-276.
[5]G.Q.Chen and Y.G.Lu,A study of approaches to applying the theory of compensated compactness,Chinese Science Bulletin,34(1989),15-19.
[6]G.Q.Chen and Y.G.Lu,Convergence of the approximation solutions to isentropic dynamics,Acta Math.Sci.,10(1990),39-46.
[7]Z.Cheng,Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity,J.Math.Anal.Appl.,340(2008),558-568.
[8]Z.Cheng,Global entropy solutions to a variant of the compressible Euler equations,Appl.Math.Lett.,21(2008),410-415.
[9]Z.Cheng,The application of the kinetic formulation to a system of quadratic flux,accepted by Acta Math.Sci..
[10]Z.Cheng,On application of kinetic formulation of the Le Roux system,accepted by Proc.Edinburgh Math.Soc..
[11]Z.Cheng,Relaxation limit for a symmetrically hyperbolic system,submitted.
[12]K.N.Chueh,C.C.Conley and J.A.Smoller,Positive invariant regions for systems of nonlinear equations,Indiana.Univ.Math.J.,26(1977),373-392.
[13]X.Q.Ding,G.Q.Chen and P.Z.Luo,Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics Ⅰ,Acta Math.Sci.,4(1985),483-500.
[14]R.J.DiPerna,Global solutions to a class of nonlinear hyperbolic systems of equations,Comm.Pure Appl.Math.,26(1973),1-28.
[15]R.J.DiPerna,Convergence of approximate solutions to conservation laws,Arch. Rat.Mech.Anal.,82(1983),27-70.
[16]S.Earnshaw,On the mathematical theory of sound,Philos.Trans.,150(1858),1150-1154.
[17]J.Glimm,Solutions in the large for nonlinear hyperbolic systems of equations,Comm.Pure Appl.Math.,18(1965),697-715.
[18]F.J.James,Y.J.Pang and B.Perthame,Kinetic formulation for chromatography and some other hyperbolic systems,J.Math.Pure.Appl.,74(1995),367-385.
[19]P.T.Kan,On the Cauchy problem of a 2×2 system of nonstrictly hyperbolic conservation laws[Ph D thesis],New York Univ.,1989.
[20]A.Y.LeRoux,Numerical stability for some equations of gas dynamics,Mathematics of Computation,37(1981),435-446.
[21]P.X.Lin,Young measuses and an application of compensated compactness to onedimensional nonlinear elastodynamics,Trans.Am.Math.Soc.,329(1992),377-413.
[22]P.L.Lions,B.Perthame and P.E.Souganidis,Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,Comm.Pure Appl.Math.,49(1996),599-638.
[23]P.L.Lions,B.Perthame and E.Tadmor,Kinetic formulation of the isentropic gas dynamics and p-system,Commun.Math.Phys.,163(1994),415-431.
[24]M.Liu and Z.Chang,Conservation laws Ⅲ:relaxation limit,Revista Colom.Mater.,41(2007),107-115.
[25]Y.G.Lu,Convergence of solutions to nonlinear dispersive equations without convexity conditions,Appl.Anal.,31(1989),239-246.
[26]Y.G.Lu,Convergence of the viscosity method for a nonstrictly hyperbolic system,Acta Math.Sci.,12(1992),349-360.
[27]Y.G.Lu,Singular limits of stiff relaxation and dominant diffusion for nonlinear systems,J.Diff.Eqs.,179(2002),687-713.
[28]Y.G.Lu,Hyperbolic conservation laws and the compensated compactess method,128,Chapman and Hall,CRC Press,New York,2002.
[29]Y.G.Lu,Existence of global entropy solutions to a nonstrictly hyperbolic system,Arch.Rat.Mech.Anal.,178(2005),287-299.
[30]Y.G.Lu,Global weak solution for a symmetrically hyperbolic system,Appl.Math.Lett.,19(2006),522-526.
[31]Y.G.Lu,I.Mantilla and L.Rendon,Convergence of approximated solutions to a nonstrictly hyperbolic system,Advanced Nonlinear Studies,1(2001),65-79.
[32]F.Murat,Compacite par compensation,Ann.Scuola Norm.Sup.Pisa,5(1978),489-507.
[33]B.Perthame,Kinetic formulations,Oxford Univ.Press,2002.
[34]M.E.Schonbek,Convergence of solutions to nonlinear dispersive equations,Comm.Partial Diff.Eqs.,7(1982),959-1000.
[35]D.Serre,La Compacite par compensation pour les systemes hyperboliques non lineaires de deux equations a une dimension d'space,J.Math.Pures Appl.,65(1986),423-468.
[36]D.Serre,Systems of conservation laws,Cambridge Univ.Press,2000.
[37]J.Smoller,Shock waves and reaction-diffusion equations,Springer-Verlag,BerlinHeidelberg-New York,1983.
[38]M.Tao,Z.Cheng and J.Yan,Conservation laws Ⅱ:relaxation limit,Revista Colom.Mater.,41(2007),91-106.
[39]T.Tartar,Compensated compactness and applications to partial differential equations,Nonlinear Analysis and Mechanics,Heriot-Watt symposium,Ⅳ(R.J.Knops ed.),Pitman Res.Notes Math.Ser.,Longman Sci.Tech.,Harlow,1979.
[40]T.Tartar,The compensated compactness method applied to systems of conservation laws,Systems of Nonlinear Partial Differential Equations(J.M.Ball,ed.),NATO ASI Series,Reidel,1983.
[41]J.Yan,Z.Cheng and M.Tao,Conservation laws Ⅰ:viscosity solutions,Revista Colom.Mater.,41(2007),81-90.
[42]T.Yang,C.Z.Zhu and H.J.Zhao,Compactness framework of L~p approximate solutions for scalar conservation laws,J.Math.Anal.Appl.,220(1998),164-186.
[43]应隆安,滕振寰,双曲型守恒律方程及其差分方法,北京:科学出版社,1991.
[44]K.Yosida,Functional analysis,Springer,New York,1968.
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