变密度流体力学方程的适定性及精确解的研究
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摘要
本文的研究内容主要包括两方面.第一,在假设初始密度ρ0有界(即0 < m <ρ0 第二,利用分离变量法,构建了N-维等温欧拉方程组的显式解和N-维等温无压力带有摩擦阻尼的欧拉方程组的显式解.同时,证明了这些解也满足相应的Navier-Stokes方程.特别地,解在有限时刻T发生爆破.
The research of this dissertation includes two aspects. In the first place, under theassumption that the initial density is bounded away from zero we establish the local well-posedness in some critical Besov spaces for the compressible magneto-hydrodynamic equa-tions with density-dependent viscosities in RN(N≥2) by constructing a sequence of smoothsolutions, and using a compactness argument the convergence of the sequence is proved.The compressible magneto-hydrodynamic equations is:
In the second place, we use the separation method to construct a family of analyticalsolutions to the N-dimensional isothermal Euler equations, and a family of analytical solu-tions to the N-dimensional the pressureless isothermal Euler equations with frictional damp-ing. We also prove that the analytical solutions of Euler equations satisfy the correspondingNavier-Stokes equations. In particular, the solutions may blow up in a finite time T.
The research of this dissertation includes two aspects. In the first place, under theassumption that the initial density is bounded away from zero we establish the local well-posedness in some critical Besov spaces for the compressible magneto-hydrodynamic equa-tions with density-dependent viscosities in RN(N≥2) by constructing a sequence of smoothsolutions, and using a compactness argument the convergence of the sequence is proved.The compressible magneto-hydrodynamic equations is:
In the second place, we use the separation method to construct a family of analyticalsolutions to the N-dimensional isothermal Euler equations, and a family of analytical solu-tions to the N-dimensional the pressureless isothermal Euler equations with frictional damp-ing. We also prove that the analytical solutions of Euler equations satisfy the correspondingNavier-Stokes equations. In particular, the solutions may blow up in a finite time T.
引文
[1] Chen Qionglei, Miao Changxing, Zhang Zhifei, Well-poseness in critical Spaces for compress-ible Navier-Stokes equations with density dependent viscosities. arXIV: 0811.4215v1 November2008.
[2] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases.Comm. Partial Di?erential Equations, 26(2001), 1183-1233.
[3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations. Invent.Math., 141(2000), 579-614.
[4] R. Danchin, On the uniquness in critical spaces for compressible Navier-Stokes equations. No-linear Differential Equations Appl., 12(2005), 111-128.
[5] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constantdensity. Comm. Partial Differential Equations, 32(2007), 1373-1397.
[6] Haspot,Well-posedness in critical spaces for the system of Navier-Stokes compressible. arXIV :0904.1354v1 [math.AP] 8 Apr 2009.
[7] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens etéquations de Navier-Stokes. J. Differential Equations, 121(1992), 314-328.
[8] J.-Y. Chemin, Perfect incompressible fluids. Oxford University Press, New York, 1998.
[9] J.-Y. Chemin, Théorèmes d’unicitépour le système de Navier-Stokes tridimensionnel. J.d’Analyse Math., 77(1999), 27-50.
[10] Bian Dongfen, Yuan Baoquan, Local well-posedness in critical spaces for compressible MHDequations. preprint.
[11] Bian Dongfen, Yuan Baoquan, Well-posedness in super critical spaces for the compressible MHDequations. preprint.
[12] J.-M. BONY. Calcul symbolique et propagation des singularités pour leséquation aux dérivéespartielles non linéaires. Ann. Sci. (E|′)cole Norm. Sup. (4), 14(1981), 209-246.
[13] Lu Ming, Du Yi, Yao Zhengan, Blow-up criterion for compressible MHD equations.J.Math.Anal.Appl., 379(2011), 425-438.
[14] Hu Xianpeng, Wang Dehua, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. submitted for publication.
[15] Hu Xianpeng, Wang Dehua, Global solutions to thethree-dimensional full compressible magne-tohydrodynamic flows. Comm. Math. Phys., (2008), in press.
[16] H. Abidi,T. Hmidi, Résultats d’existence globale pour le système de la magnethohydrody-namique inhomogene. Ann. Math. Blaise Pascal 14(2007), 103-148.
[17] B.Desjardins, C. Lebris, Remarks on a nonhomogeneous model of magnetohydrodynamics. Dif-ferential and Integral Equations. 11(1998), 3,377-394.
[18] J.-F. Gerbeau, C. Lebris, Existence of solution for a density-dependent magnetohydrodynamicequation. Adv. Differential Equations, 2(3)(1998), 427-452.
[19] H. Abidi, M. Paicu, Globale existence for the MHD system in critical spaces. arXIV:0806.3417vl[math-AP] 20 Jun 2008.
[20] H. Fujita and T. Kato, On the Navier-Stokes initial value problem . I. Arch. Rational Mech. Anal.,16(1964), 269-315.
[21] Chen Guiqiang and Wang Dehua, The cauchy Problem for Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics. Vol. I, 421-543, North-Holland, Amsterdam,2002.
[22] Li Tianhong, Some special solutions of the multidimensional Euler equations in R~N. Comm. PureAppl. Anal., 4(4)(2005)757-762.
[23] P.L. Lions, Mathematical topics in fluid mechanics, Volume 1, 2. 1998, Oxford:Clarendon Press,1998.
[24] Yuen Manwai, Analytical solutions to the Navier-stokes equations. arⅩⅣ: 0811.0377v3 [math-ph] 19 Feb 2009.
[25] Yuen Manwai, Analytical blowup solutions to the 2-dimension isothermal Euler-Possion equa-tions of gaseous starsⅡ, arⅩⅣ: 0906.0176v1.
[26] Yuen Manwai, Analytical blowup solutions to the isothermal Euler-Possion equations of gaseousstars in R~N. arXIV: 0906.0178v1.
[27] D.Serre, Solutions classiques globales deséquations d’Euler pour un ?uid parfait compressible.Ann. Inst. Fourier(Grenoble)47, 139-153(1997).
[28] C. Dafermos, Hyperbolic conservation laws in continuum physics. Second edition. Grundlehrender Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], 325.Springer-Verlag, Berlin, 2005.
[29] K. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7,345392(1954).
[30] M. Grassin, Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math.,J.47, 1397-1432(1998).
[31] T.Kato, The cauchy problem for quasi-linear symmetric hyperbolic system. Arch. Rational Mech.Anal., 58, 181-205(1975).
[32] P.Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation.Comm. Pure Appl. Math., 7(1954), 159-193.
[33] T. C. Sideris, Formation of singularities in three-dimensional compressible ?uids. Comm. Math.Phy., 101, 475-485(1985).
[34] Chen Guiqiang and Wang Dehua, The Cauchy problem for the Euler equations for compressible fluids[J]. Handb. Math. Fluid Dynam., 2002. 1: 7-15.
[35] Zhu Xusheng, Wang Weike, The regular solutions of the isentropic Euler equations with degen-erate linear damping [J].Chinese Ann. Math. Ser B, 2005, 26(4): 583-598.
[36] Yuen Manwai, Analytical blowup solutions to the 2-dimension isothermal Euler-Possion equa-tions of gaseous stars. J. Math. Anal. Appl., 341(2008), 445-456.
[37] Lawrencr C. Evans, Partial di?erential equations. Graduate Studies in Mathematics Volume 19.
[38]苗长兴,Fourier局部化方法及其在流体力学方程中的应用.北京应用物理与计算数学研究所.
[39]原保全, Boussinesq方程组在空间中局部解的存在性和延拓准则.数学学报(中文版),0583-1431(2010), 03-0455-14.
[40]苗长兴,调和分析及其在偏微分方程中的应用(第二版).科学出版社.
[41]张恭庆,林渠源,泛函分析讲义.上册,北京大学出版社.
[42]王元明,徐军祥,索伯列夫空间讲义.东南大学出版社.
[2] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases.Comm. Partial Di?erential Equations, 26(2001), 1183-1233.
[3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations. Invent.Math., 141(2000), 579-614.
[4] R. Danchin, On the uniquness in critical spaces for compressible Navier-Stokes equations. No-linear Differential Equations Appl., 12(2005), 111-128.
[5] R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constantdensity. Comm. Partial Differential Equations, 32(2007), 1373-1397.
[6] Haspot,Well-posedness in critical spaces for the system of Navier-Stokes compressible. arXIV :0904.1354v1 [math.AP] 8 Apr 2009.
[7] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens etéquations de Navier-Stokes. J. Differential Equations, 121(1992), 314-328.
[8] J.-Y. Chemin, Perfect incompressible fluids. Oxford University Press, New York, 1998.
[9] J.-Y. Chemin, Théorèmes d’unicitépour le système de Navier-Stokes tridimensionnel. J.d’Analyse Math., 77(1999), 27-50.
[10] Bian Dongfen, Yuan Baoquan, Local well-posedness in critical spaces for compressible MHDequations. preprint.
[11] Bian Dongfen, Yuan Baoquan, Well-posedness in super critical spaces for the compressible MHDequations. preprint.
[12] J.-M. BONY. Calcul symbolique et propagation des singularités pour leséquation aux dérivéespartielles non linéaires. Ann. Sci. (E|′)cole Norm. Sup. (4), 14(1981), 209-246.
[13] Lu Ming, Du Yi, Yao Zhengan, Blow-up criterion for compressible MHD equations.J.Math.Anal.Appl., 379(2011), 425-438.
[14] Hu Xianpeng, Wang Dehua, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. submitted for publication.
[15] Hu Xianpeng, Wang Dehua, Global solutions to thethree-dimensional full compressible magne-tohydrodynamic flows. Comm. Math. Phys., (2008), in press.
[16] H. Abidi,T. Hmidi, Résultats d’existence globale pour le système de la magnethohydrody-namique inhomogene. Ann. Math. Blaise Pascal 14(2007), 103-148.
[17] B.Desjardins, C. Lebris, Remarks on a nonhomogeneous model of magnetohydrodynamics. Dif-ferential and Integral Equations. 11(1998), 3,377-394.
[18] J.-F. Gerbeau, C. Lebris, Existence of solution for a density-dependent magnetohydrodynamicequation. Adv. Differential Equations, 2(3)(1998), 427-452.
[19] H. Abidi, M. Paicu, Globale existence for the MHD system in critical spaces. arXIV:0806.3417vl[math-AP] 20 Jun 2008.
[20] H. Fujita and T. Kato, On the Navier-Stokes initial value problem . I. Arch. Rational Mech. Anal.,16(1964), 269-315.
[21] Chen Guiqiang and Wang Dehua, The cauchy Problem for Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics. Vol. I, 421-543, North-Holland, Amsterdam,2002.
[22] Li Tianhong, Some special solutions of the multidimensional Euler equations in R~N. Comm. PureAppl. Anal., 4(4)(2005)757-762.
[23] P.L. Lions, Mathematical topics in fluid mechanics, Volume 1, 2. 1998, Oxford:Clarendon Press,1998.
[24] Yuen Manwai, Analytical solutions to the Navier-stokes equations. arⅩⅣ: 0811.0377v3 [math-ph] 19 Feb 2009.
[25] Yuen Manwai, Analytical blowup solutions to the 2-dimension isothermal Euler-Possion equa-tions of gaseous starsⅡ, arⅩⅣ: 0906.0176v1.
[26] Yuen Manwai, Analytical blowup solutions to the isothermal Euler-Possion equations of gaseousstars in R~N. arXIV: 0906.0178v1.
[27] D.Serre, Solutions classiques globales deséquations d’Euler pour un ?uid parfait compressible.Ann. Inst. Fourier(Grenoble)47, 139-153(1997).
[28] C. Dafermos, Hyperbolic conservation laws in continuum physics. Second edition. Grundlehrender Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], 325.Springer-Verlag, Berlin, 2005.
[29] K. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7,345392(1954).
[30] M. Grassin, Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math.,J.47, 1397-1432(1998).
[31] T.Kato, The cauchy problem for quasi-linear symmetric hyperbolic system. Arch. Rational Mech.Anal., 58, 181-205(1975).
[32] P.Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation.Comm. Pure Appl. Math., 7(1954), 159-193.
[33] T. C. Sideris, Formation of singularities in three-dimensional compressible ?uids. Comm. Math.Phy., 101, 475-485(1985).
[34] Chen Guiqiang and Wang Dehua, The Cauchy problem for the Euler equations for compressible fluids[J]. Handb. Math. Fluid Dynam., 2002. 1: 7-15.
[35] Zhu Xusheng, Wang Weike, The regular solutions of the isentropic Euler equations with degen-erate linear damping [J].Chinese Ann. Math. Ser B, 2005, 26(4): 583-598.
[36] Yuen Manwai, Analytical blowup solutions to the 2-dimension isothermal Euler-Possion equa-tions of gaseous stars. J. Math. Anal. Appl., 341(2008), 445-456.
[37] Lawrencr C. Evans, Partial di?erential equations. Graduate Studies in Mathematics Volume 19.
[38]苗长兴,Fourier局部化方法及其在流体力学方程中的应用.北京应用物理与计算数学研究所.
[39]原保全, Boussinesq方程组在空间中局部解的存在性和延拓准则.数学学报(中文版),0583-1431(2010), 03-0455-14.
[40]苗长兴,调和分析及其在偏微分方程中的应用(第二版).科学出版社.
[41]张恭庆,林渠源,泛函分析讲义.上册,北京大学出版社.
[42]王元明,徐军祥,索伯列夫空间讲义.东南大学出版社.