流体力学方程组的Cauchy问题
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摘要
本文主要研究了流体力学方程组的Cauchy问题。本文由以下几章组成：
第一章为绪论。在本章中我们简要回顾了一阶拟线性双曲组和与本文相关的一些偏微分方程的研究历史及一些重要结果，并叙述了本文的主要结论。
第二章研究了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。在初值是一个常状态的小扰动并且初速度的旋度等于零的假设下，我们证明了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。
第三章研究了两维的可压非等熵Euler方程的Cauchy问题。当初值是一个常状态的小扰动时，我们给出光滑球对称解的生命跨度的精确估计。在第四章中我们证明了Minkowski空间R~(1+3)中具有慢衰减初值的极值曲面方程Cauchy问题整体光滑解的存在性和唯一性。
在第五章中我们证明了广义Boussinesq方程Cauchy问题整体解的存在性和唯一性。在适当的假设下，我们进一步证明当t趋向无穷大时，广义Boussinesq方程Cauchy问题整体小解的L_∞范数趋向于0。
第六章研究了R~3空间中的不可压magneto-micropolar流体方程组和R~n(n =2, 3)空间中的具有部分粘性的不可压magneto-micropolar流体方程组的Cauchy问题，得到了光滑解的爆破准则。
This thesis mainly concerns the Cauchy problems for fluid flows. This thesis isorganized as follows:
In Chapter 1, we review some of the history and important known results on thestudies of first-order quasilinear hyperbolic systems, and other partial differential equationsrelated to the topic of the thesis. We also state the main results in this thesis.
In Chapter 2, we investigate the two-dimensional compressible isentropic Euler equationsfor Chaplygin gases. Under the assumption that the initial data is close to a constantstate and the vorticity of the initial velocity vanishes, we prove the global existence of thesmooth solution to the Cauchy problem for two-dimensional flow of Chaplygin gases.
In Chapter 3, we study the spherically symmetric Eulerian flows of ideal polytropicgases with variable entropy in two space dimensions. Under the assumption that theinitial data is close to a constant state, we give a precise estimate on the life-span ofsmooth solutions.
In Chapter 4, we prove the global existence of classical solutions to the Cauchyproblem for the minimal surface equation in the Minkowski space ?1+3 with slow decayinitial data.
In Chapter 5, we show the existence and the uniqueness of global solution for theCauchy problem for the generalized Boussinesq equation. Under suitable assumptions, wealso prove that the L~∞norm of small solution to the Cauchy problem for the generalizedBoussinesq equation decays to zero as ?? tends to the infinity.
In Chapter 6, we investigate the Cauchy problem for incompressible magneto-micropolarfluid equations in R~3 and incompressible magneto-micropolar fluid equations with partialviscosity in R~n(n = 2, 3). In this Chapter we derive blow-up criteria of smooth solutions.
第一章为绪论。在本章中我们简要回顾了一阶拟线性双曲组和与本文相关的一些偏微分方程的研究历史及一些重要结果，并叙述了本文的主要结论。
第二章研究了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。在初值是一个常状态的小扰动并且初速度的旋度等于零的假设下，我们证明了两维可压等熵Euler方程Cauchy问题光滑解的整体存在性。
第三章研究了两维的可压非等熵Euler方程的Cauchy问题。当初值是一个常状态的小扰动时，我们给出光滑球对称解的生命跨度的精确估计。在第四章中我们证明了Minkowski空间R~(1+3)中具有慢衰减初值的极值曲面方程Cauchy问题整体光滑解的存在性和唯一性。
在第五章中我们证明了广义Boussinesq方程Cauchy问题整体解的存在性和唯一性。在适当的假设下，我们进一步证明当t趋向无穷大时，广义Boussinesq方程Cauchy问题整体小解的L_∞范数趋向于0。
第六章研究了R~3空间中的不可压magneto-micropolar流体方程组和R~n(n =2, 3)空间中的具有部分粘性的不可压magneto-micropolar流体方程组的Cauchy问题，得到了光滑解的爆破准则。
This thesis mainly concerns the Cauchy problems for fluid flows. This thesis isorganized as follows:
In Chapter 1, we review some of the history and important known results on thestudies of first-order quasilinear hyperbolic systems, and other partial differential equationsrelated to the topic of the thesis. We also state the main results in this thesis.
In Chapter 2, we investigate the two-dimensional compressible isentropic Euler equationsfor Chaplygin gases. Under the assumption that the initial data is close to a constantstate and the vorticity of the initial velocity vanishes, we prove the global existence of thesmooth solution to the Cauchy problem for two-dimensional flow of Chaplygin gases.
In Chapter 3, we study the spherically symmetric Eulerian flows of ideal polytropicgases with variable entropy in two space dimensions. Under the assumption that theinitial data is close to a constant state, we give a precise estimate on the life-span ofsmooth solutions.
In Chapter 4, we prove the global existence of classical solutions to the Cauchyproblem for the minimal surface equation in the Minkowski space ?1+3 with slow decayinitial data.
In Chapter 5, we show the existence and the uniqueness of global solution for theCauchy problem for the generalized Boussinesq equation. Under suitable assumptions, wealso prove that the L~∞norm of small solution to the Cauchy problem for the generalizedBoussinesq equation decays to zero as ?? tends to the infinity.
In Chapter 6, we investigate the Cauchy problem for incompressible magneto-micropolarfluid equations in R~3 and incompressible magneto-micropolar fluid equations with partialviscosity in R~n(n = 2, 3). In this Chapter we derive blow-up criteria of smooth solutions.
引文
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2] R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250.
[3] G. Ahmadi and M. Shahinpoor,Universal stability of magneto-micropolar fluid motions,Internat. J. Engrg. Sci., 12 (1974), 657-663.
[4] D. G. Akmel, Global existence and decay for solution to the bad Boussinesq equation in twospace dimensions, Applicable Analysis, 83 (2004), 17-36.
[5] S. Alinhac, Une solution approch′ee en grand temps des′equations d’Euler compressiblesaxisymmetric en dimension deux, Comm. Partial Diff. Equ., 17 (1992), 447-490.
[6] S. Alinhac, Temps de vie des solutions r′eguli`eres d’Euler compressibles axisymmetric endimension deux, Invent. Math., 111 (1993), 627-670.
[7] S. Alinhac, Blowup for nonlinear hyperbolic equations, Boston, Birkhauser, (1995).
[8] S. Alinhac, Explosion g′eom′etrque pour des syst′emes quasi-lin′eaires, American Journal ofMathematics, 117 (1995), 987-1017.
[9] S. Alinhac, Stability of Geometric blowup, Arch. Rat. Mech. Anal., 150 (1999), 97-125.
[10] S. Alinhac, Remarks on the blowup rate of classical solution to quasilinear multidimentionalhyperbolic systems, J. Math. Pure Appl., 79 (2000), 839-854.
[11] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,Invent. Math., 145 (2001), 597-618.
[12] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, General solutions of nonlinearequations in the geometric theory of the relativistic string, Comm. Math. Phys., 84, 471(1982).
[13] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the3-D Euler equations , Comm. Math. Phys. 94 (1984), 61-66.
[14] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces, Grundlehren der Mathematischen Wissenschaften,Springer-Verlag , Berlin, 1976.
[15] M. Bordeman and J. Hoppe, The dynamics of relativistic membranes: Reduction to 2-dimensional fluid dynamics, Phys. Lett. B, 317 (1993), 315-320.
[16] M. J. Boussinesq, Essai sur la th′eorie des eaux courantes, M′emoires pr′esent′es par diverssavants′a I,Acad′emie des Sciences Inst. France, s′eries 2 (1877), 1-680.
[17] S. Brendle, hypersurfaces in Minkowski space with vanishing mean curvature, Comm. PureAppl. Math., 28 (2002), 1249-1279.
[18] R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energydissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.
[19] M. Cannone, Q. L. Chen and C. X. Miao A losing estimate for the ideal MHD equationswith application to blow-up criterion, SIAM J. Math. Anal., 38 (2007), 1847-1859.
[20] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations,J. Math. Phys., 44 (2003), 6132-6139.
[21] J. Y. Chemin, perfect incompressible Fluids, Oxford Lecture Ser. Math. Appl., 14, TheClarendon Press/ Oxford Univ. Press, New York, 1998.
[22] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,Comm. Pure Appl. Math., 39 (1986), 267-282.
[23] D. Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs inMathematics, European Mathematical Society, 2007.
[24] D. Christodoulou, The formation of black holes in general relativity, EMS Monographs inMathematics, European Mathematical Society, 2009.
[25] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Appl. Math. Sci. 21,Spring-Verlag, 1976.
[26] Wen-Rong Dai and De-Xing Kong, Asymptotic behavior of global classical solutions ofgeneral qusilinear hyperbolic systems with weakly linearly degeneracy, Chinese Ann. Math.Ser. 27B (2006), 263-286.
[27] Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classicalsolutions of qusilinear hyperbolic systems with linearly degenerate characteristic fields, J.Differential Equations, 235 (2007), 127-165.
[28] R. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems ofconservation laws, Indiana U. Math. J., 24 (1975), 1047-1071.
[29] R. DiPerna, Decay of solutions to nonlinear hyperbolic systems of conservation laws with aconvex extension, Arch. Rat. Mech. Anal., 60 (1977), 1-46.
[30] B.-Q. Dong and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensionalmicropolar flows, J. Math. Phys., 50 (2009), 103525-1-103525-13.
[31] AC Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16(1966), 1-18.
[32] Sadek Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the MorreyCampanatospace, Nonlinear Differ. Equ. Appl., in press.
[33] G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of themicropolar fluid equations, Internat. J. Eng. Sci., 15 (1977), 105-108.
[34] R. T. Glassey, Existence in the large for□u =F(u) in two space dimensions, Math. Z.,178 (1981), 233-261.
[35] P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressibleEuler equations with variable entropy in three space dimensions, Arch. Rat. Mech.Anal., 177 (2005), 479-511.
[36] P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygingases with variable entropy, J. Math. Pure Appl., 87 (2007), 91-117.
[37] M. Grassin, Global smooth solutions to Euler equations for perfect gas, Indiana. Univ. Math.J., 47 (1998), 1397-1432.
[38] C. He and Z. P. Xin, Partial regularity of suitable weak sokutions to the incompressiblemagnetohydrodynamics equations, J. Funct. Anal., 227 (2005), 113-152.
[39] J. Hoppe, Some classical solutions of relativistic membrane equations in 4-space-time dimensions,Phys.Letter B, 329 (1994), 10-14.
[40] L. H¨ormander, The lifespan of classical solutions of nonlinear hyperbolic equations, LectureNotes in Math., 1256 (1987), 214-280.
[41] L. H¨ormander, ??1 ???∞estimates for the wave operator, Analyse Math′ematique et Applications,Gauthier-Villars, pairs, 1988, 211-234.
[42] L. H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations, Math′ematiqueand Applications 26, Spring-Verlag, 1997.
[43] A. Hoshiga and H. Kubo, Global solvability for systems of nonlinear wave equations withmultiple speeds in two space dimensions, Diff. Int. Equations, 17 (2004), 593-622.
[44] A. Hoshiga, Existence and blowing up of solutions to systems of quasilinear wave equationsin two space dimensions, Adv. Math. Sci. Appl., 15 (2005), 69-110.
[45] A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations withquadratic nonlinearities in 2-dimensional space, Funkcialaj Ekvacioj, 69 (2006), 357-384.
[46] W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterraequation with large initial data, SIAM Journal on Mathematical Analysis, 16 (1985) 110-134.
[47] Shou-Jun Huang and De-Xing Kong, Equations for the motion of relativistic torus in theMinkowski space ?1+??, J. Math. Phys., 48 (2007), 103508-1-103508-15.
[48] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm.Pure Appl. Math., 27 (1974), 377-405.
[49] V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinearequation of parabolic and hyperbolic types, Journal of Soviet Mathematics, 10 (1978), 53-70.
[50] T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Rat.Mech. Anal., 58 (1975), 181-205.
[51] S. Klainerman, Global existence for nonlinear wave equation , Comm. Pure Appl. Math.,33 (1980), 43-101.
[52] S. Klainerman, Uniform decay estimates and the lorentz invariance of the classical waveequations, Comm. Pure Appl. Math., 38 (1985), 321-332.
[53] S. Klainerman, The null condition and global existence to nonlinear wave equations, Lecturesin Appl. Math. 23, Amer. Math. Soc., 1986, 293-326.
[54] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ???+1,Comm. Pure Appl. Math., 40 (1987), 111-116.
[55] De-Xing Kong, Cauchy problem for quasilinear hyperbolic systems with higher order disspativeterms, Nonlinear Differential Equations and Applications, 4 (1997), 477-489.
[56] De-Xing Kong and Mikio Tsuji, Global Solutions for 2×2 Hyperbolic Systems with LinearlyDegenerate Characteristics, Funkcialaj Ekvacioj, 42 (1999), 129-155.
[57] De-Xing Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slowdecay initial data, Chinese Ann. Math., 21B (2000), 413-440.
[58] De-Xing Kong, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, theMathematical Society of Japan, Tokyo, 2000.
[59] De-Xing Kong, Formation and propagation of singularities for 2×2 quasilinear hyperbolicsystems, Trans. Amer. Math. Soc., 354 (2002), 3155-3179.
[60] De-Xing Kong, Global structure stability of Riemann solutions of quasilinear hyperbolicsystems of conservation laws: shocks and contact discontinuities, J. Differential Equations,188 (2003), 242-271.
[61] De-Xing Kong and Tong Yang, Asymptotic behavior of global classical solutions of quasilinearhyperbolic systems, Comm. Partial Differential Equations, 28 (2003), 1203-1220.
[62] De-Xing Kong, A nonlinear geometric equation related to electrodynamics, Europhys. Lett.,66 (2004), 617-623.
[63] De-Xing Kong, Global structure instability of Riemann solutions of quasilinear hyperbolicsystems of conservation laws: rarefraction waves, J. Differential Equations, 219 (2005),421-450.
[64] De-Xing Kong, Qing-You Sun and Yi Zhou, The equation for time-like extremal surfacesin Minkowski space ?2+??, J. Math. Phys., 47 (2006), 013503.
[65] De-Xing Kong, Qiang Zhang and Qing Zhou, The dynamics of relativistic strings movingin the Minkowski space ?1+??, Comm. Math. Phys., 269 (2007), 153-174.
[66] De-Xing Kong and Qiang Zhang, Solution formula and time periodicity for the motion ofrelativistic strings in the Minkowski space ??1+??, Phys. D. 238 (2009), 902-922.
[67] DeXing Kong，KeFeng Liu and YuZhu Wang, Global existence of smooth solutions to twodimensionalcompressible isentropic Euler equations for Chaplygin gases, Science in China(Series A), 53 (2010), 719-738.
[68] M. Kovalyov, Long-time bahaviour of solutions of systems of nonlinear wave equations,Comm. Partial Differential Equations, 12 (1987), 471-501.
[69] H. Kozono and Y. Taniuchi, limiting case of the Sobolev inequality in BMO with applicationto the Euler Equations, Comm. Math. Phys., 214 (2000), 191-200.
[70] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spacesand regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002),251-278.
[71] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957),537-556.
[72] Z. Lei and Y. Zhou, BKM criterion and global weak solutions for Magnetohydrodynamicswith zero viscosity, DCDS-A 25 (2009), 575-583.
[73] H. A. Levine and B. O. Sleeman, A note on the non-existence of global solutions of initialboundary value problems for the Boussinesq equation ?????? = ?????? + 3?????????? ? 12(??2)????, J.Math. Anal., 107 (1985), 206-210.
[74] Ta-Tsien Li and Wen-Ci Yu, Boundary value problems for quasilinear hyperbolic systems,Duke University Mathematics Series V, (1985).
[75] T.-T. Li and Y. M. Chen, Global Classical Solution for Nonlinear Evolution Equations,Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 45. LongmanScientific and Technical: New York, 1992.
[76] T.-T. Li and Y. Zhou, Life-span of classical solution to fully nonlinear wave equations -II,Nonlinear Analysis, Theory, Method and Applications, 19 (1992), 833-853.
[77] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations in twospacedimensions II, J. Partial Differential Equations, 6 (1993), 17-38.
[78] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations in twospace, J. Math. Pures Anal., 73 (1994), 223-249.
[79] T.-T. Li and Y. Zhou, Nonlinear stability for two space dimensional wave equations withhigher order perturbation, Nonlin. world, 1 (1994), 35-58.
[80] Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutionsfor general qusilinear hyperbolic systems, Comm. Partial Differential Equations, 19(1994), 1263-1317.
[81] Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinearnon-strictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229.
[82] Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general qusilinearhyperbolic systems with decay initial data, Nonlinear Analysis, 28 (1997),1299-1322.
[83] Ta-Tsien Li and De-Xing Kong, Initial value problem for general quasilinear hyperbolicsystems with characteristics with constant multiplicity, J. Partial Differential Equations,10 (1997), 299-322.
[84] Ta-Tsien Li and De-Xing Kong, Global classical solutions with small amplitude for generalquasilinear hyperbolic systems, in New Approaches in Nonlinear Analysis, Hardronic Press(1999), 203-237.
[85] Ta-Tsien Li and De-Xing Kong, Breakdown of classical solutions to quasilinear hyperbolicsystems, Nonlinear Analysis, 40 (2000), 407-437.
[86] F. Linares and M. Scialom, Asymptotic behavior of solutions of a generalized Boussinesqtype equation, Nonlinear Analysis, 25 (1995), 1147-1158.
[87] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, Journalof Differential Equations, 106 (1993), 257-293.
[88] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initialdata, Comm. Pure Appl. Math., 43 (1990), 445-472.
[89] H. Lindblad, A remark on global existence for small initial data of the minimal surfaceequation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
[90] Tai-Ping Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolicpartial differential equations, J. Differential Equations, 33 (1979), 92-111.
[91] Y. Liu, Existence and blow up of a nonlinear Pochhammer-Chree equation, Indiana UniversityMathematics Journal, 45 (1996), 797-816.
[92] Y. Liu, Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation,Journal of Functional Analysis, 147 (1997), 51-68.
[93] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several SpaceVariables, Applied Mathematical Sciences 53, Springer, 1984.
[94] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press:Cambridge, 2002.
[95] W. H. Meeks III, A. Ros and H. Rosenberg, The Global Theory of Minimal Surfaces inFlat Spaces, Lecture Notes in Mathematics, 1775, Fondazione CIME, (2002).
[96] M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strongsolutions, Math. Nachr., 188 (1997), 301-319.
[97] E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: globalexistence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.
[98] C. Rogers and W. F. Shadwick, B¨acklund Transformations and Their Applications, AcademicPress, New York, London, 1982.
[99] B. L. Roˇzdestvenski and N. N. Janenko, Systems of Quasilinear Equations and Their Applicationsto Gas Dynamics, Translations of Mathematical Monographs 55, Amer. Math.Soc., Providence, RI, 1983.
[100] M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemannproblem, Ind. Univer. Math. J., 34 (1985), 533-589.
[101] G. Schneider and C. W. Eugene, Kawahara dynamics in dispersive media, Physica D, 152-153 (2001), 108-110.
[102] D. Serre, Richness and the classification of quasilinear hyperbolic systems, I.M.A., 29(1989), 315-333.
[103] D. Serre, Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000.
[104] T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm.Math. Phys., 101 (1985), 475-485.
[105] T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Eulerequations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.
[106] T. C. Sideris, Delayed singularity formation in 2-D compressible flow, Amer. J. Math., 119(1997), 371-422.
[107] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, InternationalPress, 1995.
[108] E. M. Stein, Singular Integrals and Differentiability Properties of Function, Princeton University:Princeton, NJ, 1970.
[109] M. E. Taylor, Partial Differential Equations, Vol. 1: Basic Theory, New York: Springer-Verlag, 1996.
[110] M. E. Taylor, Partial Differential Equations, Vol. 2: Qualitative Studies of Linear Equations,New York: Springer-Verlag, 1996.
[111] M. E. Tayler, Partial Differential Equations III, Nonlinear Equations, Springer: New York,1996.
[112] H. Triebel, Theory of Function Spaces, Birkha¨user, Boston, 1983.
[113] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compact datain two space dimensions, J. Differential Equations, 104 (1993), 332-360.
[114] S. B. Wang and G. W. Chen, Small amplitude solutions of the generlized IMBq equation,J. Math. Anal. Appl., 264 (2002), 846-866.
[115] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation,Nonliear Analysis TMA, 64 (2006), 159-173.
[116] Y. Wang and C. L. Mu, Global existence and blow-up of the solutions for the multidimensionalgeneralized Boussinesq equation, Math. Meth. Appl. Sci., 30 (2007), 1403-1417.
[117] Y. Wang and C. L. Mu, Blow-up and scattering of solution for a generalized Boussinesqequation, Applied Mathematics and Computation, 188 (2007), 1131-1141.
[118] Yu-Zhu Wang, Global existence and asymptotic behaviour of solutions for the generalizedBoussinesq equation, Nonlinear Analysis, 70 (2009), 465-482.
[119] Yu-Zhu Wang, Global existence of classical solutions to the minimal surface equation in twospace dimensions with slow decay initial data, Journal Mathematical Physics, 50 (2009),103506-01-103506-01-14.
[120] N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in abounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526.
[121] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magnetomicropolarfluid equations, Math. Meth. Appl. Sci., 31 (2008), 1113-1130.
[122] Y. Zhou, Regularity criteria for the 3D MHD equations in term of the pressure, Internat.J. Non-linear Mech, 41 (2006), 1174-1180.
[123] Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,Chinese Ann. Math., 25B (2004), 37-56.
[124] Y. Zhou and Z. Lei, Logarithmically improved criterion for Euler and Navier-Stokes Equations,preprint, (2008).
[125]姜礼尚，孔德兴，陈志浩，应用偏微分方程讲义，高等教育出版社，2008，1-275。
[126]周忆，完全非线性波动方程的经典解的整体存在性及生命区间，博士论文，复旦大学，1992。125
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[2] Wen-Rong Dai, De-Xing Kong and Ke-Feng Liu, Dissipative hyperbolic geometric flow,Asian J. Math., 12 (2008), 345-364.
[3] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initialdata, Comm. Pure Appl. Math. 43 (1990), 445-472.
[4] H. Lindblad, A remark on global existence for small initial data of the minimal surfaceequation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
[5] L. H¨ormander, L~1-L~∞estimates for the wave operators, in Analyse Math′ematique etApplications, Gauthier-Villars, Pairs, 1988.
[6] L.H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations,Math′ematique and Applications, 26, Spring-Verlag, 1997.
[7] De-Xing Kong, Life-span of classical solutions to quasilinear reducible hyperbolic systemsand its applications, Chin. Ann. of Math. 13A (1992), 188-195.
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[10] De-Xing Kong, Ke-Feng Liu and De-Liang Xu, The hyperbolic geometric flow onRiemann surfaces , Comm. in Partial Differential Equation 34 (2009), 553-580.
[11] S. Klainerman, Global existence for nonlinear wave equation, Comm. Pure Appl.Math. 33 (1980), 43-101.
[12] S. Klainerman, Uniform decay estimates and the lorentz invariance of the classicalwave equations, Comm. Pure Appl. Math. 38 (1985), 321-332.
[13] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space???+1, Comm. Pure Appl. Math. 40 (1987), 111-116.
[14] M. Kovalyov, Long-time behaviour of solutions of systems of nonlinear wave equations,Comm. in Partial Differential Equation 12 (1987), 471-501.
[15] T.-T. Li and Y.-M. Chen, Initial value problems for nonlinear wave equations , Comm.in Partial Differential Equation 13 (1988), 383-422.
[16] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations intwo space dimensions , J. Math. Pures Appl. 2 (1994), 223-249.
[17] T.-T. Li and Y. Zhou, Life-span of classical solutions to fully nonlinear wave equationsII , Nonlinear Analysis, Theory, Methods and Applications 19 (1992), 833-853.
[18] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis, II,International Press, 1995.
[19] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compactdata in two space dimensions, J. Differential Equations 104 (1993), 332-360.
[2] R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250.
[3] G. Ahmadi and M. Shahinpoor,Universal stability of magneto-micropolar fluid motions,Internat. J. Engrg. Sci., 12 (1974), 657-663.
[4] D. G. Akmel, Global existence and decay for solution to the bad Boussinesq equation in twospace dimensions, Applicable Analysis, 83 (2004), 17-36.
[5] S. Alinhac, Une solution approch′ee en grand temps des′equations d’Euler compressiblesaxisymmetric en dimension deux, Comm. Partial Diff. Equ., 17 (1992), 447-490.
[6] S. Alinhac, Temps de vie des solutions r′eguli`eres d’Euler compressibles axisymmetric endimension deux, Invent. Math., 111 (1993), 627-670.
[7] S. Alinhac, Blowup for nonlinear hyperbolic equations, Boston, Birkhauser, (1995).
[8] S. Alinhac, Explosion g′eom′etrque pour des syst′emes quasi-lin′eaires, American Journal ofMathematics, 117 (1995), 987-1017.
[9] S. Alinhac, Stability of Geometric blowup, Arch. Rat. Mech. Anal., 150 (1999), 97-125.
[10] S. Alinhac, Remarks on the blowup rate of classical solution to quasilinear multidimentionalhyperbolic systems, J. Math. Pure Appl., 79 (2000), 839-854.
[11] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,Invent. Math., 145 (2001), 597-618.
[12] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, General solutions of nonlinearequations in the geometric theory of the relativistic string, Comm. Math. Phys., 84, 471(1982).
[13] J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the3-D Euler equations , Comm. Math. Phys. 94 (1984), 61-66.
[14] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces, Grundlehren der Mathematischen Wissenschaften,Springer-Verlag , Berlin, 1976.
[15] M. Bordeman and J. Hoppe, The dynamics of relativistic membranes: Reduction to 2-dimensional fluid dynamics, Phys. Lett. B, 317 (1993), 315-320.
[16] M. J. Boussinesq, Essai sur la th′eorie des eaux courantes, M′emoires pr′esent′es par diverssavants′a I,Acad′emie des Sciences Inst. France, s′eries 2 (1877), 1-680.
[17] S. Brendle, hypersurfaces in Minkowski space with vanishing mean curvature, Comm. PureAppl. Math., 28 (2002), 1249-1279.
[18] R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energydissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.
[19] M. Cannone, Q. L. Chen and C. X. Miao A losing estimate for the ideal MHD equationswith application to blow-up criterion, SIAM J. Math. Anal., 38 (2007), 1847-1859.
[20] D. Chae and H. Huh, Global existence for small initial data in the Born-Infeld equations,J. Math. Phys., 44 (2003), 6132-6139.
[21] J. Y. Chemin, perfect incompressible Fluids, Oxford Lecture Ser. Math. Appl., 14, TheClarendon Press/ Oxford Univ. Press, New York, 1998.
[22] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,Comm. Pure Appl. Math., 39 (1986), 267-282.
[23] D. Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs inMathematics, European Mathematical Society, 2007.
[24] D. Christodoulou, The formation of black holes in general relativity, EMS Monographs inMathematics, European Mathematical Society, 2009.
[25] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Appl. Math. Sci. 21,Spring-Verlag, 1976.
[26] Wen-Rong Dai and De-Xing Kong, Asymptotic behavior of global classical solutions ofgeneral qusilinear hyperbolic systems with weakly linearly degeneracy, Chinese Ann. Math.Ser. 27B (2006), 263-286.
[27] Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classicalsolutions of qusilinear hyperbolic systems with linearly degenerate characteristic fields, J.Differential Equations, 235 (2007), 127-165.
[28] R. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems ofconservation laws, Indiana U. Math. J., 24 (1975), 1047-1071.
[29] R. DiPerna, Decay of solutions to nonlinear hyperbolic systems of conservation laws with aconvex extension, Arch. Rat. Mech. Anal., 60 (1977), 1-46.
[30] B.-Q. Dong and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensionalmicropolar flows, J. Math. Phys., 50 (2009), 103525-1-103525-13.
[31] AC Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16(1966), 1-18.
[32] Sadek Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the MorreyCampanatospace, Nonlinear Differ. Equ. Appl., in press.
[33] G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of themicropolar fluid equations, Internat. J. Eng. Sci., 15 (1977), 105-108.
[34] R. T. Glassey, Existence in the large for□u =F(u) in two space dimensions, Math. Z.,178 (1981), 233-261.
[35] P. Godin, The lifespan of a class of smooth spherically symmetric solutions of the compressibleEuler equations with variable entropy in three space dimensions, Arch. Rat. Mech.Anal., 177 (2005), 479-511.
[36] P. Godin, Global existence of a class of smooth 3D spherically symmetric flows of Chaplygingases with variable entropy, J. Math. Pure Appl., 87 (2007), 91-117.
[37] M. Grassin, Global smooth solutions to Euler equations for perfect gas, Indiana. Univ. Math.J., 47 (1998), 1397-1432.
[38] C. He and Z. P. Xin, Partial regularity of suitable weak sokutions to the incompressiblemagnetohydrodynamics equations, J. Funct. Anal., 227 (2005), 113-152.
[39] J. Hoppe, Some classical solutions of relativistic membrane equations in 4-space-time dimensions,Phys.Letter B, 329 (1994), 10-14.
[40] L. H¨ormander, The lifespan of classical solutions of nonlinear hyperbolic equations, LectureNotes in Math., 1256 (1987), 214-280.
[41] L. H¨ormander, ??1 ???∞estimates for the wave operator, Analyse Math′ematique et Applications,Gauthier-Villars, pairs, 1988, 211-234.
[42] L. H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations, Math′ematiqueand Applications 26, Spring-Verlag, 1997.
[43] A. Hoshiga and H. Kubo, Global solvability for systems of nonlinear wave equations withmultiple speeds in two space dimensions, Diff. Int. Equations, 17 (2004), 593-622.
[44] A. Hoshiga, Existence and blowing up of solutions to systems of quasilinear wave equationsin two space dimensions, Adv. Math. Sci. Appl., 15 (2005), 69-110.
[45] A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations withquadratic nonlinearities in 2-dimensional space, Funkcialaj Ekvacioj, 69 (2006), 357-384.
[46] W. J. Hrusa, Global existence and asymptotic stability for a semilinear hyperbolic Volterraequation with large initial data, SIAM Journal on Mathematical Analysis, 16 (1985) 110-134.
[47] Shou-Jun Huang and De-Xing Kong, Equations for the motion of relativistic torus in theMinkowski space ?1+??, J. Math. Phys., 48 (2007), 103508-1-103508-15.
[48] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm.Pure Appl. Math., 27 (1974), 377-405.
[49] V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinearequation of parabolic and hyperbolic types, Journal of Soviet Mathematics, 10 (1978), 53-70.
[50] T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Rat.Mech. Anal., 58 (1975), 181-205.
[51] S. Klainerman, Global existence for nonlinear wave equation , Comm. Pure Appl. Math.,33 (1980), 43-101.
[52] S. Klainerman, Uniform decay estimates and the lorentz invariance of the classical waveequations, Comm. Pure Appl. Math., 38 (1985), 321-332.
[53] S. Klainerman, The null condition and global existence to nonlinear wave equations, Lecturesin Appl. Math. 23, Amer. Math. Soc., 1986, 293-326.
[54] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ???+1,Comm. Pure Appl. Math., 40 (1987), 111-116.
[55] De-Xing Kong, Cauchy problem for quasilinear hyperbolic systems with higher order disspativeterms, Nonlinear Differential Equations and Applications, 4 (1997), 477-489.
[56] De-Xing Kong and Mikio Tsuji, Global Solutions for 2×2 Hyperbolic Systems with LinearlyDegenerate Characteristics, Funkcialaj Ekvacioj, 42 (1999), 129-155.
[57] De-Xing Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slowdecay initial data, Chinese Ann. Math., 21B (2000), 413-440.
[58] De-Xing Kong, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, theMathematical Society of Japan, Tokyo, 2000.
[59] De-Xing Kong, Formation and propagation of singularities for 2×2 quasilinear hyperbolicsystems, Trans. Amer. Math. Soc., 354 (2002), 3155-3179.
[60] De-Xing Kong, Global structure stability of Riemann solutions of quasilinear hyperbolicsystems of conservation laws: shocks and contact discontinuities, J. Differential Equations,188 (2003), 242-271.
[61] De-Xing Kong and Tong Yang, Asymptotic behavior of global classical solutions of quasilinearhyperbolic systems, Comm. Partial Differential Equations, 28 (2003), 1203-1220.
[62] De-Xing Kong, A nonlinear geometric equation related to electrodynamics, Europhys. Lett.,66 (2004), 617-623.
[63] De-Xing Kong, Global structure instability of Riemann solutions of quasilinear hyperbolicsystems of conservation laws: rarefraction waves, J. Differential Equations, 219 (2005),421-450.
[64] De-Xing Kong, Qing-You Sun and Yi Zhou, The equation for time-like extremal surfacesin Minkowski space ?2+??, J. Math. Phys., 47 (2006), 013503.
[65] De-Xing Kong, Qiang Zhang and Qing Zhou, The dynamics of relativistic strings movingin the Minkowski space ?1+??, Comm. Math. Phys., 269 (2007), 153-174.
[66] De-Xing Kong and Qiang Zhang, Solution formula and time periodicity for the motion ofrelativistic strings in the Minkowski space ??1+??, Phys. D. 238 (2009), 902-922.
[67] DeXing Kong，KeFeng Liu and YuZhu Wang, Global existence of smooth solutions to twodimensionalcompressible isentropic Euler equations for Chaplygin gases, Science in China(Series A), 53 (2010), 719-738.
[68] M. Kovalyov, Long-time bahaviour of solutions of systems of nonlinear wave equations,Comm. Partial Differential Equations, 12 (1987), 471-501.
[69] H. Kozono and Y. Taniuchi, limiting case of the Sobolev inequality in BMO with applicationto the Euler Equations, Comm. Math. Phys., 214 (2000), 191-200.
[70] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spacesand regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002),251-278.
[71] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957),537-556.
[72] Z. Lei and Y. Zhou, BKM criterion and global weak solutions for Magnetohydrodynamicswith zero viscosity, DCDS-A 25 (2009), 575-583.
[73] H. A. Levine and B. O. Sleeman, A note on the non-existence of global solutions of initialboundary value problems for the Boussinesq equation ?????? = ?????? + 3?????????? ? 12(??2)????, J.Math. Anal., 107 (1985), 206-210.
[74] Ta-Tsien Li and Wen-Ci Yu, Boundary value problems for quasilinear hyperbolic systems,Duke University Mathematics Series V, (1985).
[75] T.-T. Li and Y. M. Chen, Global Classical Solution for Nonlinear Evolution Equations,Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 45. LongmanScientific and Technical: New York, 1992.
[76] T.-T. Li and Y. Zhou, Life-span of classical solution to fully nonlinear wave equations -II,Nonlinear Analysis, Theory, Method and Applications, 19 (1992), 833-853.
[77] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations in twospacedimensions II, J. Partial Differential Equations, 6 (1993), 17-38.
[78] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations in twospace, J. Math. Pures Anal., 73 (1994), 223-249.
[79] T.-T. Li and Y. Zhou, Nonlinear stability for two space dimensional wave equations withhigher order perturbation, Nonlin. world, 1 (1994), 35-58.
[80] Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutionsfor general qusilinear hyperbolic systems, Comm. Partial Differential Equations, 19(1994), 1263-1317.
[81] Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinearnon-strictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229.
[82] Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general qusilinearhyperbolic systems with decay initial data, Nonlinear Analysis, 28 (1997),1299-1322.
[83] Ta-Tsien Li and De-Xing Kong, Initial value problem for general quasilinear hyperbolicsystems with characteristics with constant multiplicity, J. Partial Differential Equations,10 (1997), 299-322.
[84] Ta-Tsien Li and De-Xing Kong, Global classical solutions with small amplitude for generalquasilinear hyperbolic systems, in New Approaches in Nonlinear Analysis, Hardronic Press(1999), 203-237.
[85] Ta-Tsien Li and De-Xing Kong, Breakdown of classical solutions to quasilinear hyperbolicsystems, Nonlinear Analysis, 40 (2000), 407-437.
[86] F. Linares and M. Scialom, Asymptotic behavior of solutions of a generalized Boussinesqtype equation, Nonlinear Analysis, 25 (1995), 1147-1158.
[87] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, Journalof Differential Equations, 106 (1993), 257-293.
[88] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initialdata, Comm. Pure Appl. Math., 43 (1990), 445-472.
[89] H. Lindblad, A remark on global existence for small initial data of the minimal surfaceequation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
[90] Tai-Ping Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolicpartial differential equations, J. Differential Equations, 33 (1979), 92-111.
[91] Y. Liu, Existence and blow up of a nonlinear Pochhammer-Chree equation, Indiana UniversityMathematics Journal, 45 (1996), 797-816.
[92] Y. Liu, Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation,Journal of Functional Analysis, 147 (1997), 51-68.
[93] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several SpaceVariables, Applied Mathematical Sciences 53, Springer, 1984.
[94] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press:Cambridge, 2002.
[95] W. H. Meeks III, A. Ros and H. Rosenberg, The Global Theory of Minimal Surfaces inFlat Spaces, Lecture Notes in Mathematics, 1775, Fondazione CIME, (2002).
[96] M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strongsolutions, Math. Nachr., 188 (1997), 301-319.
[97] E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: globalexistence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.
[98] C. Rogers and W. F. Shadwick, B¨acklund Transformations and Their Applications, AcademicPress, New York, London, 1982.
[99] B. L. Roˇzdestvenski and N. N. Janenko, Systems of Quasilinear Equations and Their Applicationsto Gas Dynamics, Translations of Mathematical Monographs 55, Amer. Math.Soc., Providence, RI, 1983.
[100] M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemannproblem, Ind. Univer. Math. J., 34 (1985), 533-589.
[101] G. Schneider and C. W. Eugene, Kawahara dynamics in dispersive media, Physica D, 152-153 (2001), 108-110.
[102] D. Serre, Richness and the classification of quasilinear hyperbolic systems, I.M.A., 29(1989), 315-333.
[103] D. Serre, Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000.
[104] T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm.Math. Phys., 101 (1985), 475-485.
[105] T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Eulerequations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.
[106] T. C. Sideris, Delayed singularity formation in 2-D compressible flow, Amer. J. Math., 119(1997), 371-422.
[107] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, InternationalPress, 1995.
[108] E. M. Stein, Singular Integrals and Differentiability Properties of Function, Princeton University:Princeton, NJ, 1970.
[109] M. E. Taylor, Partial Differential Equations, Vol. 1: Basic Theory, New York: Springer-Verlag, 1996.
[110] M. E. Taylor, Partial Differential Equations, Vol. 2: Qualitative Studies of Linear Equations,New York: Springer-Verlag, 1996.
[111] M. E. Tayler, Partial Differential Equations III, Nonlinear Equations, Springer: New York,1996.
[112] H. Triebel, Theory of Function Spaces, Birkha¨user, Boston, 1983.
[113] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compact datain two space dimensions, J. Differential Equations, 104 (1993), 332-360.
[114] S. B. Wang and G. W. Chen, Small amplitude solutions of the generlized IMBq equation,J. Math. Anal. Appl., 264 (2002), 846-866.
[115] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation,Nonliear Analysis TMA, 64 (2006), 159-173.
[116] Y. Wang and C. L. Mu, Global existence and blow-up of the solutions for the multidimensionalgeneralized Boussinesq equation, Math. Meth. Appl. Sci., 30 (2007), 1403-1417.
[117] Y. Wang and C. L. Mu, Blow-up and scattering of solution for a generalized Boussinesqequation, Applied Mathematics and Computation, 188 (2007), 1131-1141.
[118] Yu-Zhu Wang, Global existence and asymptotic behaviour of solutions for the generalizedBoussinesq equation, Nonlinear Analysis, 70 (2009), 465-482.
[119] Yu-Zhu Wang, Global existence of classical solutions to the minimal surface equation in twospace dimensions with slow decay initial data, Journal Mathematical Physics, 50 (2009),103506-01-103506-01-14.
[120] N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in abounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526.
[121] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magnetomicropolarfluid equations, Math. Meth. Appl. Sci., 31 (2008), 1113-1130.
[122] Y. Zhou, Regularity criteria for the 3D MHD equations in term of the pressure, Internat.J. Non-linear Mech, 41 (2006), 1174-1180.
[123] Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,Chinese Ann. Math., 25B (2004), 37-56.
[124] Y. Zhou and Z. Lei, Logarithmically improved criterion for Euler and Navier-Stokes Equations,preprint, (2008).
[125]姜礼尚，孔德兴，陈志浩，应用偏微分方程讲义，高等教育出版社，2008，1-275。
[126]周忆，完全非线性波动方程的经典解的整体存在性及生命区间，博士论文，复旦大学，1992。125
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[3] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initialdata, Comm. Pure Appl. Math. 43 (1990), 445-472.
[4] H. Lindblad, A remark on global existence for small initial data of the minimal surfaceequation in Minkowskian space time, Proc. Am. Math. Soc., 132 (2004), 1095-1102.
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[12] S. Klainerman, Uniform decay estimates and the lorentz invariance of the classicalwave equations, Comm. Pure Appl. Math. 38 (1985), 321-332.
[13] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space???+1, Comm. Pure Appl. Math. 40 (1987), 111-116.
[14] M. Kovalyov, Long-time behaviour of solutions of systems of nonlinear wave equations,Comm. in Partial Differential Equation 12 (1987), 471-501.
[15] T.-T. Li and Y.-M. Chen, Initial value problems for nonlinear wave equations , Comm.in Partial Differential Equation 13 (1988), 383-422.
[16] T.-T. Li and Y. Zhou, Life-span of classical solutions to nonlinear wave equations intwo space dimensions , J. Math. Pures Appl. 2 (1994), 223-249.
[17] T.-T. Li and Y. Zhou, Life-span of classical solutions to fully nonlinear wave equationsII , Nonlinear Analysis, Theory, Methods and Applications 19 (1992), 833-853.
[18] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis, II,International Press, 1995.
[19] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compactdata in two space dimensions, J. Differential Equations 104 (1993), 332-360.