非线性波动方程经典解的生命跨度
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摘要
本文将讨论小初值条件下多波速的非线性波动方程组柯西问题以及具有星形障碍非线性波动方程外问题经典解的生命跨度(即局部解存在的最大时间区间)。
作者在正文中分别讨论了上述两种情况下对应的非线性项明确依赖于未知函数本身时的生命跨度。在正文的第一部分，作者证明了小初值多波速方程组柯西问题在其非线性项明确依赖于未知函数本身时经典解的生命跨度的下界估计T_∈≥C／∈~2。证明的方法是基于Klainerma[15]的广义能量积分的方法。对于单波速波动方程的柯西问题，这个方法的关键之处在于利用到波动算子(?)_t~2-Δ的洛仑兹不变性，而对于多波速方程组，由于不再具备洛仑兹不变性，此时为证明问题带来了较大的困难。为克服这个困难近年来一些数学工作者在处理方法和角度上作了许多改进和创新。Klainerman & Sideris [18]通过构造了相应的关于未知函数二阶导数的加权L~2范数估计，并且通过利用此加权估计，证明了在多波速情况下非线性项具有散度形式并且不依赖未知函数本身时的生命跨度。然而当非线性项是明显依赖于未知函数本身时，仅有关于未知函数二阶导数的加权估计是不够的，在正文的第一部分中，作者采用Klainerman & Sideris [18]中的办法，进一步建立起相应的关于未知函数一阶导数的加权L~2估计，并用其来证明了当非线性项是明显依赖于未知函数本身时解的生命跨度的下界。
在正文的第二部分，作者证明了具有星形障碍的外问题在Dirichlet边界的情况下，当非线性项是明显依赖于未知函数本身时经典解的生命跨度的下界估计T_e≥C／∈~2。在证明外问题时，作者仍然是采用广义能量积分的办法来处理。但是对于外问题，此时不但不具有洛仑兹不变性，而且由于需要考虑边界情况，从而对于算子t(?)_t+(?)x_i(?)_(xi)的使用也需要作一定的限制，此时的问题在处理上将会很复杂，而且需要更多的技巧。为此Keel，Smith & Sogge[19，21]通过构造未知函数一阶倒数的加权L_(t，x)~2估计，并将其用来处理当非线性项不明确依赖未知函数本身时的Dirichlet边值问题。而对于非线性项明显依赖于未知函数u时，还需要一些新的估计。本文在第二部分通过构造出相应的关于未知函数本身的加权L_(t，x)~2估计，并用此得到了当非线性项明确依赖于未知函数本身时的生命跨度的下界估计。
本文的结构如下，
在第一章，作者将介绍三维空间中关于非线性波动方程的研究的历史背景，以及研究的现状。并将列出本文的主要结果。
在正文的的一部分即第二章作者将推广Klainerman-Sideris中的关于多波速波动方程组柯西问题的加权估计，并利用这些估计，证明了具不同波速的非线性波动方程组小初值柯西问题经典解的生命跨度的下界估计。
作者将在正文的第二部分得到了在星形障碍外问题时经典解的生命跨度的下界估计。为证明这个结果，作者于第三章给出一些相应的线性方程的估计，并且于第四章得到了在小初值条件下对于具有星形障碍的外问题经典解的生命跨度的下界估计。
The present Ph.D. dissertation deals with the Cauchy problem for nonlinear wave equations with different propagation speeds and the initial boundary value problem to nonlinear wave equation outside of star-shaped obstacle with small initial data in three space dimensions.
Since the classical work of John [7], lots of efforts has been made to study the lifespan(the maximal existence time of unique local classical solutions) of classical solution to nonlinear wave equation with small cauchy data. Most of those efforts were devoted to study cauchy problem of the nonlinear wave equation by means of the Lorentz invariance of the wave operator (?)_t~2-△(see [7,8,9,14-18,22-26]) etc. This paper is devoted to study the lifespan of the classical solution for the nonlinear wave equations with different propagation speeds and the initial boundary problem of the nonlinear wave equation which do not have Lorentz invariance any more.
The first part of this dissertation is devoted to study the lifespan for the multiple propagation speeds systems of nonlinear wave equations with the nonlinear term explicitly depending on the unknown function u. To prove our result, we use the generalized energy method of Klainerman [15]. In the single speed case, the key of this method is to use the Lorentz invariance of the wave operator (?)_t~2-△. In the multiple speed case, the Lorentz invariance do not hold, so additional technical difficulties arise. To compensate the lack of Lorentz boosts, in [18], klainerman and Sideris developed some weighted estimates of the seconde order derivatives of the solution to handle the case that the nonlinear term do not explicitly depend on the unknown function itself. When the nonlinear term explicitly depend on the unknown function itself, some more estimates are necessary. In the first Chapter of this dissertation, the author will develop the weighted L~2 estimates of the first order derivatives of the solution to handle the case that the nonlinear term explicitly depend on the unknown function itself.
In the second part of this dissertation, the author will study the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u. The main method of our proof is still to use the generalized energy method of Klainerman [15], however, for the case that the problem outside of a star-shaped obstacle, the Lorentz invariance do not hold either, at the same time, another difficulty we encounter in the obstacle case is related to the scaling operator .In the Minkowski space case, Lpreserve the equation ((?)_t~2-△)u = 0, in the obstacle case, that the Dirichlet boundary conditions are not preserved by this operator. When we deal with the boundary conditions, the coefficient became large on the obstacle as t goes to infinity if we use L twice. To overcome this difficulty, Keel, Smith and Sogge [19,21] developed some weighted L_(t,x)~2 estimates for the first order derivatives of unknown function u and use those estimates to handle the case that the nonlinear term do not explicitly depending on the unknown function u. In the second part of this paper, the author will develop the weighted L_(t,x)~2 estimates for the unknown function itself, and by the same frame, we can handle the case that the nonlinear term explicitly depending on the unknown function u.
This dissertation is organized as follows. In the first Chapter, we will introduce the research history of nonlinear wave equation. The main results of the dissertation will also be stated in chapter 1.
In the second Chapter, the author will develop some new weighted estimates and obtain the lower bound of lifespan for the multiple speed case with the nonlinear term depending explicitly with the unknown function u itself.
In the third Chapter, the author will show some estimates for the linear wave equation both in the Minkowski space and for the case outside of star-shaped obsta- cle. And in the forth Chapter, the author will obtain the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u itself.
作者在正文中分别讨论了上述两种情况下对应的非线性项明确依赖于未知函数本身时的生命跨度。在正文的第一部分，作者证明了小初值多波速方程组柯西问题在其非线性项明确依赖于未知函数本身时经典解的生命跨度的下界估计T_∈≥C／∈~2。证明的方法是基于Klainerma[15]的广义能量积分的方法。对于单波速波动方程的柯西问题，这个方法的关键之处在于利用到波动算子(?)_t~2-Δ的洛仑兹不变性，而对于多波速方程组，由于不再具备洛仑兹不变性，此时为证明问题带来了较大的困难。为克服这个困难近年来一些数学工作者在处理方法和角度上作了许多改进和创新。Klainerman & Sideris [18]通过构造了相应的关于未知函数二阶导数的加权L~2范数估计，并且通过利用此加权估计，证明了在多波速情况下非线性项具有散度形式并且不依赖未知函数本身时的生命跨度。然而当非线性项是明显依赖于未知函数本身时，仅有关于未知函数二阶导数的加权估计是不够的，在正文的第一部分中，作者采用Klainerman & Sideris [18]中的办法，进一步建立起相应的关于未知函数一阶导数的加权L~2估计，并用其来证明了当非线性项是明显依赖于未知函数本身时解的生命跨度的下界。
在正文的第二部分，作者证明了具有星形障碍的外问题在Dirichlet边界的情况下，当非线性项是明显依赖于未知函数本身时经典解的生命跨度的下界估计T_e≥C／∈~2。在证明外问题时，作者仍然是采用广义能量积分的办法来处理。但是对于外问题，此时不但不具有洛仑兹不变性，而且由于需要考虑边界情况，从而对于算子t(?)_t+(?)x_i(?)_(xi)的使用也需要作一定的限制，此时的问题在处理上将会很复杂，而且需要更多的技巧。为此Keel，Smith & Sogge[19，21]通过构造未知函数一阶倒数的加权L_(t，x)~2估计，并将其用来处理当非线性项不明确依赖未知函数本身时的Dirichlet边值问题。而对于非线性项明显依赖于未知函数u时，还需要一些新的估计。本文在第二部分通过构造出相应的关于未知函数本身的加权L_(t，x)~2估计，并用此得到了当非线性项明确依赖于未知函数本身时的生命跨度的下界估计。
本文的结构如下，
在第一章，作者将介绍三维空间中关于非线性波动方程的研究的历史背景，以及研究的现状。并将列出本文的主要结果。
在正文的的一部分即第二章作者将推广Klainerman-Sideris中的关于多波速波动方程组柯西问题的加权估计，并利用这些估计，证明了具不同波速的非线性波动方程组小初值柯西问题经典解的生命跨度的下界估计。
作者将在正文的第二部分得到了在星形障碍外问题时经典解的生命跨度的下界估计。为证明这个结果，作者于第三章给出一些相应的线性方程的估计，并且于第四章得到了在小初值条件下对于具有星形障碍的外问题经典解的生命跨度的下界估计。
The present Ph.D. dissertation deals with the Cauchy problem for nonlinear wave equations with different propagation speeds and the initial boundary value problem to nonlinear wave equation outside of star-shaped obstacle with small initial data in three space dimensions.
Since the classical work of John [7], lots of efforts has been made to study the lifespan(the maximal existence time of unique local classical solutions) of classical solution to nonlinear wave equation with small cauchy data. Most of those efforts were devoted to study cauchy problem of the nonlinear wave equation by means of the Lorentz invariance of the wave operator (?)_t~2-△(see [7,8,9,14-18,22-26]) etc. This paper is devoted to study the lifespan of the classical solution for the nonlinear wave equations with different propagation speeds and the initial boundary problem of the nonlinear wave equation which do not have Lorentz invariance any more.
The first part of this dissertation is devoted to study the lifespan for the multiple propagation speeds systems of nonlinear wave equations with the nonlinear term explicitly depending on the unknown function u. To prove our result, we use the generalized energy method of Klainerman [15]. In the single speed case, the key of this method is to use the Lorentz invariance of the wave operator (?)_t~2-△. In the multiple speed case, the Lorentz invariance do not hold, so additional technical difficulties arise. To compensate the lack of Lorentz boosts, in [18], klainerman and Sideris developed some weighted estimates of the seconde order derivatives of the solution to handle the case that the nonlinear term do not explicitly depend on the unknown function itself. When the nonlinear term explicitly depend on the unknown function itself, some more estimates are necessary. In the first Chapter of this dissertation, the author will develop the weighted L~2 estimates of the first order derivatives of the solution to handle the case that the nonlinear term explicitly depend on the unknown function itself.
In the second part of this dissertation, the author will study the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u. The main method of our proof is still to use the generalized energy method of Klainerman [15], however, for the case that the problem outside of a star-shaped obstacle, the Lorentz invariance do not hold either, at the same time, another difficulty we encounter in the obstacle case is related to the scaling operator .In the Minkowski space case, Lpreserve the equation ((?)_t~2-△)u = 0, in the obstacle case, that the Dirichlet boundary conditions are not preserved by this operator. When we deal with the boundary conditions, the coefficient became large on the obstacle as t goes to infinity if we use L twice. To overcome this difficulty, Keel, Smith and Sogge [19,21] developed some weighted L_(t,x)~2 estimates for the first order derivatives of unknown function u and use those estimates to handle the case that the nonlinear term do not explicitly depending on the unknown function u. In the second part of this paper, the author will develop the weighted L_(t,x)~2 estimates for the unknown function itself, and by the same frame, we can handle the case that the nonlinear term explicitly depending on the unknown function u.
This dissertation is organized as follows. In the first Chapter, we will introduce the research history of nonlinear wave equation. The main results of the dissertation will also be stated in chapter 1.
In the second Chapter, the author will develop some new weighted estimates and obtain the lower bound of lifespan for the multiple speed case with the nonlinear term depending explicitly with the unknown function u itself.
In the third Chapter, the author will show some estimates for the linear wave equation both in the Minkowski space and for the case outside of star-shaped obsta- cle. And in the forth Chapter, the author will obtain the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u itself.
引文
[1] R.Agemi and K.Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, Advanves in nonlinear partial differ-ential equations and stochastics, edited by S.Kawahara and T.Yanagisawa, series on Adv. Math. for Appl. Sci., Vol.48, World Scientific, 1998, 43-86.
[2] Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, CPAM, 39(1986), 267-282.
[3] P.S.Datti, Nonlinear wave equations in exterior domains, Nonlinear analysis, 4(1990), 321-331.
[4] D.Gilbarg and Trudinger, Elliptic partial differential eqautions of second order, Springer-Verlag, second edition, Third printing, 1998.
[5] K.Hidano, An elementary proof of global or almost global existence for quasilinear wave equations, Tohaku Math. J., 56(2004), 271-287.
[6] K.Hidano, A remark on the almost global existence theorems of Keel, Smith and Sogge. 2005
[7] F.John, Blow-up for qusilinear wave equations in three space dimentions, CPAM. 34(1981), 29-51.
[8] F.John, Non-existence of global soltions of □u=((?)/(?)_t)F(u_t) in two and three space dimentions, MRC Technical Summary Report, 1984.
[9] F.John and S.Klainerman, Almost global existence to nonlinear wave equations in three space dimentions, 7(1984), 443-455.
[10] S.Katayama, Global and almost global existence for systerms of nonlinear wave equations with different propagation speeds, Diff. Intergral Eqs. 17(2004), 1043-1078 .
[11] S.Katayama, Global existence for a class of systems nonlinear wave equations in three dimentions, Chinese Ann. Math. 25B(2004), 1043-1078.
[12] S.Katayama, Global existence for systerms of wave equations with nonresonant nonlinearities and null forms, J. Differential Eqs., 209(2005), 140-171
[13] S.Katayama and K.Yokoyama, Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, preprint, 2005.
[14] S.Klainerman. Almost global existence to nonlinear wave equations in three space dimentions, CPAM, 16(t984), 443-455.
[15] S.Klainerman. Uniform decay estimate and the lorentz invariance of the classical wave eqautions, Comm. Pure and Appl. Math., 38(1985), 321-332.
[16] S.Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Applied Math, 23(1986), 293-326.
[17] S.Klainerman, Remarks on the global sobolev inequalities in the Minkovski space R~(n+1) , CPAM, 7(1987), 111-117.
[18] S.Klainerman and T.C.Sideris, On almost global existence for nonrelativistic wave equations in 3D, CPAM, 49(1996), 307-321.
[19] M.Keel, H.Smith and C.D.sogge, Almost global existence for some semilinear wave equations, J.d'Anal.Math., 87(2002), 265-279.
[20] M.Keel, H.Smith and C.D.sogge, Global existence for a quasilinear wave equations outside of Star-shaped Domains, Journal of Functional Analysis. 189(2002), 155-226.
[21] M.Keel, H.Smith and C.D.sogge, Almost global existence for quasilinear wave equations in three space dimentions, J.Am.Math.Soc., 17(2004), 109-153.
[22] Li Ta-Tsien and Chen Yunmei. Global classical solutions for nonlinear evolution equations, Longman Scientific & Technical UK. 1992.
[23] Li Ta-Tsein and Yuxin, Life span of classical solutions to fully nonlinear wave equations, Comm. Partial Diff. Eqs, 16(1991), 909-940.
[24] Li Ta-Tsein and Zhou Yi, Life span of classical solutions to fully nonlinear wave equations-Ⅱ, Nonlinear Analysis, 19(1992), 833-853.
[25] Li Ta-Tsein and Zhou Yi, A note on the life-span of classical solutions to nonlinear wave equations in four space dimentions, Indiana University Mathmatics Joural, 44(1995), 1207-1248.
[26] H.Lindblad, On the Life span of solutions of nonlinear wave equations with small data, Comm. Pure and Appl. Math., 43(1990), 445-472.
[27] P.D.Lax, C.S.Morawetz, and R.S.Philips, Exponential decay of solutions of the wave equations, CPAM, 16 (1963), 477-486.
[28] P.D.Lax, C.S.Morawetz and R.S.Philips, Scatterring theory, Academic Press , San Diego, 1989.
[29] C.S.Morawetz, The Decay of solutions of the Exterior initial-boundary value problem for the wave equation, CPAM, 14(1961), 561-568.
[30] C.S.Morawetz, Exponential decay of solutions of the wave equation , CPAM, 14(1966), 439-444.
[31] J.Metcalfe, M.Nakamura and C.D.Sogge, Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains. 2004.
[32] J.Metcalfe and C.D.Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 2004.
[33] C.S.Morawetz, J.Ralson and W.Strauss, Decay of solutions of the wave equation, CPAM, 30(1977), 447-508.
[34] Y. Shibata, Global existence of small solutions to nonlinear wave equations, Nonlinear evolution equation, 14(1981), 409-425.
[35] T.C.Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch, Rat.Mech.Anal. , 86(1984), 369-381.
[36] T.C.Sideris, Nonresonance and global existence of pretressed nonlinear elastic waves, Annals of Math., 151(2000), 849-874.
[37] C.D.Sogge, Lecture on nonlinear wave equations, International Press, Combridge, MA1995.
[38] H.Smith and Sogge, Global strichartz estimates for the nontrapping perturbations of the Laplacian, Comm.PDE, 25(2000), 2171-2183.
[39] T.C.Sideris and Shun-Yi Tu, Global existence for systerms of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33(2001), 477-488.
[40] Y.Shibata and Y.Tsutsmi, Local existence of solution for the inintial boundary value problem of fully nonlinear wave equation, Nonlinear Analysis, 11(1987), 335-365.
[2] Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, CPAM, 39(1986), 267-282.
[3] P.S.Datti, Nonlinear wave equations in exterior domains, Nonlinear analysis, 4(1990), 321-331.
[4] D.Gilbarg and Trudinger, Elliptic partial differential eqautions of second order, Springer-Verlag, second edition, Third printing, 1998.
[5] K.Hidano, An elementary proof of global or almost global existence for quasilinear wave equations, Tohaku Math. J., 56(2004), 271-287.
[6] K.Hidano, A remark on the almost global existence theorems of Keel, Smith and Sogge. 2005
[7] F.John, Blow-up for qusilinear wave equations in three space dimentions, CPAM. 34(1981), 29-51.
[8] F.John, Non-existence of global soltions of □u=((?)/(?)_t)F(u_t) in two and three space dimentions, MRC Technical Summary Report, 1984.
[9] F.John and S.Klainerman, Almost global existence to nonlinear wave equations in three space dimentions, 7(1984), 443-455.
[10] S.Katayama, Global and almost global existence for systerms of nonlinear wave equations with different propagation speeds, Diff. Intergral Eqs. 17(2004), 1043-1078 .
[11] S.Katayama, Global existence for a class of systems nonlinear wave equations in three dimentions, Chinese Ann. Math. 25B(2004), 1043-1078.
[12] S.Katayama, Global existence for systerms of wave equations with nonresonant nonlinearities and null forms, J. Differential Eqs., 209(2005), 140-171
[13] S.Katayama and K.Yokoyama, Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, preprint, 2005.
[14] S.Klainerman. Almost global existence to nonlinear wave equations in three space dimentions, CPAM, 16(t984), 443-455.
[15] S.Klainerman. Uniform decay estimate and the lorentz invariance of the classical wave eqautions, Comm. Pure and Appl. Math., 38(1985), 321-332.
[16] S.Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Applied Math, 23(1986), 293-326.
[17] S.Klainerman, Remarks on the global sobolev inequalities in the Minkovski space R~(n+1) , CPAM, 7(1987), 111-117.
[18] S.Klainerman and T.C.Sideris, On almost global existence for nonrelativistic wave equations in 3D, CPAM, 49(1996), 307-321.
[19] M.Keel, H.Smith and C.D.sogge, Almost global existence for some semilinear wave equations, J.d'Anal.Math., 87(2002), 265-279.
[20] M.Keel, H.Smith and C.D.sogge, Global existence for a quasilinear wave equations outside of Star-shaped Domains, Journal of Functional Analysis. 189(2002), 155-226.
[21] M.Keel, H.Smith and C.D.sogge, Almost global existence for quasilinear wave equations in three space dimentions, J.Am.Math.Soc., 17(2004), 109-153.
[22] Li Ta-Tsien and Chen Yunmei. Global classical solutions for nonlinear evolution equations, Longman Scientific & Technical UK. 1992.
[23] Li Ta-Tsein and Yuxin, Life span of classical solutions to fully nonlinear wave equations, Comm. Partial Diff. Eqs, 16(1991), 909-940.
[24] Li Ta-Tsein and Zhou Yi, Life span of classical solutions to fully nonlinear wave equations-Ⅱ, Nonlinear Analysis, 19(1992), 833-853.
[25] Li Ta-Tsein and Zhou Yi, A note on the life-span of classical solutions to nonlinear wave equations in four space dimentions, Indiana University Mathmatics Joural, 44(1995), 1207-1248.
[26] H.Lindblad, On the Life span of solutions of nonlinear wave equations with small data, Comm. Pure and Appl. Math., 43(1990), 445-472.
[27] P.D.Lax, C.S.Morawetz, and R.S.Philips, Exponential decay of solutions of the wave equations, CPAM, 16 (1963), 477-486.
[28] P.D.Lax, C.S.Morawetz and R.S.Philips, Scatterring theory, Academic Press , San Diego, 1989.
[29] C.S.Morawetz, The Decay of solutions of the Exterior initial-boundary value problem for the wave equation, CPAM, 14(1961), 561-568.
[30] C.S.Morawetz, Exponential decay of solutions of the wave equation , CPAM, 14(1966), 439-444.
[31] J.Metcalfe, M.Nakamura and C.D.Sogge, Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains. 2004.
[32] J.Metcalfe and C.D.Sogge, Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math., 2004.
[33] C.S.Morawetz, J.Ralson and W.Strauss, Decay of solutions of the wave equation, CPAM, 30(1977), 447-508.
[34] Y. Shibata, Global existence of small solutions to nonlinear wave equations, Nonlinear evolution equation, 14(1981), 409-425.
[35] T.C.Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch, Rat.Mech.Anal. , 86(1984), 369-381.
[36] T.C.Sideris, Nonresonance and global existence of pretressed nonlinear elastic waves, Annals of Math., 151(2000), 849-874.
[37] C.D.Sogge, Lecture on nonlinear wave equations, International Press, Combridge, MA1995.
[38] H.Smith and Sogge, Global strichartz estimates for the nontrapping perturbations of the Laplacian, Comm.PDE, 25(2000), 2171-2183.
[39] T.C.Sideris and Shun-Yi Tu, Global existence for systerms of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33(2001), 477-488.
[40] Y.Shibata and Y.Tsutsmi, Local existence of solution for the inintial boundary value problem of fully nonlinear wave equation, Nonlinear Analysis, 11(1987), 335-365.