闵可夫斯基空间中的时向极值曲面若干问题的研究
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摘要
在此博士论文中,我们主要关心弦理论及粒子物理中的一个重要模型-闵可夫斯基空间中的时向极值曲面的一些分析问题.对于闵可夫斯基空间中时向极值曲面方程初值问题、混合初边值问题的经典解的整体存在唯一性及整体解的渐近性态进行了研究。
由于物理及力学领域的需要及其它应用领域的相关研究的发展,很多时候所考察的问题最终归结为一个数学问题来解决。闵可夫斯基空间的时向极值曲面作为弦理论及粒子物理中的一个重要而非平凡的模型,在数学上对其进行相应的研究就显得较为重要。它还在流体力学、电磁场理论及黑洞理论中起一定的作用。闵可夫斯基空间中的时向极值曲面能够很好的刻画闵可夫斯基空间中的相对论弦的运动,这就更使得我们对该方程进行系统的研究。时向极值曲面方程可以通过其面积泛函的Euler-Lagrange方程得到,为一维守恒律方程组的形式。所以我们用考察拟线性双曲组的相关方法,对闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程的相关问题(初值问题(Cauchy problem)、混合初边值问题(Mixed initial boundary value problem,包括第一类(Dirichlet problem)、第二类(Neumann problem)及Robin初边值问题))的整体经典解的存在唯一性及经典解的渐近性态给出了一些有意义的结论。另一方面,闵可夫斯基空间中高维时向极值曲面的研究在几何及物理意义上的理解相应理论也起着重要的意义。
全文的结构安排如下:
第一章概要地介绍了闵可夫斯基空间的极值曲面方程有关问题的历史发展和研究进展,前人在处理与本文相关的一些偏微分方程(拟线性双曲型方程组)方面的相关工作及研究整体经典解及解的渐近性态的方法。进一步我们对闵可夫斯基空间中时向极值曲面方程相关问题的提法,处理上的大体思路给出简要说明。同时还简单陈述了本文的主要结果。
第二章在具有线性退化特征的对角型的拟线性双曲型方程组初值问题整体经典解存在的基础上,考虑其经典解的渐近性态问题。在经典解整体存在的基础上,在初值及其一阶导数的L~1∩L~∞模有界假设条件下,我们证明了,当时间t趋向于正无穷大时,整体经典解趋向于一组C~1行波解的线性组合。作为该结论的一个重要的应用,我们将该结论应用到闵可夫斯基空间中的时向极值曲面方程的相应问题上,得到了整体经典解的存在唯一性及经典解的渐近性态。
第三章研究了闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程在半无界区域内的混合初边值问题。在初值有界且边值适当小的假设条件下,我们得到该方程在半无界区域内混合初边值问题的C~2经典解的整体存在性及唯一性。进一步在整体经典解存在的基础上,在边值适当的假设条件下,我们给出了,当时间t趋向于正无穷大时,解的一阶导数趋向于一组C~1行波解。从几何角度上看,这意味着该极值曲面趋向于一个广义的圆柱;同时,该行波解也为极值曲面方程的精确解。
在第三章的基础上,我们继续研究了闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程在区域R~+×[0,1]上的混合初边值问题。在边值具有某种意义下小且衰减的假设条件下,我们得到了该问题经典解的整体存在唯一性。详细内容见本文第四章。
前面四章考虑闵可夫斯基空间R~(1+(1+n))中的二维时向极值曲面的相关问题。本文第五章中,我们考虑闵可夫斯基空间中的高维时向极值曲面的初值问题,高维时向极值曲面方程对应于一个非线性波动方程,且其非齐次项满足零条件。当初始值充分接近任意的时向平面a_0t+a_1x_1+…+a_nx_n+b=0时,我们得到了相应的非线性波动方程的整体光滑解。
The present Ph.D dissertation is devoted to the study of time-like extremal surface in Minkowski space.We prove the global existence of classical solutions of Cauchy problem and the mixed initial-boundary value problem for equation of time-like extremal surface and gets the asymptotic behavior of global classical solutions.
In the study of physics and other application areas,mosdy the problem can be reduced to the mathematical problem.As an important and non-trivial model in string theory and particle physics,extremal surface in Minkowski space is also an important investigation topic in mathematics.It also plays an important role in fluids mechanics, electromagnetism as well as in the theory of black hole.Extremal surface equation is Euler-Lagrange equation of its area functional,which is a first order system of conservation laws.Using the characteristic methods of quasilinear hyperbolic systems,we consider the related problem of the equation of time-like extremal surface in Minkowski space R~(1+(1+n)),i.e.Cauchy problem,the mixed initial-boundary value problem(including Dirichelt boundary problem,Neumann boundary problem and Robin boundary problem) and get global classical solutions and asymptotic behavior with more interesting phenomena. It also plays a unique role in our understanding in geometry and physics by studying high dimensional time-like extremal surface in Minkowski space.
The dissertation is organized as follows.
Chapter One is an introduction.It is devoted to introducing physical background and previous mathematical research works on extremal surface,especially important achievements by other scholars who treated similar problems in quasilinear hyperbolic systems or global classical solutions and its asymptotic behavior related to those of ours that are presented in the following chapters.The main problems we concerned,main results we obtained,and methods we utilized in this Ph.D.dissertation are also illustrated with com- ments.
In Chapter Two we concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields.Based on the existence results of global classical solutions,we prove that when t tends to infinity,the solution approaches a combination of C~1 travelling wave solutions,provided that L~1∩L~∞norm of the initial data as well as its derivative are bounded.Application is given to equation of the time-like extremal surface in Minkowski space.
In Chapter Three we investigate the mixed initial-boundary value problem for the equation of time-like extremal surface in Minkowski space R~(1+(1+n)) in the first quadrant. Under the assumptions that the initial data are bounded and the boundary data are small, we prove the global existence and uniqueness of the C~2 solutions of the Dirichlet problem and Neumann problem for this kind of equation.Based on the existence results on global classical solutions,we show that:as t tends to infinity,the first order derivatives of the solutions approach C~1 travelling wave solution,provided that L~1∩L~∞norm of the first and second order derivatives of the initial and boundary data are bounded.In which we can proved that geometrically this extremal surface is a generalized cylinder.Moreover, The travelling wave solutions are the exact solutions of system under consideration.This theory can also be used to study the equation of Born-Infeld type.e.g.the equation of relativistic strings moving.
In Chapter Four we study the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space R~(1+(1+n)) on the strip R~+×[0,1].Under the assumptions that the boundary data are small and decaying,we get the global existence and uniqueness of classical solutions.
In Chapter Five we show that the nonlinear wave equation corresponding to the high dimensional time-like extremal surface equation in Minkowski space have global smooth solutions for initial data sufficiently close to the arbitrary time-like plane a_0t+a_1x_1+...+ a_nx_n+b=0.
由于物理及力学领域的需要及其它应用领域的相关研究的发展,很多时候所考察的问题最终归结为一个数学问题来解决。闵可夫斯基空间的时向极值曲面作为弦理论及粒子物理中的一个重要而非平凡的模型,在数学上对其进行相应的研究就显得较为重要。它还在流体力学、电磁场理论及黑洞理论中起一定的作用。闵可夫斯基空间中的时向极值曲面能够很好的刻画闵可夫斯基空间中的相对论弦的运动,这就更使得我们对该方程进行系统的研究。时向极值曲面方程可以通过其面积泛函的Euler-Lagrange方程得到,为一维守恒律方程组的形式。所以我们用考察拟线性双曲组的相关方法,对闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程的相关问题(初值问题(Cauchy problem)、混合初边值问题(Mixed initial boundary value problem,包括第一类(Dirichlet problem)、第二类(Neumann problem)及Robin初边值问题))的整体经典解的存在唯一性及经典解的渐近性态给出了一些有意义的结论。另一方面,闵可夫斯基空间中高维时向极值曲面的研究在几何及物理意义上的理解相应理论也起着重要的意义。
全文的结构安排如下:
第一章概要地介绍了闵可夫斯基空间的极值曲面方程有关问题的历史发展和研究进展,前人在处理与本文相关的一些偏微分方程(拟线性双曲型方程组)方面的相关工作及研究整体经典解及解的渐近性态的方法。进一步我们对闵可夫斯基空间中时向极值曲面方程相关问题的提法,处理上的大体思路给出简要说明。同时还简单陈述了本文的主要结果。
第二章在具有线性退化特征的对角型的拟线性双曲型方程组初值问题整体经典解存在的基础上,考虑其经典解的渐近性态问题。在经典解整体存在的基础上,在初值及其一阶导数的L~1∩L~∞模有界假设条件下,我们证明了,当时间t趋向于正无穷大时,整体经典解趋向于一组C~1行波解的线性组合。作为该结论的一个重要的应用,我们将该结论应用到闵可夫斯基空间中的时向极值曲面方程的相应问题上,得到了整体经典解的存在唯一性及经典解的渐近性态。
第三章研究了闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程在半无界区域内的混合初边值问题。在初值有界且边值适当小的假设条件下,我们得到该方程在半无界区域内混合初边值问题的C~2经典解的整体存在性及唯一性。进一步在整体经典解存在的基础上,在边值适当的假设条件下,我们给出了,当时间t趋向于正无穷大时,解的一阶导数趋向于一组C~1行波解。从几何角度上看,这意味着该极值曲面趋向于一个广义的圆柱;同时,该行波解也为极值曲面方程的精确解。
在第三章的基础上,我们继续研究了闵可夫斯基空间R~(1+(1+n))中的时向极值曲面方程在区域R~+×[0,1]上的混合初边值问题。在边值具有某种意义下小且衰减的假设条件下,我们得到了该问题经典解的整体存在唯一性。详细内容见本文第四章。
前面四章考虑闵可夫斯基空间R~(1+(1+n))中的二维时向极值曲面的相关问题。本文第五章中,我们考虑闵可夫斯基空间中的高维时向极值曲面的初值问题,高维时向极值曲面方程对应于一个非线性波动方程,且其非齐次项满足零条件。当初始值充分接近任意的时向平面a_0t+a_1x_1+…+a_nx_n+b=0时,我们得到了相应的非线性波动方程的整体光滑解。
The present Ph.D dissertation is devoted to the study of time-like extremal surface in Minkowski space.We prove the global existence of classical solutions of Cauchy problem and the mixed initial-boundary value problem for equation of time-like extremal surface and gets the asymptotic behavior of global classical solutions.
In the study of physics and other application areas,mosdy the problem can be reduced to the mathematical problem.As an important and non-trivial model in string theory and particle physics,extremal surface in Minkowski space is also an important investigation topic in mathematics.It also plays an important role in fluids mechanics, electromagnetism as well as in the theory of black hole.Extremal surface equation is Euler-Lagrange equation of its area functional,which is a first order system of conservation laws.Using the characteristic methods of quasilinear hyperbolic systems,we consider the related problem of the equation of time-like extremal surface in Minkowski space R~(1+(1+n)),i.e.Cauchy problem,the mixed initial-boundary value problem(including Dirichelt boundary problem,Neumann boundary problem and Robin boundary problem) and get global classical solutions and asymptotic behavior with more interesting phenomena. It also plays a unique role in our understanding in geometry and physics by studying high dimensional time-like extremal surface in Minkowski space.
The dissertation is organized as follows.
Chapter One is an introduction.It is devoted to introducing physical background and previous mathematical research works on extremal surface,especially important achievements by other scholars who treated similar problems in quasilinear hyperbolic systems or global classical solutions and its asymptotic behavior related to those of ours that are presented in the following chapters.The main problems we concerned,main results we obtained,and methods we utilized in this Ph.D.dissertation are also illustrated with com- ments.
In Chapter Two we concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields.Based on the existence results of global classical solutions,we prove that when t tends to infinity,the solution approaches a combination of C~1 travelling wave solutions,provided that L~1∩L~∞norm of the initial data as well as its derivative are bounded.Application is given to equation of the time-like extremal surface in Minkowski space.
In Chapter Three we investigate the mixed initial-boundary value problem for the equation of time-like extremal surface in Minkowski space R~(1+(1+n)) in the first quadrant. Under the assumptions that the initial data are bounded and the boundary data are small, we prove the global existence and uniqueness of the C~2 solutions of the Dirichlet problem and Neumann problem for this kind of equation.Based on the existence results on global classical solutions,we show that:as t tends to infinity,the first order derivatives of the solutions approach C~1 travelling wave solution,provided that L~1∩L~∞norm of the first and second order derivatives of the initial and boundary data are bounded.In which we can proved that geometrically this extremal surface is a generalized cylinder.Moreover, The travelling wave solutions are the exact solutions of system under consideration.This theory can also be used to study the equation of Born-Infeld type.e.g.the equation of relativistic strings moving.
In Chapter Four we study the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space R~(1+(1+n)) on the strip R~+×[0,1].Under the assumptions that the boundary data are small and decaying,we get the global existence and uniqueness of classical solutions.
In Chapter Five we show that the nonlinear wave equation corresponding to the high dimensional time-like extremal surface equation in Minkowski space have global smooth solutions for initial data sufficiently close to the arbitrary time-like plane a_0t+a_1x_1+...+ a_nx_n+b=0.
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