非线性偏微分方程的概周期型粘性解
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摘要
本文研究Hamilton-Jacobi方程的回复解,主要包括两部分内容:一部分是关于Hamilton-Jacobi方程的遥远概周期粘性解的存在性和唯一性,另一部分是关于二阶非线性抛物型偏微分方程的周期、概周期和遥远概周期粘性解的存在性和唯一性。
Hamilton-Jacobi方程是流体力学、大气动力学、海洋内波动力学和光学中非常重要的数学模型之一。它在哈密顿动力学、最优控制理论以及微分对策理论中也有非常重要的应用。做为完全非线性的偏微分方程,Hamilton-Jacobi方程的经典光滑解不容易求出甚至不存在。许多应用学科的发展需要解决这个问题。上世纪八十年代初,Crandall、Evans和Lions等人打破了这个僵局,利用极值原理,创立了Hamilton-Jacobi方程的粘性解理论,极大地推进了偏微分方程弱解理论的发展。粘性解理论是Lions获得菲尔兹奖的重要内容。八十年代后期,Crandall、Lions、Ishii和Jensen等人推广了这一理论,建立了二阶Hamilton-Jacobi方程和椭圆型方程的粘性解理论。这一理论还在发展过程中,目前这方面的文献已有数千篇之多。其中非常值得注意的一个动向是,这一理论与动力系统的Arnold扩散和弱KAM理论有着深刻的联系。
Bohr等人创立的概周期函数理论领域已经得到很大的发展,产生了许多具有更广泛意义的函数类,譬如概自守函数、渐近概周期函数、弱概周期函数,一致概周期函数等。到了上世纪八十年代Sarason提出了遥远概周期函数和缓慢震荡函数。九十年代初,张传义提出了伪概周期函数。这些函数类的分析与代数性质得到广泛研究。它们在动力系统和微分方程定性理论中有着广泛的应用。
本文欲在上述两个数学发展进程的交叉领域开展研究,主要研究Hamilton-Jacobi方程的概周期型粘性解的存在唯一性。我们应用Perron方法和粘性解的比较定理,对Hamilton-Jacobi方程及二阶非线性抛物型偏微分方程(二阶Hamilton-Jacobi方程)的概周期和遥远概周期粘性解的存在性和唯一性进行了系统研究。
本文主要做了以下工作
第一,提出并证明了遥远概周期函数和缓慢震荡函数的几个性质和引理。
第二,对Hamilton-Jacobi方程,证明了其遥远概周期粘性解的存在性和唯一性,并证明了该方程的高频率的遥远概周期粘性解的渐近行为。
第三,推广了已有的Hamilton-Jacobi方程粘性解的比较定理,得到适用于有界区域上带有Dirichlet边界条件的Hamilton-Jacobi方程的比较定理,并证明了该方程的遥远概周期粘性解的存在性。
第四,推广了适用于Hamilton-Jacobi方程粘性解的比较定理,得到了适用于二阶非线性抛物型偏微分方程的比较定理,并证明了该方程的周期、概周期及遥远概周期粘性解等回复解的存在性和唯一性,研究了该方程的高频率的概周期及遥远概周期粘性解的渐近行为。
This paper studies recurrent solutions of Hamilton-Jacobi equations, it consistsmainly of two parts: one part concerns on the existence and uniqueness of remotely al-most periodic viscosity solutions of Hamilton-Jacobi equations, the other part concernson the existence and uniqueness of periodic, almost periodic and remotely almost pe-riodic viscosity solutions of second order nonlinear parabolic partial differential equa-tions.
The Hamilton-Jacobi equation is one of the most important mathematical modelsof ?uid mechanics, atmospheric dynamics, ocean internal wave dynamics and optics.Also it has very important applications in Hamiltonian dynamics, optimal control the-ory and differential games. As a fully nonlinear partial differential equations, the classi-cal smooth solution of Hamilton-Jacobi equation is not easy to find or even not exist. Itis necessary to solve this problem for the development of many application disciplines.In the early 1980s, Crandall, Evans and Lions broke this impasse, using extremum prin-ciple, they found the viscosity theory of Hamilton-Jacobi equation which had greatlyadvanced the development of weak solution theory of partial differential equations. Theviscosity theory is the key elements for Lions to get Fields Medal. In the late 1980s,Crandall, Lions, Ishii and Jensen etc promoted this theory, built the viscosity theory ofsecond order Hamilton-Jacobi equations and elliptic equations. This theory is still inthe development, thousands of articles in this area have been written so far. One of avery noteworthy trends are deep contact between this theory and the Arnold prolifera-tion and weak KAM theory of dynamic system.
The field of almost periodic function theory founded by Bohr etc has been devel-oped greatly, there are many function classes which have broader sense, for example,almost automorphic function, asymptotically almost periodic function, weakly almostperiodic function, uniformly almost periodic function etc. Up to 1980s, D. Sarasonproposed remotely almost periodic function and slowly oscillating function. In theearly 1990s, Chuanyi Zhang proposed pseudo almost periodic function. The analysisand algebra properties of these function classes have been extensively studied. Theyhave extensively applications in dynamic system and qualitative theory of differentialequations.
This paper will start the research in the cross-cutting areas of the two above mathe-matical development processes, mainly focus on the existence and uniqueness of almostperiodic type viscosity solutions of Hamilton-Jacobi equations. We will use Perron’smethod and comparison theorem of viscosity solutions to do systematic study in theexistence and uniqueness of almost periodic and remotely almost periodic viscositysolutions of Hamilton-Jacobi equations and second order nonlinear parabolic partialdifferential equations (second order Hamilton-Jacobi equations).
In this paper, some aspects of concrete work are done.
First, we propose and prove some properties and lemmas of remotely almost peri-odic functions and slowly oscillating functions.
Second, for Hamilton-Jacobi equations, we prove the existence and uniqueness oftime remotely almost periodic viscosity solutions, and also we study the asymptoticbehavior of time remotely almost periodic viscosity solutions for high frequencies.
Third, we extend the comparison theorem of viscosity solutions of Hamilton-Jacobi equations, and get a new comparison theorem of Hamilton-Jacobi equations withthe Dirichlet boundary condition in a bounded domain. Then we prove the existence oftime remotely almost periodic viscosity solutions of such equations.
Fourth, we extend the comparison theorem of viscosity solutions of Hamilton-Jacobi equations, and get a new comparison theorem of second order nonlinearparabolic partial differential equations, then we prove the existence, uniqueness andasymptotic behavior for high frequencies of time almost periodic and remotely almostperiodic viscosity solutions of such equations.
Hamilton-Jacobi方程是流体力学、大气动力学、海洋内波动力学和光学中非常重要的数学模型之一。它在哈密顿动力学、最优控制理论以及微分对策理论中也有非常重要的应用。做为完全非线性的偏微分方程,Hamilton-Jacobi方程的经典光滑解不容易求出甚至不存在。许多应用学科的发展需要解决这个问题。上世纪八十年代初,Crandall、Evans和Lions等人打破了这个僵局,利用极值原理,创立了Hamilton-Jacobi方程的粘性解理论,极大地推进了偏微分方程弱解理论的发展。粘性解理论是Lions获得菲尔兹奖的重要内容。八十年代后期,Crandall、Lions、Ishii和Jensen等人推广了这一理论,建立了二阶Hamilton-Jacobi方程和椭圆型方程的粘性解理论。这一理论还在发展过程中,目前这方面的文献已有数千篇之多。其中非常值得注意的一个动向是,这一理论与动力系统的Arnold扩散和弱KAM理论有着深刻的联系。
Bohr等人创立的概周期函数理论领域已经得到很大的发展,产生了许多具有更广泛意义的函数类,譬如概自守函数、渐近概周期函数、弱概周期函数,一致概周期函数等。到了上世纪八十年代Sarason提出了遥远概周期函数和缓慢震荡函数。九十年代初,张传义提出了伪概周期函数。这些函数类的分析与代数性质得到广泛研究。它们在动力系统和微分方程定性理论中有着广泛的应用。
本文欲在上述两个数学发展进程的交叉领域开展研究,主要研究Hamilton-Jacobi方程的概周期型粘性解的存在唯一性。我们应用Perron方法和粘性解的比较定理,对Hamilton-Jacobi方程及二阶非线性抛物型偏微分方程(二阶Hamilton-Jacobi方程)的概周期和遥远概周期粘性解的存在性和唯一性进行了系统研究。
本文主要做了以下工作
第一,提出并证明了遥远概周期函数和缓慢震荡函数的几个性质和引理。
第二,对Hamilton-Jacobi方程,证明了其遥远概周期粘性解的存在性和唯一性,并证明了该方程的高频率的遥远概周期粘性解的渐近行为。
第三,推广了已有的Hamilton-Jacobi方程粘性解的比较定理,得到适用于有界区域上带有Dirichlet边界条件的Hamilton-Jacobi方程的比较定理,并证明了该方程的遥远概周期粘性解的存在性。
第四,推广了适用于Hamilton-Jacobi方程粘性解的比较定理,得到了适用于二阶非线性抛物型偏微分方程的比较定理,并证明了该方程的周期、概周期及遥远概周期粘性解等回复解的存在性和唯一性,研究了该方程的高频率的概周期及遥远概周期粘性解的渐近行为。
This paper studies recurrent solutions of Hamilton-Jacobi equations, it consistsmainly of two parts: one part concerns on the existence and uniqueness of remotely al-most periodic viscosity solutions of Hamilton-Jacobi equations, the other part concernson the existence and uniqueness of periodic, almost periodic and remotely almost pe-riodic viscosity solutions of second order nonlinear parabolic partial differential equa-tions.
The Hamilton-Jacobi equation is one of the most important mathematical modelsof ?uid mechanics, atmospheric dynamics, ocean internal wave dynamics and optics.Also it has very important applications in Hamiltonian dynamics, optimal control the-ory and differential games. As a fully nonlinear partial differential equations, the classi-cal smooth solution of Hamilton-Jacobi equation is not easy to find or even not exist. Itis necessary to solve this problem for the development of many application disciplines.In the early 1980s, Crandall, Evans and Lions broke this impasse, using extremum prin-ciple, they found the viscosity theory of Hamilton-Jacobi equation which had greatlyadvanced the development of weak solution theory of partial differential equations. Theviscosity theory is the key elements for Lions to get Fields Medal. In the late 1980s,Crandall, Lions, Ishii and Jensen etc promoted this theory, built the viscosity theory ofsecond order Hamilton-Jacobi equations and elliptic equations. This theory is still inthe development, thousands of articles in this area have been written so far. One of avery noteworthy trends are deep contact between this theory and the Arnold prolifera-tion and weak KAM theory of dynamic system.
The field of almost periodic function theory founded by Bohr etc has been devel-oped greatly, there are many function classes which have broader sense, for example,almost automorphic function, asymptotically almost periodic function, weakly almostperiodic function, uniformly almost periodic function etc. Up to 1980s, D. Sarasonproposed remotely almost periodic function and slowly oscillating function. In theearly 1990s, Chuanyi Zhang proposed pseudo almost periodic function. The analysisand algebra properties of these function classes have been extensively studied. Theyhave extensively applications in dynamic system and qualitative theory of differentialequations.
This paper will start the research in the cross-cutting areas of the two above mathe-matical development processes, mainly focus on the existence and uniqueness of almostperiodic type viscosity solutions of Hamilton-Jacobi equations. We will use Perron’smethod and comparison theorem of viscosity solutions to do systematic study in theexistence and uniqueness of almost periodic and remotely almost periodic viscositysolutions of Hamilton-Jacobi equations and second order nonlinear parabolic partialdifferential equations (second order Hamilton-Jacobi equations).
In this paper, some aspects of concrete work are done.
First, we propose and prove some properties and lemmas of remotely almost peri-odic functions and slowly oscillating functions.
Second, for Hamilton-Jacobi equations, we prove the existence and uniqueness oftime remotely almost periodic viscosity solutions, and also we study the asymptoticbehavior of time remotely almost periodic viscosity solutions for high frequencies.
Third, we extend the comparison theorem of viscosity solutions of Hamilton-Jacobi equations, and get a new comparison theorem of Hamilton-Jacobi equations withthe Dirichlet boundary condition in a bounded domain. Then we prove the existence oftime remotely almost periodic viscosity solutions of such equations.
Fourth, we extend the comparison theorem of viscosity solutions of Hamilton-Jacobi equations, and get a new comparison theorem of second order nonlinearparabolic partial differential equations, then we prove the existence, uniqueness andasymptotic behavior for high frequencies of time almost periodic and remotely almostperiodic viscosity solutions of such equations.
引文
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