Nonlinear Partial Differential Equations, Almost Periodic Type Of Viscosity Solutions

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.