非线性发展方程整体解及其吸引子相关问题研究

本项目拟研究物理、力学、材料科学等学科中出现的下列重要的非线性发展方程解的整体适定性、能控性及其无穷维动力系统吸引子的存在性、上半连续性和维度估计等问题：（1）三维不可压NS方程Cauchy问题球（柱）对称解的整体存在性和渐近性（含衰减率）；（2）三维可压非等熵NS方程组解的Green 函数逐点估计；（3）一维和高维球（柱）对称辐射流体方程组解的整体适定性 （整体存在性、渐近性（衰减率估计））；（4）无穷维动力系统吸引子理论（含拉回吸引子存在性、上半连续性）及其在某些非线性发展方程中的应用；（5）非线性双曲系统的能控性。这些问题均是近十几年来国际上广受关注的、极具挑战的、国际前沿的重要的学术问题。因此，对这些问题开展深入细致的研究，不仅需要在理论和应用上创新，而且必将对非线性泛函分析、非线性发展方程及其无穷维动力系统的理论和应用产生重要影响。新的研究方向必然需要新的数学方法与技巧的创新。

The project will study the problems of the global wellposedness of solutions, controability and the existence, upper semicontinuity and dimensional estimates of attractors of infinite-dimensional dynamical systems for the following important evolutionary equations arising from the fields of sciences of physics, mechanics and material science: (1) The global existence and asymptotics (including the decay rates) of spherically (cylindrically) symmetric solutions to the Cauchy problem of 3D incompressible NS equation; (2) The pointwise estiamtes of Green functions of solutions to the Cauchy problem for 3D non-isentropic compressible NS equations; (3) The global wellposedness (global existence and asymptotic behavior (estimates of decay rates) ) of solutions to one and higher dimensional spherical (cylindrical) radiative fluid equations; (4) The attractor theories (including the existence and upper semicontinuity of pullback attractors) for infinite-dimensional dynamical systems and their applications to some nonlinear evolutionary equations; (5) The controability for nonlinear hyperbolic systems. These problems are all very active, internationally extensive conerns received, challengable and important acadmic ones in the international frontiers in recent more than ten years. Therefore, a deepgoing detailed study undertaken on these problems not only needs creativities in both theories and applications, but also necessarily creats important impacts on the theories and applications for nonlinear functional analysis, nonlinear evolutionary equations and their infinite-dimensional dynamical systems. New research directions must need the creativities in mathematical methods and techniques.