Qualitative analysis focused on some nonlinear systems

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

The PI proposes to study qualitative properties of solutions to some well-known nonlinear partial differential equations. These equations arise naturally from geometry, fluid dynamics, physics, chemistry, and biology. One is about the global stability of solutions with reasonable datum to the three dimensional incompressible Navier-Stokes equations. The Navier-Stokes system of equations is the guiding system for the dynamics of the incompressible flows. This global stability problem is closely related to the `millennium' open question of whether the three dimensional Navier-Stokes equations can develop a finite time singularity from reasonable initial data. Many physical phenomena are heavily involved with fluid flows. The understanding of the incompressible fluid flows is certainly the first and the fundamental step. The other is the Hardy-Littlewood-Sobolev type which arises in geometric analysis, functional analysis, and in quantum physical problems. The PI proposes to work on fluid flows which are axisymmetric and to derive certain asymptotic estimate of the solutions. This is a possible way to show that for an axisymmetric smooth initial data with compact support the corresponding solution is globally well-posed (and thus develops no singularity). This project also explores other ways of studying the local and global structures of axisymmetric Navier-Stokes equations as well as the full Navier-Stokes equations and seek applications in related research fields. For the Hardy-Littlewood-Sobolev type systems, classification of solutions leads to discoveries of new structures of well-known nonlinear partial differential and integral equations arising from the mathematical and physical sciences. Liouville type theorems, with the Lane-Emden conjecture as a special case, are widely used for establishing various forms of a priori estimates for solutions. This is the core of nonlinear analysis of differential and integral equations or systems. The essential uniqueness are closely related to certain physical bound states and their quantization. Deriving new and intrinsic estimates point-wise or in integral forms, and getting asymptotic estimates of some key integral play central rules in this project. On both problems, the PI emphasizes on finding new and intrinsic connections of seemingly independent phenomena and on providing some new angles to the problems. The proposed research projects also actively involve graduate students and young mathematicians. And it provides a solid training for them in mathematical analysis, physical modeling and numerical simulation. The ongoing informal seminar `analysis of nonlinear partial differential equations' that meets weekly to train the graduate students and junior faculty members is also an important part of the project.