Bifurcation, uniqueness and regularity for differential equations with critical and supercritical drifts

具有临界和超临界漂移的微分方程的分岔、唯一性和正则性

基本信息

批准号：
RGPIN-2018-04137
负责人：
Tsai, TaiPeng
金额：
$ 1.46万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711876
关键词：
Bifurcation uniqueness regularity differential equations

项目摘要

We are motivated by the following problems for the 3D incompressible Navier-Stokes equations (NS). For NS with large initial data, the uniqueness of weak solutions and the global in time regularity of strong solutions are both outstanding open questions. A third outstanding open question is the existence of a stationary solution of the boundary value problem (BVP) with large boundary data when the boundary has multiple components. The key issue for all these 3 problems is the lack of understanding of the (nonlinear) drift term, which competes with the diffusion. For the regularity question, known examples, with the drift term modified, show that it is not sufficient to only consider energy estimates and imbedding. The uniqueness question is closely related to the BVP of the Leray system obtained from NS by a similarity transform. The study of the BVP relies on the spectral analysis of the linearized operators, for which the structure of the drift term again is significant. We propose a systematic study of the questions of bifurcation of BVP, uniqueness and regularity for a sequence of related systems of PDEs with drift terms. We first consider an elliptic or parabolic differential equation for a scalar function defined in $R^n$ with a drift term in the equation. The coefficient of the drift will be assumed either critical (belonging to weak $L^n$), or supercritical (belonging to $L^q$, $q

我们的动机是解决 3D 不可压缩纳维-斯托克斯方程 (NS) 的以下问题。对于具有大量初始数据的 NS，弱解的唯一性和强解的全局时间规律性都是悬而未决的问题。第三个悬而未决的问题是，当边界具有多个分量时，具有大量边界数据的边值问题 (BVP) 是否存在平稳解。所有这三个问题的关键问题是缺乏对与扩散竞争的（非线性）漂移项的理解。对于规律性问题，已知的例子（修改了漂移项）表明仅考虑能量估计和嵌入是不够的。唯一性问题与通过相似变换从 NS 获得的 Leray 系统的 BVP 密切相关。 BVP 的研究依赖于线性算子的谱分析，其中漂移项的结构也很重要。我们提出了对带有漂移项的偏微分方程相关系统序列的 BVP 分岔、唯一性和规律性问题进行系统研究。我们首先考虑 $R^n$ 中定义的标量函数的椭圆或抛物线微分方程，方程中包含漂移项。漂移系数将被假定为临界（属于弱 $L^n$）或超临界（属于 $L^q$、$q