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

本文主要研究三维不可压N-S方程(NS)的以下问题。对于具有大初始数据的NS，弱解的唯一性和强解的全局时间正则性都是突出的开放问题。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)的定常解的存在性。所有这三个问题的关键问题都是对(非线性)漂移项的缺乏理解，它与扩散竞争。对于正则性问题，已知的例子表明，修改了漂移项后，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得到的Leray系统的边值问题密切相关。边值问题的研究依赖于线性化算子的谱分析，而漂移项的结构对线性化算子的谱分析又具有重要意义。 系统地研究了一类具有漂移项的相关偏微分方程组边值问题的分支、唯一性和正则性问题。我们首先考虑定义在$R^n$中的具有漂移项的标量函数的椭圆型或抛物型微分方程。假定漂移系数为临界(属于弱$L^n$)或超临界(属于$L^q$，$q

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