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

我们的动机是以下问题的三维不可压缩Navier-Stokes方程（NS）。对于具有大初值的NS，弱解的唯一性和强解的全局正则性都是悬而未决的问题。第三个悬而未决的问题是存在一个稳定的解决方案的边界值问题（BVP）与大边界数据时，边界有多个组件。所有这3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。对于正则性问题，已知的例子，与漂移项修改，表明它是不够的，只考虑能量估计和嵌入。唯一性问题是密切相关的Leray系统的边值问题从NS通过相似变换。边值问题的研究依赖于线性化算子的谱分析，漂移项的结构对线性化算子的谱分析同样重要。本文系统地研究了一类带漂移项的相关偏微分方程组的边值问题的分支、唯一性和正则性问题。我们首先考虑一个椭圆或抛物型微分方程的标量函数定义在$R^n$与漂移项的方程。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

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