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性问题。第三个突出的开放问题是当边界有多个分量时，具有大边界数据的边值问题（BVP）的平稳解的存在性。所有这三个问题的关键问题是缺乏对（非线性）漂移项的理解，它与扩散相竞争。对于正则性问题，已知的修正漂移项的例子表明，仅考虑能量估计和嵌入是不够的。唯一性问题与由NS进行相似变换得到的Leray系统的BVP密切相关。BVP的研究依赖于线性化算子的谱分析，其中漂移项的结构也是重要的。******系统地研究了一类带漂移项的偏微分方程相关系统的分岔问题、唯一性问题和正则性问题。我们首先考虑在R^n$中定义的标量函数的椭圆型或抛物型微分方程，方程中有漂移项。漂移系数将被假设为临界（属于弱$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