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

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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