Remarks on the error of hypothetical non-unique solutions to the 3D Navier-Stokes equations

Zachary Bradshaw, University of Arkansas
Fine Hall 314

In-Person Talk 

In the past several years evidence has emerged that uniqueness fails for the Navier-Stokes equations in the natural energy class. Classical uniqueness criteria imply such solutions would satisfy certain properties. For example, non-unique solutions cannot be finite in scaling-critical space-time Lebesgue norms—the Prodi-Serrin scaling class. It is not clear if the same is true for the error, which is the difference of two non-unique solutions. This points to the possibility that the error can in some sense be better than the background flows. Given that the error encodes information about the severity of non-uniqueness (e.g. a small error amounts to a sort of approximate stability), it is natural to explore its properties. In this talk we will state several results in this direction. First, we discuss bounds on the rate at which the error can grow—the so called ``separation rate.’’ This work is drawn from collaborative work between P. Phelps and the speaker. We will then explore which spatial scales are necessarily active as the error evolves.