Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems

Speaker: Trevor Leslie (University of Wisconsin, Madison)

We consider the Euler Alignment Model with smooth, slowly decaying interaction protocol. It has been known since the work of Carrillo, Choi, Tadmor and Tan in 2016 that a certain conserved quantity '$$e$$' governs the global-in-time existence or finite-time blowup of sufficiently regular solutions. We give an interpretation of the quantity e and use it to analyze the structure of the limiting density profile. We draw two striking conclusions: First, the singular support of the limiting density measure (where 'aggregation' occurs) is precisely the image of the initial zero set of $$e$$, under the limiting flow map. This allows us to reverse-engineer mass concentration sets of a specified topological genus, for example. Second, the smoothness of $$e$$ at time zero controls the size of mass concentration set: If $$e_0$$ is $$C^k$$, then the mass concentration set has Hausdorff dimension at most $$1/(k+1)$$. We show that this bound is sharp by means of an explicit example. This is joint work with Lear, Shvydkoy, and Tadmor. If time allows, we will also discuss the role of e in the limiting dynamics for the case of strongly singular interaction protocols.

Time: March 25, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan