Regularity and Small-Scale Creations in Chemotaxis and Incompressible Fluids

Abstract

In this dissertation, we concern ourselves with two fundamental phenomena in nature: aggregation phenomena in biology and motions of incompressible fluids. The two phenomena, to say the least, are ubiquitous in the world where we live, and both of them display rich and complicated dynamics. Roughly speaking, we are interested in the interactions between these two phenomena, with an emphasis on their long-time behaviors that are characterized by either stability or instability. The dissertation is thus arranged into two integral parts. The first part, covered by Chapter 2, 3, and 4, addresses the stability issue of an array of chemotaxis-fluid systems. We study how singularity formation in the Keller-Segel equation, a nonlocal aggregation-diffusion equation that describes chemotaxis, can be prevented when it is additionally coupled to certain incompressible flows which are forced by buoyancy. The second part, which comprises Chapter 5, investigates the instability phenomenon in the context of incompressible, free-boundary Euler equation subject to the effect of surface tension. More precisely, we present the construction of a scenario that realizes double-exponential vorticity gradient growth.