The Euler Equations in Function Spaces of Generalized Smoothness (Zachary Radke, OSU)

Abstract: In this talk, we will describe a well/ill-posedness result for the 2D incompressible Euler equations. We investigate solutions in a setting logarithmically smoother than previously done, in a hope to identify the key dynamics leading to a breakdown of regularity in 2D fluid flow. When order of the logarithmic derivative is sufficiently large one obtains global well posedness, however, below this threshold, one can construct initial data for which the corresponding solution blows up instantaneously in the logarithmic Sobolev norm. In this sense, the result is sharp at this logarithmic scale, but by no means is the story completed by it so we will discuss ways to dive deeper.