Maximal Double-Exponential Growth for the Euler Equation on the Half-Plane
Andrej Zlatos
TL;DR
This work analyzes the 2D incompressible Euler equation on the half-plane and proves that smooth solutions can exhibit double-exponential growth of the vorticity gradient. It establishes a sharp upper bound on the growth rate via a detailed velocity decomposition for odd-in-$x_1$ vorticities, yielding a universal differential inequality whose solution gives a double-exponential bound with rate $2/\pi$. The authors then construct explicit smooth, compactly supported initial data (odd in $x_1$) whose evolution attains this growth rate, demonstrating that $2/\pi$ is the maximal possible rate on the half-plane; the construction extends to uniformly smooth domains with symmetry. Together, the results provide, for the first time on unbounded or half-plane domains, both the attainability and optimality of maximal gradient growth for the 2D Euler equation.
Abstract
We show that smooth solutions to the Euler equation on the half-plane can exhibit double-exponential growth of their vorticity gradients. We also determine the maximal possible growth rate and construct solutions that saturate it. These are the first such results on an unbounded resp. any 2D domain.