Symmetries and Critical Phenomena in Fluids

Abstract

We show that given an initial vorticity which is bounded and $m$-fold rotationally symmetric for $m \ge 3$, there is a unique global solution to the 2D Euler equation on the whole plane. That is, in the well-known $L^1 \cap L^\infty$ theory of Yudovich, the $L^1$ assumption can be dropped upon having an appropriate symmetry condition. This contains a class of radially homogeneous solutions to the 2D Euler equation, which gives rise to a new 1D fluid model. We discuss several interesting properties of this 1D system. Using this framework, we construct time quasi-periodic solutions to the 2D Euler equation exhibiting pendulum-like behavior. We also exhibit solutions with compact support for which angular derivatives grow linearly in time or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi-geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions which are merely Lipschitz continuous near the origin - though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered and we extract a 1D model which bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the 1D model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation which are compact support. While the study of special infinite energy solutions to fluid models is classical, this appears to be the first case where they can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite-energy solutions for long time and have a direct bearing on the global regularity problem for finite-energy solutions.