Symmetries and Critical Phenomena in Fluids

Abstract

We study two‐dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well‐posedness result that allows for the data and solutions to be scale‐invariant. These scale‐invariant solutions are new and their study seems to have far‐reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two‐dimensional Euler squation. That is, in the well‐known L1∩L∞ theory of Yudovich, the L1‐assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0‐homo‐geneous in space. Such vorticity is shown to satisfy a new one‐dimensional evolution equation on 𝕊1. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi‐periodic solutions to the two‐dimensional Euler equation exhibiting pendulum‐like behavior. Finally, using the analysis of the one‐dimensional equation, we exhibit strong solutions to the two‐dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) 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 that 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 one‐dimensional model that bears great resemblance to the so‐called De Gregorio model. We then show that finite‐time singularity formation for the one‐dimensional model implies finite‐time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions 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 direct bearing on the global regularity problem for finite‐energy solutions. © 2019 Wiley Periodicals, Inc.

Uniformly Rotating Smooth Solutions for the Incompressible 2D Euler Equations

“…A classical theorem of Yudovich [27] asserts that if the initial data is in L 1 ∩ L ∞ , then there is uniqueness within that class. In a very recent preprint, [16], the L 1 assumption can be dropped upon having an appropriate symmetry (m-fold) condition.…”

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

“…In addition, it is well-known that Yudovich [57] obtained the existence and uniqueness of global weak solutions if the initial vorticity ω 0 lies in L 1 ∩ L ∞ for the domain D = R 2 (see [47] for the bounded domain case). One can refer to [3,4,6,11,18,19,20,26,39,40,42,56,58] and the references therein for other related interesting and important aspects concerning with the two-dimensional incompressible Euler equations.…”

“…When 0 < α < 1 2 , (1.5) is the so-called inviscid modified or generalized SQG in the literature (see [38] and references therein). There have been a number of mathematical studies on inviscid SQG and inviscid modified SQG equations and we refer the readers to 2 [9,10,12,13,14,15,16,17,19,23,25,28,29,36,37,38,50,52] and the references therein for more details.…”

Preservation of log-Hölder coefficients of the vorticity in the transport equation

“…To clarify, in general (without the odd-odd assumption) it is possible that |ω(x, t)| ≃ log( 1 |x| ) −γ and u(x, t) ∈ L 1 t Lip hold at the same time for any γ > 0. This happens for instance the vorticity satisfies a rotational symmetry; see [3].…”