Strong illposedness for SQG in critical Sobolev spaces

Abstract

We prove that the inviscid surface quasi-geostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data H 2 (T 2 ) without any solutions in L ∞ t H 2 . Moreover, we prove strong critical norm inflation for C ∞ -smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with two-dimensional incompressible Euler equations.

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

“…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.…”

A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces

“…As Bourgain and Li are emphasizing in the introduction of [3], the 3D case is much more involved compared to the 2D case due to the presence of the vortex stretching term, lack of L p conservation of the vorticity, absence of Yudovich theory, and the issues arising from dealing with the non-local norm H 5/2 . A short proof for the 2D case was given in [8], and this simplified argument became a basis for further developments in the well-posedness theory, including continuous-in-time loss of Sobolev regularity ( [9]), enstrophy growth in the Navier-Stokes equations ( [12,11]), and strong ill-posedness for active scalar equations ( [10]). Some further developments in this direction are given in [17,5].…”

“…This follows from the embedding H d/2 ⊂ BM O and the singular integral operator bound L ∞ → BM O. Moreover, the BM O bound implies that the equation (1.4) possesses at most one solution belonging to the class (3.1) and the velocity u(t, •) is log-Lipschitz (see [10,1]): there exists a constant C > 0 such that sup…”

“…Let us illustrate the main ideas of the proof, which have been inspired by the previous works [8,12,10]. The interested reader can find more information there; in the current setup, the main difference is the presence of vortex stretching.…”

“…There are two competing effects; while the squeezing of the vorticity weakens the induced velocity gradient, the vortex stretching term strengthens it. As in our previous work [10], we identify a timescale for each of the bubbles during which it essentially remains the same, and obtain a lower bound on the velocity gradient based on such a stability statement. It turns out that, together with the help of the vortex stretching term, such a lower bound is sufficient to prove strong ill-posedness results.…”