Global smoothness for a 1D supercritical transport model with nonlocal velocity

Abstract

We are concerned with a nonlocal transport 1D-model with supercritical dissipation γ ∈ (0, 1) in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the famous 2D dissipative quasigeostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when γ ∈ (0, 1/2). On the other hand, as stated by Kiselev (2010), in the supercritical subrange γ ∈ [1/2, 1) it is an open problem to know whether its solutions are globally regular. We show global existence of non-negative H 3/2 -strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth non-negative initial data, the model has a unique global smooth solution provided that γ ∈ [γ 1 , 1) where γ 1 depends on the H 3/2 -initial data norm. Our approach is inspired on that of Coti Zelati and Vicol (IUMJ, 2016).

Soliton solutions to a (2+1)-dimensional nonlocal NLS equation

“…In [29], Li and Rodrigo proved the finite-time blowup of smooth solutions to (1.3) in the supercritical case 0 < γ < 1 2 (see also [24,31,33] for different proofs for this blow-up result). Recently, for each smooth nonnegative initial data, Ferreira and Moitinho [18] obtained the existence of global classical solutions to (1.3) for γ ∈ (γ 1 , 1) with γ 1 depending on the H 3 2 -norm of the initial data. For the remaining case 1 2 ≤ γ < 1, whether smooth solutions to (1.3) may blow up in finite time is currently still open.…”

Formation of singularities for multi-dimensional transport equations with nonlocal velocity

“…Another proof of this dissipative singularity result via the telescopic sum argument is presented in [30]. Recently, in [19], Ferreira and Moitinho obtained the existence of global classical solutions to (1.6) with γ ∈ (γ 1 , 1), where γ 1 depends on some norm of the initial data. The question of global regularity vs finite time singularity for the dissipative CCF model (1.6) with 1 2 ≤ γ < 1 is open as well.…”

The singularities for a periodic transport equation

“…in [4], where the authors concluded that the solutions would develop into gradient blow-up as κ = 0 and exist globally as κ > 0, α > 1. References [5,6] are recommended for some recent results on this transport equation.…”