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 ) \gamma \in (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 well-known 2D dissipative quasi-geostrophic 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 ) \gamma \in (0,1/2) . On the other hand, as stated by Kiselev (2010), in the supercritical subrange γ ∈ [ 1 / 2 , 1 ) \gamma \in \lbrack 1/2,1) it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative H 3 / 2 H^{3/2} -strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that γ ∈ [ γ 1 , 1 ) \gamma \in \lbrack \gamma _{1},1) where γ 1 \gamma _{1} depends on the H 3 / 2 H^{3/2} -initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).