On the local and global existence of solutions to 1d transport equations with nonlocal velocity

Bae, Hantaek; Granero-Belinchón, Rafael; Lazar, Omar

doi:10.3934/nhm.2019019

Cited by 3 publications

(2 citation statements)

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“…Moreover, they conjectured that solutions obtained as vanishing viscosity approximations could be bounded in C 1/2 , for all t > 0, which would possibly yield Hölder regularization effects for the case 1/2 ≤ γ < 1 and then would solve the global regularity conjecture in [14, p. 251] (see Conjectures 7.1 and 7.2 in [16]). In [2], Bae, Granero-Belinchón and Lazar considered the inviscid case and developed a theory of global weak super solutions for (1.1) with non-negative data θ 0 ∈ L 1 ∩ L 2 .…”

Section: Introductionmentioning

confidence: 99%

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

Ferreira

Moitinho

2020

Proc. Amer. Math. Soc.

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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).

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Section: Introductionmentioning

confidence: 99%

“…Let us remark that in view of (1.2) the non-negative condition is relatively common for (1.1) and have been assumed in several works (see, e.g., [2,3,7,11,14]). Also, it is worth mentioning that, in consonance with the conjectures in [16], we obtain boundedness of solutions in C α for α > 1 − γ.…”

Section: Introductionmentioning

confidence: 99%