On a 1D transport equation with nonlocal velocity and supercritical dissipation

Do, Tam

doi:10.1016/j.jde.2014.01.037

Cited by 16 publications

(16 citation statements)

References 11 publications

(31 reference statements)

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“…It is still an open problem to know whether regular solutions blow-up or exist globally in time in the case 1/2 ≤ α < 1. The eventual regularity of smooth solution in the spirit of previous works for the supercritical SQG equation, has been obtained by Do in [16]. We also note that the local well posedness in H 2 has been obtained by Bae and Granero-Belinchón in [1] in the inviscd case.…”

Section: Introductionsupporting

confidence: 67%

On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

Lazar

2016

Journal of Differential Equations

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We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space H k (w λ,κ )∩L ∞ , where k = max(0, 3/2− α) and w λ,κ is a given family of Muckenhoupt weights, we prove a global existence result in the subcritical case α ∈ (1, 2). We also prove a local existence theorem for large data in H 2 (w λ,κ ) ∩ L ∞ in the supercritical case α ∈ (0, 1). The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.

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Section: Introductionsupporting

confidence: 67%

On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

Lazar

2016

Journal of Differential Equations

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“…Notice that for L = |D| α and u = H(θ) with H the 1D Hilbert transform, an estimate similar to (3.13) was obtained in [20,Lemma 2.7].…”

Section: )mentioning

confidence: 59%

“…Up to now, the problem concerning the global regularity of solution for the supercritical CCF equation with α ∈ [1/2, 1[ is still open. We mention that Do in [20] solved the eventual regularity of the global weak solution for all the supercritical case α ∈]0, 1[ by applying the method of [25], and also proved the global well-posedness result of the CCF equation at some slightly supercritical cases. The incompressible porous media equation is the equation (1.1) with the following velocity field…”

Section: Introductionmentioning

confidence: 91%

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On the Regularity Issues of a Class of Drift-Diffusion Equations with Nonlocal Diffusion

Miao¹,

Xue²

2019

SIAM J. Math. Anal.

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In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by some zero-order pseudo-differential operators. Through using the method of nonlocal maximum principle in a unified way, we prove the global well-posedness result in some slightly supercritical cases, and show the eventual regularity result in the supercritical type cases. The time after which the solution is smoothly regular in the supercritical type cases can be evaluated appropriately, so that we can prove a type of global result recently obtained by [17] and also show the global regularity of vanishing viscosity solution at some logarithmically supercritical cases.2010 Mathematics Subject Classification. 76D03, 35Q35, 35Q86.

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“…In the range 1 2 ≤ γ < 1, to the best of our knowledge, the formation of singularity in finite time or global smoothness is an open problem (stated by [14, p. 251]), even for sign restriction on the initial data, i.e., θ 0 ≥ 0 or θ 0 ≤ 0. In [10], for 0 < γ < 1, Do obtained eventual regularization of solutions for non-negative initial data. He also obtained global regularity for a modified 1D model with Λ γ θ replaced by…”

Section: Introductionmentioning

confidence: 99%

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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On a 1D transport equation with nonlocal velocity and supercritical dissipation

Cited by 16 publications

References 11 publications

On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

On the Regularity Issues of a Class of Drift-Diffusion Equations with Nonlocal Diffusion

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

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