TY - JOUR

T1 - On a transport equation with nonlocal drift

AU - Silvestre, Luis

AU - Vicol, Vlad

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2016

Y1 - 2016

N2 - In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions (Formula Presented) where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion (Formula Presented) where Λ = (-Λ)½, and ½ ≤ γ < 1. Our results also apply to the model with velocity field u = Λs Hθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C(s+1)/2, for all positive time.

AB - In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions (Formula Presented) where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion (Formula Presented) where Λ = (-Λ)½, and ½ ≤ γ < 1. Our results also apply to the model with velocity field u = Λs Hθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C(s+1)/2, for all positive time.

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U2 - 10.1090/tran6651

DO - 10.1090/tran6651

M3 - Article

AN - SCOPUS:84958811669

SN - 0002-9947

VL - 368

SP - 6159

EP - 6188

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -