Remarks on a multi-dimensional transport equation with nonlocal velocity
Abstract
In this paper, we revisit a multi-dimensional nonlocal transport equation of the form ∂tρ+gℛaρ· ∇ρ= 0, ρ(·,0)=ρ0, where the transform ℛa is defined by ℛaρ(x,t)=(Γ((n+1)/2))/(π(n+1)/2)P.V.∫ℝn((x−y)/(|x−y|n+1)−(x−y)/((|x−y|2+a2)(n+1)/2))ρ(y,t)dy. This model was first introduced in [W. Zhang, J. Differential Equations 467 (2026) 114275] as a natural generalization of the well-known Kiselev-Sarsam equation proposed in [A. Kiselve and N.A. Sarsam, Nonlinearity 38 (2025) 055018]. By using an improved pointwise inequality for the transform ℛa, we obtain a better and more natural version of a nonlinear integral inequality proved in Zhang [31] for the transform ℛa. In addition, we also establish a new weighted nonlinear inequality for the transform ℛa implying the finite-time blow-up for a class of smooth radial initial data.
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BibTeX
@misc{zhang2026remarks,
title = {{Remarks on a multi-dimensional transport equation with nonlocal velocity}},
author = {Wanwan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04632},
eprint = {2610.04632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04632},
}
RIS
TY - UNPB
TI - Remarks on a multi-dimensional transport equation with nonlocal velocity
AU - Wanwan Zhang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04632
UR - https://arxiv.org/abs/2610.04632
AB - In this paper, we revisit a multi-dimensional nonlocal transport equation of the form ∂_tρ+gℛ_aρ· ∇ρ= 0, ρ(·,0)=ρ_0, where the transform ℛ_a is defined by ℛ_aρ(x,t)=(Γ((n+1)/2))/(π^(n+1)/2)P.V.∫_ℝ^n((x−y)/(|x−y|^n+1)−(x−y)/((|x−y|^2+a^2)^(n+1)/2))ρ(y,t)dy. This model was first introduced in [W. Zhang, J. Differential Equations 467 (2026) 114275] as a natural generalization of the well-known Kiselev-Sarsam equation proposed in [A. Kiselve and N.A. Sarsam, Nonlinearity 38 (2025) 055018]. By using an improved pointwise inequality for the transform ℛ_a, we obtain a better and more natural version of a nonlinear integral inequality proved in Zhang [31] for the transform ℛ_a. In addition, we also establish a new weighted nonlinear inequality for the transform ℛ_a implying the finite-time blow-up for a class of smooth radial initial data.
ER -