2025/11/06 by Zhang, Wanwan
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2511.04660
In this paper, we propose and study a multi-dimensional nonlocal active scalar equation of the form ∂tρ+gRaρ⋅ ∇ρ= 0,~ρ(⋅,0)=ρ0, where the transform Ra is defined by Raf(x)=\fracΓ((n+1)/(2))π(n+1)/(2)P.V.∫ℝn(\fracx-y|x-y|n+1-\fracx-y(|x-y|2+a2)(n+1)/(2))f(y)dy. This model can be viewed as a natural generalization of the well-known Kiselev-Sasarm equation, which was introduced in [13] as a one-dimensional model for the two-dimensional incompressible porous media equation. We show the local well-posedness for this multi-dimensional model as well as the gradient blow-up in finite time for a class of radial initial data.