Finite time blow-up for a multi-dimensional model of the Kiselev-Sarsam equation
Wanwan Zhang
TL;DR
The paper introduces a multi-dimensional Kiselev-Sarsam-type equation $\partial_t\rho + g\,\mathcal{R}_a\rho \cdot \nabla\rho = 0$ with a Calderón-Zygmund-type velocity operator $\mathcal{R}_a$ whose kernel smoothly interpolates between the Riesz transform and the zero operator as the parameter $a$ varies. It establishes local well-posedness in $H^s(\mathbb{R}^n)$ for $s>\frac{n}{2}+1$ and proves finite-time gradient blow-up for radial, smooth, compactly supported, nondecreasing initial data with $\rho_0(0)<0$, by combining a Beale-Kato-Majda-type criterion with sharp nonlinear lower bounds on the transport term. The analysis relies on Fourier multiplier properties of $\mathcal{R}_a$, Kato-Ponce commutator estimates, and a radial monotonicity preservation argument, culminating in a differential inequality that enforces blow-up in finite time. The results extend the classical KS mechanism to higher dimensions and provide concrete nonlinear inequalities that govern singularity formation for this nonlocal active-scalar model. $\,$
Abstract
In this paper, we propose and study a multi-dimensional nonlocal active scalar equation of the form \begin{eqnarray*} \partial_tρ+g\mathcal{R}_aρ\cdot \nablaρ= 0,~ρ(\cdot,0)=ρ_{0}, \end{eqnarray*} where the transform $\mathcal{R}_a$ is defined by \begin{eqnarray*} \mathcal{R}_af(x)=\frac{Γ(\frac{n+1}{2})}{π^{\frac{n+1}{2}}}P.V.\int\limits_{\mathbb{R}^n}\Big(\frac{x-y}{|x-y|^{n+1}}-\frac{x-y}{(|x-y|^2+a^2)^{\frac{n+1}{2}}}\Big)f(y)dy. \end{eqnarray*} 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.