The singularities for a periodic transport equation

TL;DR

This paper investigates a 1D periodic transport equation with nonlocal flux and fractional dissipation, establishing local well-posedness and demonstrating finite-time singularity formation both with and without dissipation.

Contribution

It provides the first analysis of singularity formation in a fractional dissipative transport equation with nonlocal flux on a periodic domain.

Findings

Solutions develop singularities in finite time without dissipation.

Weak fractional dissipation does not prevent finite-time blowup.

Local well-posedness is established in Sobolev space H^3.

Abstract

In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+\kappa\Lambda^{\alpha}u=0,\quad (t,x)\in R^{+}\times S, $$ where $\kappa\geq0$, $0<\alpha\leq1$ and $S=[-\pi,\pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $\kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $\kappa\Lambda^{\alpha}u$, we also prove that the finite time blowup would occur.