Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

The singularities for a periodic transport equation

Pages: 1751 – 1760

(Fast Communication)

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n6.a13

Authors

Yong Zhang (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

Fengquan Li (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

Fei Xu (School of Mathematical Sciences, Dalian University of Technology, Dalian, China)

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 $\kappa \geq 0, 0 \lt \alpha \leq 1$ 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 a singularity in finite time. By adding a weak dissipation term $\kappa \Lambda^\alpha u$, we also prove that the finite-time blowup would occur.

Keywords

singularity, nonlocal flux, fractional dissipation, odd initial data

2010 Mathematics Subject Classification

35A01, 35Q35, 76B03, 76B15

Received 11 January 2021

Accepted 1 May 2021

Published 2 August 2021