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# 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

#### 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

This project was supported by the National Natural Science Foundation of China (No. 11571057).

Received 11 January 2021

Accepted 1 May 2021

Published 2 August 2021