The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

Linrui Li,Shu Wang

doi:10.1155/2020/2757398

Abstract

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

Highlights

We study the formation of singularities for the following coupled Burgers–Constantin–Lax–Majda system with the nonlocal term:
It is known that the solutions to two systems, both the CLM system and the Burgers system, have the characteristic of finite-time blow-up singularity formation for the smooth solutions with the smooth initial data
E rest of this paper is as follows. e main results of this paper are given in Section 2, and Section 3 is devoted to the proofs of the main results of Section 2

Summary

Introduction

It is known that the solutions to two systems, both the CLM system and the Burgers system, have the characteristic of finite-time blow-up singularity formation for the smooth solutions with the smooth initial data. We recall some related problems on the finite-time blow-up singularity regimes about some models with the Hilbert transform and the generalized Burgers equations. Vx Hω, as another simplified model of the 3D vorticity version of incompressible inviscid Euler flow in [4, 5] and obtained some numeric results which implied that the convection term vωx seems to prevent the appearance of finite-time blow-up singularity.

Results

Conclusion