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

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Linrui Li ◽  
Shu Wang
Keyword(s):  
Initial Data ◽  
Finite Time ◽  
Smooth Solution ◽  
Blow Up ◽  
Nonlocal Term ◽  
Convection Term ◽  
Large Initial Data ◽  
Singularity Formation ◽  
Time Singularity ◽  
Finite Time Singularity

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.

scholarly journals Hall MHD and Electron Inertia Effects in Current Sheet Formation at a Magnetic Neutral Line

2015 ◽  
Vol 5 (2) ◽  
pp. 109-125 ◽  
Author(s):  
Yuri E. Litvinenko ◽  
Liam C. McMahon
Keyword(s):  
Current Sheet ◽  
Finite Time ◽  
Numerical Solutions ◽  
Neutral Line ◽  
Singularity Formation ◽  
Time Singularity ◽  
Similarity Reduction ◽  
Electron Inertia ◽  
Magnetic Neutral Line ◽  
Finite Time Singularity

AbstractAn exact self-similar solution is used to investigate current sheet formation at a magnetic neutral line in incompressible Hall magnetohydrodynamics. The collapse to a current sheet is modelled as a finite-time singularity in the solution for electric current density at the neutral line. We establish that a finite-time collapse to the current sheet can occur in Hall magnetohydrodynamics, and we find a criterion for the finite-time singularity in terms of the initial conditions. We derive an asymptotic solution for the singularity formation and a formula for the singularity formation time. The analytical results are illustrated by numerical solutions, and we also investigate an alternative similarity reduction. Finally, we generalise our solution to incorporate resistive, viscous and electron inertia terms.

Finite time blow-up of complex solutions of the conserved Kuramoto–Sivashinsky equation in ℝd and in the torus 𝕋d, d ⩾ 1

2019 ◽  
Vol 149 (5) ◽  
pp. 1175-1188
Author(s):  
Léo Agélas
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Besov Space ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Large Initial Data ◽  
Complex Solutions ◽  
Complex Valued ◽  
Sivashinsky Equation

AbstractWe consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

Finite-time singularity formation in Hele-Shaw systems

1993 ◽  
Vol 47 (6) ◽  
pp. 4182-4196 ◽  
Author(s):  
Todd F. Dupont ◽  
Raymond E. Goldstein ◽  
Leo P. Kadanoff ◽  
Su-Min Zhou
Keyword(s):  
Finite Time ◽  
Singularity Formation ◽  
Time Singularity ◽  
Finite Time Singularity
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