Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation

Abstract

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457\dots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at $a=1/2$ as well as prove the existence of a leading order complex singularity for general values of $a$ in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For $a_c<a\le 1$, we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For $a\gtrsim 1.3$, we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for $a<a_c$ which are similar to the real line case. For $a_c<a\le 0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $a\ge 1,$ we identify a global existence of solutions.

arXiv comment49 pages, 29 figures

References in corpus (12)

• The AAA algorithm for rational approximation
• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
• A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
• Dispersion-managed soliton in a strong dispersion map limit
• Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation
• New conformal mapping for adaptive resolving of the complex singularities of Stokes wave
• Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation
• Beyond leading order logarithmic scaling in the catastrophic self-focusing (collapse) of a laser beam in Kerr media
• Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Constants of Motion
• On the Constantin-Lax-Majda Model with Convection
• Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation
• Logarithmic scaling of the collapse in the critical Keller-Segel equation

Cited by in corpus (6)

• Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation
• On the Slightly Perturbed De Gregorio Model on $S^1$
• Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model
• On self-similar finite-time blowups of the De Gregorio model on the real line
• Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation
• 1D Model for the 3D Magnetohydrodynamics