Collapse Versus Blow-Up and Global Existence in the Generalized Constantin-Lax-Majda Equation
Abstract
The question of finite-time singularity formation versus 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 ac=0.6890665337007457 … below which there is finite-time singularity formation that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We prove the existence of a leading-order power-law complex singularity for general values of a in the analytical continuation of the solution from the real spatial coordinate into the complex plane and identify the power-law exponent. This singularity controls the leading-order behavior of the collapsing solution. We prove that this singularity can persist over time, without other singularity types present, provided a =0 or 1/2. This enables the construction of exact analytical solutions for these values of a. For other values of a, this leading-order singularity must coexist with other singularity types over any nonzero interval of time. For ac<a ≤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 ≳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 <ac which are similar to the real line case. For ac<a ≤0.95 , we find new blow-up solutions which are neither expanding nor collapsing. For a ≥1 , we identify a global existence of solutions.
- Publication:
-
Journal of NonLinear Science
- Pub Date:
- October 2021
- DOI:
- 10.1007/s00332-021-09737-x
- arXiv:
- arXiv:2010.01201
- Bibcode:
- 2021JNS....31...82L
- Keywords:
-
- Constantin-Lax-Majda equation;
- Collapse;
- Blow-up;
- Self-similar solution;
- Nonlinear Sciences - Pattern Formation and Solitons;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Physics - Fluid Dynamics
- E-Print:
- 49 pages, 29 figures