TY - JOUR

T1 - Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation

AU - Lushnikov, Pavel M.

AU - Silantyev, Denis A.

AU - Siegel, Michael

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/10

Y1 - 2021/10

N2 - 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.

AB - 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.

KW - Blow-up

KW - Collapse

KW - Constantin–Lax–Majda equation

KW - Self-similar solution

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U2 - 10.1007/s00332-021-09737-x

DO - 10.1007/s00332-021-09737-x

M3 - Article

AN - SCOPUS:85112446626

SN - 0938-8974

VL - 31

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 5

M1 - 82

ER -