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

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 a_c= 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 a_c< 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< a_c which are similar to the real line case. For a_c< 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.