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تسجيل مستخدم جديدCollapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation
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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 $a_c=0.6890665337007457ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457ldots here then ldots doesnt 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 $a_c<aleq1$, 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 $a_c<ale0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $ age 1,$ we identify a global existence of solutions.
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