Analysis Seminar

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Nonlinear eigenvalue problem for collapse in the Generalized Constantin-Lax-Majda Equation with and without dissipation

Speaker: Pavel Lushnikov, University of New Mexico

Location: Warren Weaver Hall 1302

Date: Thursday, October 19, 2023, 11 a.m.

Synopsis:

We analyze the dynamics of singularities and finite time blowup of
generalized Constantin-Lax-Majda equation which corresponds to
non-potential effective motion of fluid with competing convection and
vorticity stretching terms. Both non-viscous fluid and fluid with various
types of dissipation including usual viscosity are considered. An infinite
families of exact solutions are found together with the different types of
complex singularities approaching the real line in finite times. A
nonlinear eigenvalue problem is formulated and solved to determine the
rate of blow up and the corresponding self-similar solutions. Both
solutions on the real line and periodic solutions are considered. In the
periodic geometry, a global-in-time existence of solutions  is proven when
the data is small and dissipation is strong enough. The found analytical
solutions on the real line allow finite-time singularity formation for
arbitrarily small data, even for various form of dissipation, thereby
illustrating a critical difference between the problems on the real line
and the circle. The analysis is complemented by accurate numerical
simulations, which are able to track the formation and motion
singularities in the complex plane. The computations validate and extend
the analytical theory.