Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation with and without dissipation

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• Pavel Lushnikov (University of New Mexico)
• Friday 09 September 2022, 09:00-10:00
• Seminar Room 1, Newton Institute.

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HY2W02 - Analysis of dispersive systems

Co-authors: names and affiliations: David M. Ambrose (Drexel University)Michael Siegel (New Jersey Institute of Technology)Denis A. Silantyev (University of Colorado, Colorado Springs)

We  analyze the dynamics of singularities and finite time blowup ofgeneralized Constantin-Lax-Majda equation which corresponds tonon-potential effective motion of fluid with competing convection andvorticity stretching terms. Both non-viscous fluid and fluid with varioustypes of dissipation including usual viscosity are considered. An infinitefamilies of exact solutions are found together with the different types ofcomplex singularities approaching the real line in finite times. Bothsolutions on the real line and periodic solutions are considered. In theperiodic geometry, a global-in-time existence of solutions  is proven whenthe data is small and dissipation is strong enough. The found analyticalsolutions on the real line allow finite-time singularity formation forarbitrarily small data, even for various form of dissipation, therebyillustrating a critical difference between the problems on the real lineand the circle. The analysis is complemented by accurate numericalsimulations, which are able to track the formation and motionsingularities in the complex plane. The computations validate and extendthe analytical theory.    

This talk is part of the Isaac Newton Institute Seminar Series series.

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