Analysis Seminar

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Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations

Speaker: Matei Coiculescu, NYU, Courant

Location: Warren Weaver Hall 1302

Date: Thursday, September 24, 2026, 11 a.m.

Synopsis:

We investigate homothetic solutions to the Navier-Stokes equations. These are forward self-similar solutions U with the property that U and \beta*U are both solutions of the Navier-Stokes equations for some nontrivial \beta. We prove rigidity theorems for these solutions, which can only arise from (-1)-homogeneous data that are also stationary solutions of the Euler equations. We also construct nontrivial examples and show the existence of unstable approximate eigenvalues for some homothetic solutions. Finally, in 2D, our rigidity theorem excludes essentially any homothetic solution besides the Oseen vortex, which we then show to be spectrally stable by a general stability criterion akin to the Rayleigh stability condition.