Analysis and PDE Seminar: Sharp fronts for models of incompressible fluids

Friday, February 19 at 5:00pm to 6:00pm

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Calvin Khor, Beijing Normal University

Abstract: In the pursuit of understanding vortex filaments for the 3D Euler equation (solutions with concentrated vorticity), we describe sharp fronts and almost sharp fronts for the SQG equation, which are analogues of vortex filaments that can be discussed in a rigourous setting. We further discuss a singular generalisation of the SQG model for which given an almost-sharp front, a special `spine’ curve can be chosen in a measure theoretic way that formally survives in the limit. In the second part of the talk, we discuss a different system, called the Navier—Stokes—Boussinesq system, for which we cannot write the explicit evolution equation for the contour dynamics of the sharp front. Nevertheless, the regularity of a sharp front can be shown to be preserved globally in time. We show that this can be extended to the setting with fractional dissipation, until right before the equation with critical dissipation, which does not seem to have a well-posedness theory for temperature patch solutions.

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