Convex integration has its roots in the work of Nash in geometry and evolved very recently to a powerful new tool that can prove non-uniqueness for many equations. We know from undergraduate PDE course that one-dimensional Burgers' equation can be proven to exhibit finite-time shock via characteristics; however, such an approach was limited and could not shed much light on other complicated models, a primary example being the Navier-Stokes equations due to being vector-valued and the presence of its pressure term, etc. Via convex integration we now know non-uniqueness at a relatively low regularity level for many systems of equations; examples include Euler equations, Navier-Stokes equations, MHD system, Boussinesq system, active scalars such as the surface quasi-geostrophic equations and porous media equations, etc. This phenomenon has very recently spilled over to non-uniqueness in the compressible case and stochastic case, the latter being when the equation is forced by random noise. The purpose of this talk is to give an overview of the convex integration technique and review recent developments and open problems, primarily focused on the stochastic case.

日時 : 2024年１月15日（月）　15時～16時
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締切日：2024年１月４日（木）17時厳守