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Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Friday, January 10, 2020 - 12:00 to 12:50

Location:

STAG 160

Event Link:

Local Speaker:

Abstract:

The blowup phenomena of reaction-diffusion equations were studied by Kaplan, Fujita, Levine and others in the 1960s. In 1975, McKean found a representation of solution to the KPP-Fisher equation through a branching process, leading to a probabilistic approach to study the regularity of solutions to a family of semilinear parabolic equations. In 1997, Le Jan and Sznitman used ideas similar to McKean's for the Navier-Stokes equations (NSE). The branching process associated with NSE is known to be explosive in stochastic sense (Dascaliuc, Pham, Thomann, Waymire 2019). This property appeals the idea of non-uniqueness of solutions. However, this remains a challenging problem. In several toy models (e.g. the alpha-Riccati equation, the cheap NSE), one can indeed construct more than one solution. I will discuss the nonuniqueness, blowup, and stochastic explosion of these models. Joint work with Radu Dascaliuc, Enrique Thomann, and Ed Waymire.