Non-uniqueness of weak solutions to nonlinear heat equations
A talk in the CRC Seminar series by
Irfan Glogic from Bielefeld
| Abstract: | H. Jia and V. Sverak showed in 2015 that existence of forward self-
similar solutions that are linearly unstable can be used to establish non-uniqueness
of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equa-
tions. Although the aforementioned (non-)uniqueness question remains open to
this day, a number of works have since utilized the ideas of Jia-Sverak to demon-
strate non-uniqueness for various fluid dynamics equations, albeit with non-zero
forcing terms. In this talk, we consider the unforced focusing power nonlinearity
heat equation, and rigorously implement the Jia-Sverak method, thereby show-
ing non-uniqueness of local solutions for the full range of supercritical Lebesgue
spaces. In particular, we rigorously verify the (analogue of the) spectral assumption made by Jia-Sverak
for the Navier-Stokes equations. At the end, we discuss potential implications of this result for more general
classes of evolution equations. This is joint work with M. Hofmanova (Bielefeld), T. Lange (Pisa), and
E. Luongo (Bielefeld). Within the CRC this talk is associated to the project(s): B7 |