Talk

Convergence to a traveling wave for the Burgers-FKPP equation

  • Jing An (MPI MiS, Leipzig)
E1 05 (Leibniz-Saal)

Abstract

We consider the long time behavior of solutions to the Burgers-FKPP equation ut+βuux=uxx+uu2.

The Burgers-FKPP equation solutions exhibit a phase transition phenomenon from being pulled to pushed as β increases, and the analysis at the transition case β=2 is quite delicate. We show the convergence to a traveling wave for the whole spectrum of β. In particular, when β2, we introduce a weighted Hopf-Cole transform to construct upper and lower barriers in the self-similar variables for the linearized equation on the half line. This new transform differentiates the transition case β=2 from β<2, as its boundary condition approaches a positive constant rather than zero. In that case, capturing the exact logarithmic delay in the reference frame is essential, and the problem boils down to providing a temporal decay estimate for a spatially inhomogeneous conservation law. I will describe how we get this temporal decay rate by combining a weighted dissipation inequality with a weighted Nash-type inequality, which probably is the most novel part of the work.

This is joint work with Chris Henderson and Lenya Ryzhik.

Upcoming Events of this Seminar

  • Monday, 14.07.25 tba with Alexandra Holzinger
  • Tuesday, 15.07.25 tba with Anna Shalova
  • Tuesday, 12.08.25 tba with Sarah-Jean Meyer
  • Friday, 15.08.25 tba with Thomas Suchanek
  • Friday, 22.08.25 tba with Nikolay Barashkov
  • Friday, 29.08.25 tba with Andreas Koller