Abstract for the talk on 28.07.2021 (11:00 h)

Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

Jing An (MPI MIS, Leipzig)
Convergence to a traveling wave for the Burgers-FKPP equation
28.07.2021, 11:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, E1 05 (Leibniz-Saal)

We consider the long time behavior of solutions to the Burgers-FKPP equation

\(u_t +\beta u u_x = u_{xx} + u-u^2.\)

The Burgers-FKPP equation solutions exhibit a phase transition phenomenon from being pulled to pushed as \(\beta\) increases, and the analysis at the transition case \(\beta=2\) is quite delicate. We show the convergence to a traveling wave for the whole spectrum of \(\beta\). In particular, when \(\beta\leq 2\), 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 \(\beta=2\) from \(\beta<2\), 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.

Physical attendance in seminars is restricted to 25 participants.
If you want to attend this talk, you need to register in advance using this special form. Registrations will be accepted on first come, first served basis. External participants from Leipzig University need to fill in their name, address, email, and phone number.
Please also check our general Corona rules page. Participants must wear a face mask, and they are encouraged to use the corona rapid tests made available by the institute.

21.07.2021, 00:11