Speaker
Dr. Atanas Stefanov, Professor, Department of Mathematics, CAS, University of Alabama-Birmingham
Title
Mathematics Seminar Series
Subtitle
Asymptotic stability for kink solutions for the viscous Boussinesq problem and dispersive-diffusive Burger models
Physical Location
Allen 14
Abstract:
Traveling kinks are special solutions to certain 1D PDE’s posed on the real line, having two different values at both infinities. Applications include combustion dynamics, invasive species, water wave problems etc. An important question is whether or not such coherent structures are dynamically stable. That is, if one starts close to such special solutions, do we stay close to the solution (or translation thereof) forever?
We consider two recent works on kinks, arising in the Boussinesq model and also dispersive-diffusive Burger models, such as KdV-Burger. We show that the unique monotone inks are asymptotically stable for the Boussinesq. For the KdV-Burger, we show more: that the monotone kinks are actually attracting sets - every data, including large perturbations produces a solution, which converges to a translate of the kink, with explicit time decay rates. The Boussinesq result uses fairly straightforward energy estimates (and a bit of spectral theory), while for the KdV-Burger, we built upon recent works about attractivity and we bootstrap the time decay rates.
Note:
Contact Prof. Shantia Yarahmadian at syarahmadian@math.msstate.edu or (662) 325-7143 for additional information.