PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Leonid Berlyand.

Title:Kuramoto-Sivashinsky equation: A hidden coercivity in the forced Burgers' equation
Seminar:Marker Lecture Series
Speaker:Dr. Felix Otto, Max Planck Institute for Mathematics in the Sciences, Leipzig
The Kuramoto-Sivashinsky equation, i. e. $$ \partial_t u+\partial_x({\textstyle\frac{1}{2}}u^2) +\partial_x^2u+\partial_x^4u\;=\;0 $$ is a ``normal form'' for many processes which lead to complex dynamics in space and time. Numerical simulations show that after an initial layer, the statistical properties of the solution are independent of the initial data and the system size \(L\) (defined by the period \(u(t,x+L)=u(t,x)\)). More precisely, the energy \(\int u^2\,dx\) is equally distributed over all wave numbers \(|k|\ll 1\).
PDE theory is far from a rigorous understanding of these phenomena. Over the past 20 years, bounds on the space-time average \(\langle\langle(|\partial_x|^\alpha u)^2\rangle\rangle^{1/2}\) of (fractional) derivatives \(|\partial_x|^\alpha u\) of \(u\) in terms of \(L\) have been established and improved. I will present the bound $$ \langle\langle(|\partial_x|^\alpha u)^2\rangle\rangle^{1/2}\;=\;O(\ln^{5/3} L) $$ for \(1/3< \alpha\le 2\). This is the first result in favor of an extensive behavior --- albeit only up to a logarithm and for a restricted range of fractional derivatives.
The proof essentially relies on an extension of Oleinik's principle to the inhomogeneous inviscid Burgers' equation \(\partial_tu+\partial_x(\frac{1}{2}u^2)\;=\;f\). From this extension we learn that the quadratic term \(\partial_x(\frac{1}{2}u^2)\), which is conservative, effectively behaves like a coercive term in the sense that we obtain a priori estimates as if \(\int \partial_x(\frac{1}{2}u^2)\,u\,dx\;\sim\; \int||\partial_x|^{1/3}u|^3\,dx\).

Room Reservation Information

Room Number:MB114
Date:10 / 24 / 2013
Time:03:35pm - 04:35pm