Kevin Zumbrun lectures in UNC PDE mini-school

DSC_0160This past week we held the first of three NSF supported graduate mini-schools in Partial Differential Equations (October 2-4, 2013) for this academic year.  The main speaker was Kevin Zumbrun of Indiana University, who spoke about stability of periodic waves in systems of hyperbolic conservation laws.  Fortunately, we had a great turn out for all the lectures as we drew a large audience from both the pure and applied faculty, neighboring universities, other departments and a host of our graduate students.  A special thank you to the Elaine Bullock and Sunny Oakley from the math department staff for setting up the snacks and coffee for everyone each afternoon.

DSC_0162 Kevin’s first lecture discussed the existence of periodic solutions and, motivated by ideas of Whitham, a modified approach towards studying their modulation and stability within equations related to fluid flow.  In the second lecture, he discussed spectral stability and various analytic and numerical methods for computing the Floquet spectrum for related periodic problems.  Finally, he discussed the rigorous and demanding perturbation theory involved in studying these equations on long or global time scales.

In addition, there were three supporting lectures by Kevin’s graduate students Blake Barker and Soyeun Jung of Indiana and a postdoc Fang Yu of Penn State.  Blake spoke Wednesday about numerically assisted proofs of spectral stability and rigorous bounds on numerical computations using interval arithmetic, and Soyeun spoke Thursday about pointwise bounds for linearized operators arising as perturbations of periodic solutions in reaction-diffusion equations.  The final supporting talk of the mini-school was given by Fang Yu, who spoke about some Nash-Moser techniques for studying stability of shock-like solutions in 3d Euler flows.

Lastly,DSC_0165 the mini-school was nicely complemented by a related talk in the Applied Math Seminar by Tom Beale from Duke (which was kindly re-scheduled to a later time so as to allow everyone to attend both it and the mini-school).  In a fitting end to the week, Tom aptly spoke about efficient numerical methods for studying fluid flows in 2 and 3 dimension.

More details about the workshop in general and all the talk slides can be found at

Congratulations and thank yous to Profs. Hans Christianson and Jeremy Marzuola for the NSF grant (also with department alumna Prof. Anna Mazzucato [Ph.D., 2000] of Penn State) that supported this effort in addition to the summer 2012 conference in honor of Michael Taylor, and to the entire UNC PDE group for putting these mini-schools together.

The second PDE mini-school will occur November 20-21, featuring lectures from Chris Sogge on eigenfunctions of the Laplacian on manifolds.