On the Stability of Roll Waves
DateTuesday, April 3, 2018 - 12:30pm
AbstractRoll-waves are a well observed hydrodynamic instability occurring in inclined thin film flow, often mathematically described as periodic traveling wave solutions of either the viscous or inviscid St. Venant system. In this talk, I will discuss recent progress concerning the stability of both viscous and, if time allows, inviscid roll-waves in a variety of asymptotic regimes, including near the onset of hydrodynamic instability and large-Froude number analysis. This is joint work with Blake Barker (BYU), Pascal Noble (University of Toulouse), L. Miguel Rodrigues (University of Rennes), Zhao Yang (IU) and Kevin Zumbrun (IU).
Conformal Mapping in 2D Coulombic Potential Problems
DateTuesday, April 10, 2018 - 12:30pm
AbstractFor physical systems governed by Coulombic potential in 2D, one can set up the problem in complex plane and use the freedom of conformal mappings to help solve the problem. In this talk I will mention some examples following this idea and focus on the example of a uniform flow past two close-to-touching discs. By using the method of image charges combined with conformal transformations in the numerical calculation of the potential, we can reach a solution with high accuracy and stability more efficiently compared to existing methods.
Nonlinear waves and singularities in nonlinear optics, plasmas, hydrodynamics and biology
DateTuesday, April 17, 2018 - 12:30pm
AbstractMany nonlinear systems of partial differential equations have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which is called by collapse. Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability with diverse singularity regularization mechanisms become important such as optical breakdown and formation of plasma in nonlinear optical media, excluded volume constraints in bacterial aggregation or dissipation of breaking water waves. Collapses occur in numerous physical and biological systems including a nonlinear Schrodinger equation, Keller-Segel equation and many others. Wavebreaking is another example of spontaneous formation of singularities corresponding to the breaking of initially smooth smooth fluid's free surface. The recent progress in collapse theory will be reviewed with multiple applications discussed ranging from laser fusion to optical beam combining. The 2D dynamics of fluid's free surface will be also addressed through the motion of singularities outside of fluid with wavebreaking resulting from the approach of these singularities to the fluid's surface. New exact solutions are found which have arbitrary number of poles coupled with branch cuts. The residues of these poles are integrals of motion. They commute with respect of the non-canonical Poisson bracket which provides strong arguments towards complete integrability of free surface hydrodynamics
Limit shapes for Gibbs ensembles of partitions
DateTuesday, April 24, 2018 - 12:30pm
AbstractI will discuss the problem of limit shapes for Gibbs ensembles of partitions of integers and sets. This problem is related to the aggregate size distribution in various models of aggregation and polymerization and to invariant measures of zero range and coagulation-fragmentation processes. I will show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.