Georgia Institute of Technology College of Sciences

The phenomenon of wave run-up has the capital importance for the beach erosion, coastal protection and flood hazard estimation. In the present talk we will discuss two particular aspects of the wave run-up problem. In this talk we focus on the wave run-up phenomena on a sloping beach. In the first part of the talk we present a simple stochastic model of the bottom roughness. Then, we quantify the roughness effect onto the maximal run-up height using Monte-Carlo simulations. A critical comparison with more conventional approaches is also performed.In the second part of the talk we study the run-up of simple wave groups on beaches of various geometries. Some resonant amplification phenomena are unveiled. The maximal run-up height in resonant cases can be 20 times higher than in regular situations. Thus, this work can provide a possible mechanism of extreme tsunami run-up conventionally ascribed to "local site effects".References:Dutykh, D., Labart, C., & Mitsotakis, D. (2011). Long wave run-up on random beaches. Phys. Rev. Lett, 107, 184504.Stefanakis, T., Dias, F., & Dutykh, D. (2011). Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett., 107, 124502.

An implicit method [1, 2], TARDIS (Transient Advection Reaction Diffusion Implicit Simulations), has been developed that successfully couples the compressible flow to the comprehensive chemistry and multi-component transport properties. TARDIS has been demonstrated in application to two fundamental combustion problems of great interest. First, TARDIS was used to investigate stretched laminar flame velocities in eight flame configurations: outwardly and inwardly propagating H2/air and CH4/air in cylindrical and spherical geometries. Fractional power laws are observed between the velocity deficit and the flame curvature Second, the response of transient outwardly propagating premixed H2/air and CH4/air flames subjected to joint pressure and equivalence ratio oscillations were investigated. A fuller version of the abstract can be obtained from http://www.math.gatech.edu/~rll6/malik_abstract-Apr-2012.docx [1] Malik, N.A. and Lindstedt, R.P. The response of transient inhomogeneous flames to pressure fluctuations and stretch: planar and outwardly propagating hydrogen/air flames. Combust. Sci. Tech. 82(9), 2010. [2] Malik, N. A. “Fractional powers laws in stretched flame velocities in finite thickness flames: a numerical study using realistic chemistry”. Under review, (2012). [3] Markstein, G.H. Non-steady Flame Propagation. Pergamon Press, 1964. [4] Weis,M., Zarzalis, N., and Suntz, R. Experimental study of markstein number effects on laminar flamelet velocity in turbulent premixed flames. Combust. Flame, 154:671--691, 2008.

Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains"). It recently transpired that the difference between this "natural" number n of nodal domains and the actual values can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: (1) stability of the nodal partitions of a domain in R^d with respect to a perturbation of the partition boundaries and (2) stability of a graph eigenvalue with respect to a perturbation by magnetic field. In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. Based on preprints arXiv:1107.3489 (joint with P.Kuchment and U.Smilansky) and arXiv:1110.5373

This talk is concerned with new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in a range of parameters for which no explicit results of symmetry have previously been known. The method proceeds via spectral estimates. This is joint work with Jean Dolbeault and Maria Esteban.