PostDoctoral Research Fellowship
grant DMS-1103765 was very successful in both funding successful research in the original proposed topic area and in giving the PI access to the perfect environment at NYU to continue his training and make successful advances in several other fields. Due to these successes, the PI now holds a tenure-track Assistant Professorship at the University of Maryland, College Park, a top-10 US research department in the PI's field of expertise. The proposed topic of the grant was the study of a certain class of mathematical models which posses "critical mass", that is, where the behavior predicted by the model changes dramatically from one type to another if a certain quantity is above a special threshold. The class of models studied by the PI over the course of the grant mostly arise in mathematical biology for the collective motion of micro-organisms. The existing mathematical theory focused mainly on a few important representative models, however, these special models do not share the same behavior as some more accurate physical models. The PI worked to expand our mathematical understanding of the wider class of models, often by rigorously proving certain limit behaviors of the more complicated models reduced back to simpler models. This required the development and refinement of numerous mathematical tools. The research papers published or in review which are related to this topic and partially supported by this grant are J. Bedrossian, Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models. Under peer review. Preprint, arXiv:1403.4124, 2014 J. Bedrossian, N. Masmoudi, Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued initial data. To appear in Arch. Rat. Mech. Anal.. Preprint, arXiv:1205.1551, 2012. J. Bedrossian, and I. Kim. "Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion." SIAM Journal on Mathematical Analysis 45.3 (2013): 934-964. J. Azzam, J. Bedrossian, Bounded mean oscillation and the uniqueness of active scalar equations. To appear in Trans. Amer. Math. Soc.. Preprint, arXiv:1108.2735, 2012. J. Bedrossian, N. Rodr?guez, Inhomogeneous Patlak-Keller-Segel Models and aggregation equations with nonlinear diffusion in Rd., Disc. Cont. Dyn. Sys. B, 19.5 (2014): 1279 - 1309 In addition to this work, the PI collaborated with other researchers at NYU and made significant advances on pattern formation in thin films and especially, in the mixing and stability of fluids and plasmas. The latter field is now the PI's current research focus (and has secured the NSF Applied Math grant DMS-1413177 for further funding on that topic). Papers published or under peer review in these fields partially supported by the PostDoctoral Fellowship are: J. Bedrossian, N. Masmoudi, V. Vicol, Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow . Under peer review, Preprint arXiv:1408.4754, 2014. J. Bedrossian, N. Masmoudi, C. Mouhot, Landau damping: paraproducts and Gevrey regularity. Under peer review. Preprint, arXiv:1311.2870, 2013. J. Bedrossian, N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Under peer review. Preprint, arXiv:1306.5028, 2013. J. Bedrossian, R.V. Kohn Blister patterns and energy minimization in compressed thin films on compliant substrates. To appear in Comm. Pure Appl. Math.. Preprint, arXiv:1304.0284, 2013 Intellectual Merit The mathematics developed under the support of the grant has broadened our understanding of how critical mass behavior changes in non-ideal, more realistic settings and also the relevance of critical mass behavior in unusual settings. The research on mixing and stability supported by this grant has proved the relevance of certain theoretical physical mechanisms and greatly enhanced our understanding of them. The research performed by N. Masmoudi, C. Mouhot and the PI in 2013 on Landau damping has already been recently applied to relativistic plasma physics (B. Young, arXiv:1408.2666 2014). The PI is now moving forward under the support of DMS-1413177 to continue his advances in the field of mixing and stability in fluids and plasmas and now has several active research projects. Broader Impacts The mathematics developed by the support of the grant deepens our understanding of certain mathematical processes and physical phenomena. For example, by having a better theoretical understanding of the behavior of critical mass collective motion, applied researchers will have a better idea of how to design and implement experiments and what to look for in physical systems. The same goes for mixing and stability in fluids, which may have eventual influence in the design of cyclotrons for medical use and confined plasma devices for experimental physics or power generation. Another obvious, but important, broader impact is the training of the PI as an independent researcher, STEM educator and future mentor for the next generation of STEM students in the US.