Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data

The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the B\'enard system are also discussed.Comment: 12 page

Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.Comment: 39 pages, 1 figur

Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion

Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions $d\geq 2$ and in all of space for $d\geq 3$, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page

Mercury release and speciation in chemical looping combustion of coal

In the in situ Gasification Chemical Looping Combustion of coal (iG-CLC), the fuel is gasified in situ in the fuel reactor and gasification products are converted to CO2 and H2O by reaction with the oxygen carrier. This work is the first study on mercury release in Chemical Looping Combustion of coal. The fraction of the mercury in coal vaporized in the fuel reactor depended mainly on the fuel reactor temperature and the coal type. In the fuel reactor, mercury was mainly emitted as Hg0 in the gas phase and the amount increased with the temperature. In the air reactor, mercury was mostly emitted as Hg2+. In a real CLC system, mercury emissions to the atmosphere will decrease compared to conventional combustion as only mercury released in the air reactor will reach the atmosphere. However, measures should be taken to reduce Hg0 in the CO2 stream before the purification and compression steps in order to avoid operational problems.The authors thank the Government of Aragón and La Caixa (2012-GA-LC-076 project) and the Spanish Ministry for Science and Innovation (ENE2010-19550 project) for the financial support. P. Gayán thanks CSIC for the financial support of the project 201180E102. The authors also thank to Alcoa Europe-Alúmina Española S.A. for providing the Fe-enriched sand fraction used in this work. G. Galo is acknowledged for his contribution to the experimental results.Peer reviewe

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.Comment: 41 pages. To appear in Probab. Theory Related Field

X-Ray Second Harmonic Generation

We report clear experimental evidence for second harmonic generation at hard x-ray wavelengths. Using a 1.7 Å pumping beam generated by a free electron laser, we observe second harmonic generation in diamond. The generated second harmonic is of order 10 times the background radiation, scales quadratically with pump pulse energy, and is generated over a narrow phase-matching condition. Of importance for future experiments, our results indicate that it is possible to observe nonlinear x-ray processes in crystals at pump intensities exceeding 1016 W/cm2