Schedule-at-a-Glance

22 PRIMA 2013 Abstractsmeasure in configuration space (averaging out the momentumvariable). We study the conditions under whichthis projected measure can have atoms, and give an examplein which the projected measure is singular withrespect to the Lebesgue measure and diffuse. Finally, wediscuss applications of our results to the classical limit ofthe SchrŽdinger equation.Scalar conservation laws and kinetic formulationBenoît PerthamePierre-and-Marie-Curie University, CNRS/INRIA andInstitut Universitaire de France, Franceperthame@ann.jussieu.frWe present a theory of Scalar Conservation Laws withrough fluxes. The difficulty is that, even with BV estimates,the equations does not make sense in disctributionalsense. Our approach is based on the kinetic formulationof conservations laws.This theory leads naturally to consider Kinetic AvergaingLemmas with random fluxes. On simple situations wewill show the differences int he gain of regularity.This talk is based on works with P.-L. Lions and P. E.Souganidis.Kinetic description of bacterial motion bychemotaxis and asymptoticsNicolas VaucheletPierre-and-Marie-Curie University, Francevauchelet@ann.jussieu.frChemotaxis is the phenomenon in which cells direct theirmotion according to a chemical present in their environment.Since experimental observations have shown thatthe motion of bacteria (e.g. Escherichia Coli) is due tothe alternation of ‘runs and tumbles’, mathematical modellingthanks to a kinetic description has been proposed.The starting point of the study is the so-called Othmer-Dunbar-Alt model governing the dynamics of the distributionfunction f of cells at time t, position x and velocityv, assumed to have a constant modulus c > 0, as wellas the concentration S(t, x) of the involved chemical. Ageneral formulation for this model can be written as⎧⎪⎨⎪⎩∂ tf + v · ∇ xf = Q(f),Q(f) = ∫ (|v ′ |=c T [S](v ′ , v)f(v ′ )−T [S](v, v ′ )f(v) ) ∫ dv ′ ,−∆S + S = ρ(t, x) := f(t, x, v) dv.|v|=cThe third equation describes the dynamics of the chemicalagent which diffuses in the domain. It is producedby the cells themselves with a rate proportional to thedensity of cells ρ and disappears with a rate proportionalto S. The transport operator on the left-hand side of thefirst equation stands for the unbiased movement of cells(‘runs’), while the right-hand side governs ‘tumbles’, thatis chemotactic orientation, or taxis, through the turningkernel T [S](v ′ , v), which is the rate of cells changing theirvelocity from v ′ to v.In this work, we focus on positive chemotaxis whichmeans that the involved chemical S is attracting cells,called chemoattractant. Existence results for this systemhas been obtained for various assumptions on the turningkernel. Macroscopic model can be derived from thissystem of equations after a rescaling. When the taxis issmall compared to the unbiased movement of cells, thescaling must be of diffusive type, so that the limit equationsare of diffusion or drift-diffusion type. When taxisdominates the unbiased movements, a hyperbolic limitcan be derived, leading to aggregation equations. Moreover,experimental observations have shown the formationof traveling pulse of bacteria in micro-channel. The abovemathematical modeling allows to recover this behaviour,which motivates the interest on such system.Viscous wave propagation at interfaceShih-Hsien YuNational University of Singapore, Singaporematysh@nus.edu.sgIn this talk we implement the LY algorithm to derive atwo-sided master relationship. This gives rise the operatorto construct the surface wave at the interface and theconstruction of the Green’s function of a simple variablecoefficient problem for viscous conservation laws.Special Session 11Mathematical Fluid Dynamics and RelatedTopicsGlobal solutions of the equivariant heat-flowStephen GustafsonUniversity of British Columbia, Canadagustaf@math.ubc.caThe harmonic map heat-flow is a canonical, parabolic,geometric evolution equation, which is (energy) criticalin two dimensions. I will present a global regularity resultfor the equivariant, higher-degree, above-thresholdheat-flow into the sphere, which extends to higher energiesearlier work with Nakanishi and Tsai on the heatand Schroedinger flows. This is joint work with DimitriosRoxanas.Well-posedness of coupled kinetic-fluid modelsfor flockingSeung-Yeal HaSeoul National University, Koreasyha@snu.ac.krIn this talk, I will present several kinetic-fluid models forflocking. More precisely, I will discuss the dynamics ofCucker-Smale flocking particles interacting with neighboringcompressible and incompressible fluid. For suchcoupled kinetic-fluid models, we will show global existenceof weak and strong solutions and asymptotic flockingestimates for small initial data and large viscosities.This is a joint work with H.O.Bae (Ajou Univ), Young-PilChoi (Imperial College of London) and Moon-Jin Kang(SNU).Global classical and weak solutions to the 3Dfully compressible Navier-Stokes-Fourier systemXiangdi HuangAcademy of Mathematics and System Sciences, CAS,Chinaxdhuang@amss.ac.cnWe establish the global existence and uniqueness of classicalsolutions to the three-dimensional fully compressibleNavier-Stokes-Fourier system with smooth initial datawhich are of small energy but possibly large oscillationswhere the initial density is allowed to vanish. Moreover,for the initial data, which may be discontinuous and containvacuum states, we also obtain the global existenceof weak solutions. These results generalize previous oneson classical and weak solutions for initial density being