Schedule-at-a-Glance

21 PRIMA 2013 AbstractsSpecial Session 10Kinetic EquationsOn some kinetic models for Bose Einstein condensationRadjesvarane AlexandreFrench Naval Academy, France and Shanghai Jiao TongUniversity, Chinaradjesvarane.alexandre@ecole-navale.frWe present some models based on kinetic equations fordescribing Bose Einstein condensation. We will discussabout possible blow up or not of some of the partial differentialequations. This is a joint work with Jie LIAOand Chunjin LIN.Angular averaging, propagation of exponentialtails and grazing collisions for solutions ofthe Boltzmann equationIrene GambaUniversity of Texas at Austin, USAgamba@math.utexas.eduWe will discuss the interplay on the collision kernels propertiesof the Boltzmann equation and the generation andpropagation of summability of moments of the solutionfor the homogeneous initial value problem. Such summabilityyields global bounds for the solution of the Boltzmannequation by exponentially weighted norms in L 1and pointwise, where the exponent depend on the initialstate norms, the rate of the intra-molecular potentials aswell as the integrability properties on the sphere (angularaveraging) for the scattering angle cross-section. We willalso discuss the impact of these estimates in open problems,such as the grazing collision limits to the Landauequation for Coulombic interactions. We will also shownumerical simulations of these limits for different crosssections.L p -scattering and uniform stability of kineticequationsSeung-Yeal HaSeoul National University, Koreasyha@snu.ac.krIn this talk, we will review recent progress on the L p -scattering and uniform stability of several kinetic equationswith self-consistent forces. For the Vlasov equationwith a self-consistent force, we will show that theCoulomb’s potential in three dimensions is critical in thesense that if spatial dimension is larger than three, thereexists a L 1 -scattering, whereas for low dimensions lessthan equal to three, there is no L 1 -scattering. We alsopresent a framework for the L p -stability of kinetic equations.This is a joint work with Sun-Ho Choi (NUS) andQinghua Xiao (SNU).On the spatially homogeneous Boltzmann andLandau equationsLingbing HeTsinghua University, Chinalbhe@math.tsinghua.edu.cnIn this talk, I will present some recent progress on thelower and upper bounds for the Boltzmann collision operator.As one application, we can build a unified frameworkto establish the well-posedness results for bothBoltzmann and Landau equations. As another application,we will revisit the so-called "grazing collisions limit".It is shown that when almost all collisions are grazing,that is, the deviation angle θ of the collision is limitednear zero (i.e., θ ≤ ɛ), the solution f ɛ of the Boltzmannequation with initial data f 0 can be globally or locallyexpanded asf ɛ = f + O(ɛ),for non Coulomb potential, orf ɛ = f + O(| log ɛ| −1 ),for Coulomb potential, where the function f is the solutionof Landau equation, which is associated to the grazingcollisions limit of Boltzmann equation, with the sameinitial data f 0 . These give the rigorous justification ofthe Landau approximation in the spatially homogeneouscase.New regularity estimates for transport equationsPierre-Emmanuel JabinUniversity of Maryland, USApjabin@umd.eduWe present new regularity estimates which are propagatedby transport equations with rough coefficients.Those estimates provides compactness on the density,which is a key ingredient to obtain existence of solutionsfor models from fluid mechanics (of the compressibleNavier-Stokes type) to chemotaxis. The correspondingspaces are defined at a logarithmic scale in terms of numberof derivatives and can thus be seen as intermediarybetween the usual L p and Sobolev spaces.Condensation and regularity for Boson BoltzmannequationXuguang LuTsinghua University, Chinaxglu@math.tsinghua.edu.cnWe study the spatially homogeneous Boltzmann equationfor Bose-Einstein particles for the hard sphere model. Weconsider the case where the initial datum of a solution ofthe equation is a function so that there is no condensation(Dirac mass) at the initial state. We show that if theinitial datum is not very singular near the origin (the zeropointof particle energy) and if the kinetic temperatureis sufficiently high, then the solution is also a functionfor all time, and it is a unique classical mild solutionof the equation; whereas if the initial datum is singularenough near the origin but still Lebesgue integrable, thenthe condensation of a corresponding solution continuouslystarts to occur from the initial time to every later time.Hamiltonian propagation of mono-kineticmeasures with rough initial profilesPeter MarkowichKing Abdullah University of Science and Technology,Kingdom of Saudi ArabiaP.A.Markowich@damtp.cam.ac.ukConsider in the phase space of classical mechanics aRadon measure which is a probability density carried bythe graph of a Lipschitz continuous (or even less regular)vector field. We study the structure of the push-forwardof such a measure by a Hamiltonian flow. In particular,we provide an estimate on the number of folds in thesupport of the transported measure that is the image ofthe initial graph by the flow. We also study in detail thetype of singularities in the projection of the transported