Schedule-at-a-Glance

57 PRIMA 2013 Abstractsand fountain theorem, we establish the existence of nontrivalsolutions for a class of quasilinear elliptic problemswith variable exponents. Also we show the existence ofpositivity of the infimum of all eigenvalues for the aboveproblem.Overdetermined boundary value problemswith strongly nonlinear elliptic PDEFengquan Li (with Boqiang Lv, Weilin Zou)Dalian University of Techology, Chinafqli@dlut.edu.cnWe consider the strongly nonlinear elliptic Dirichlet problemin a connected bounded domain, overdetermined withthe constant Neumann condition F (∇u) = c on theboundary. Here F is convex and positively homogeneousof degree 1, and its polar F ∗ represents the anisotropicnorm on R n . We prove that, if this overdeterminedboundary value problem admits a solution in a suitableweak sense, then Ω must be of Wulff shape.Stability of boundary layers for the nonisentropiccompressible circularly symmetric2D flowCheng-Jie LiuShanghai Jiao Tong University, Chinacjliusjtu@gmail.comIn this paper, we study the asymptotic behavior of the circularlysymmetric solution to the initial boundary valueproblem of the compressible non-isentropic Navier-Stokesequations in a two-dimensional exterior domain with impermeableboundary condition when the viscosities andthe heat conduction coefficient tend to zero. By multiscaleanalysis, we obtain that away from the boundarythe compressible non-isentropic viscous flow can be approximatedby the corresponding inviscid flow, and nearthe boundary there are boundary layers for the angularvelocity, density and temperature in the leading orderexpansions of solutions, while the radial velocity andpressure do not have boundary layers in the leading order.The boundary layers of velocity and temperature aredescribed by a nonlinear parabolic coupled system. Weprove the stability of boundary layers and rigorously justifythe asymptotic behavior of solutions in the L ∞ −normfor the small viscosities and heat-conduction limit in theLagrangian coordinates, as long as the strength of theboundary layers is suitably small. Finally, we show thatthe similar asymptotic behavior of the small viscositiesand heat conduction limit holds in the Eulerian coordinatesfor the compressible non-isentropic viscous flow.Dynamics in immune reactions during woundhealing processesJianzhong Su a) , Larrissa Owens a) and Akif Ibraguimov b)a) University of Texas at Arlington, USAb) Texas Tech Universitysu@uta.eduWe propose a partial differential equation model adaptedfrom the principles of wound healing studies and analyzeit to gain insights regarding the dynamics of immunecells/proteins following the insertion of a foreign body.Specifically we look at the multiple roles of macrophagesand the conditions for stabilizing/destabilizing the equilibriumstate. Furthermore, we investigate the impact ofdiffusion and chemotaxis on the stability and the transientbehavior of the system.On the radiation field for wave equationsFang WangPrinceton University, USAfangwang@math.princeton.eduDefinitions for the radiation field by Lax-Phillips andFriedlander are introduced for standard wave equationon Minkowski space-time, which can be generalized tocurved space-time (such as Schiwarzschild space-time)and nonlinear wave equations (such as Einstein vacuumequations). Regularities of the radiation field are studied.Mapping property for Moller wave operator, whichmaps initial data to the radiation field, are investigated.In the case that the Molloer wave operator is an (or alocal) isomorphism, the characteristic initial problem isconsidered.Relationship between ω-limit Sets and MinimalSetsLarry WangSouthern Polytechnic State University, USAlwang@spsu.eduIn this paper, we study the relation between ω-limit setsand minimal sets. It is known that every minimal set is aω-limit set. However, not every ω-limit set is a minimalset, although it contains a minimal set. We would like toknow under what condition an ω-limit set turns out tobe a minimal set. We establish a necessary and sufficientcondition under which every ω-limit set is minimal.The regularity of semi-hyperbolic patches atsonic lines for the pressure gradient equationin gas dynamicsQin WangShanghai Jiao Tong University, Chinamathwq@sjtu.edu.cnWe study the uniform regularity of semi-hyperbolicpatches of self-similar solutions near sonic lines to a generalRiemann problem for the pressure gradient equation.This type of solutions, in which one family of characteristicsstarts on a sonic line and ends on a transonicshock wave, is common for the Riemann problems for theEuler system in two space dimensions. The global existenceof smooth solutions was established in Song andZheng(2009), but the smoothness near the sonic lines isnot clear. We establish that the smooth solutions are uniformlysmooth up to their sonic boundaries and the soniclines are C 1 continuous.Constructing global Lyapunov function forcomplex systemsXinAn WangShanghai Jiao Tong University, Chinawangxinan@sjtu.edu.cnLyapunov function, as one of the most significant conceptsin dynamical systems, has been widely applied inmany disciplines. However, the classical Lyapunov functionin the theory of ordinary differential equations areusually restricted to the analysis of the stability near theequilibriums of the system, and no general constructivemethod has been found. Thus, it cannot be applied tomore complex dynamical behaviors, such as multi-stablestates, periodic attractor like limit cycle, and chaos. Recently,a series of works based on the construction of potentialfunction in physics for stochastic dynamical processdemonstrate potential function plays a critical rolein the quantitative analysis of dynamical systems. In deterministiccases, the potential function is the Lyapunov