Schedule-at-a-Glance

19 PRIMA 2013 Abstractscritical slope u x = 1. We show that, for a fixed relationof the asymptote at x = ±∞, there exists a unique travelingwave solution. Furthermore, such traveling wavesolutions are local attractors. Nearby solutions approachthe traveling wave asymptotically as t → ∞. This is ajoint work with Graziano Guerra, see [3].[1] D. Amadori and W. Shen, Front tracking approximationsfor slow erosion, Discrete Contin. Dyn. Syst., 32(2012), no. 5, 1481–1502.[2] R. M. Colombo, G. Guerra, and W. Shen, Lipschitzsemigroup for an integro–differential equation for slowerosion, Quart. Appl. Math., 70 (2012), 539–578.[3] G. Guerra, and W. Shen, Existence and Stability ofTraveling Waves for an Integro-differential Equation forSlow Erosion Preprint, 2013.[4] W. Shen and T.Y. Zhang, Erosion profile by a globalmodel for granular flow, Arch. Ration. Mech. Anal., 204(2012), no. 3, 837–879.On the existence of Meyer type transonicflows and a degenerate change type equationZhouping XinThe Chinese University of Hong Kong, Hong Kong, Chinazpxin@ims.cuhk.edu.hkHistorically, two types of smooth steady compressibleirrotational transonic flows were conjectured for 2-dimensional de Laval nozzles: Taylor type and Meyertype. However, it has known since Lipman Bers that Taylortype transonic flows may not exist in general, whilethe existence of Meyer type transonic flows with suitablephysical boundary condition has been a long standingopen problem. Such a flow is governed by a quasilinearequation which changes type and becomes degenerateupon crossing sonic state where the flow may havesingularities. In this talk, I will report some progress onstudies of such problems. First, by investigating the propertiesstructure of the sonic curve, in particular its exceptionalpoints, for C 2 -smooth transonic flows in a generalnozzle, we show the instability of Taylor type flowsand Meyer type flows with non-empty exceptional points.Then we identify a class of de Laval nozzles and suitablephysical boundary conditions such that Meyer type transonicflow exists with sonic curve everywhere exceptional.Some global properties of such a solution and related openproblems will be discussed.Weakly nonlinear geometric optics for hyperbolicsystems of conservation lawsYongqian ZhangFudan University, Chinayongqianz@fudan.edu.cnIn this talk I will present a new approach to analyze thevalidation of weakly nonlinear geometric optics for entropysolutions of nonlinear hyperbolic systems of conservationlaws whose eigenvalues are allowed to have constantmultiplicity and corresponding characteristic fieldsto be linearly degenerate. The approach is based on ourcareful construction of more accurate auxiliary approximationto weakly nonlinear geometric optics, the propertiesof wave front-tracking approximate solutions, the behaviorof solutions to the approximate asymptotic equations,and the standard semigroup estimates. This is ajoint work with Guiqiang Chen and Wei Xiang of Universityof Oxford.Special Session 9Inverse ProblemsInverse scattering problems in wave propagationGang BaoZhejiang University, China and Michigan State University,USAdrbaogang@gmail.comRecent progress of our research group on inverse scatteringproblems in wave propagation will be reported. Issueson uniqueness/stability and numerical solution for the inverseproblems will be discussed.Multi-scale full waveform inversion for seismicimagingMichael P. LamoureuxUniversity of Calgary, Canadamikel@ucalgary.caThe seismic imaging problem entails recovering an imageof the earth’s subsurface from data that is recorded onthe surface of the earth, determined by the propagation ofseismic (vibrational) waves through the body of the earth.The 2D or 3D acoustic wave equation is commonly usedas a simplified mathematical model for this seismic wavepropagation. The full waveform inverse problem aims todeduce the physical parameters of the (acoustic) mediumof propagation from recorded data of impulsive waves thatare transmitted or reflected through the medium, andthus form an image of the subsurface.In this study, we demonstrate a numerical algorithmthat uses factorization in the PDE solver of the 2D acousticwave model, a multi-scale approach to the inverse solution,and a projection-based linearization search for thesolution to the inverse problem. The multi-scale approachis used to decrease the rank of the inverse problem, thusdecreasing the ill-posedness and under-determinedness ofthe solution. With a few examples, we show the robustproperties of the inversion algorithm, a fast numericalconvergence rate, and the advantages of multi-scaling.Joint with Gary F. Margrave, Vladimir ZubovOn the inverse problems for the coupled continuumpipe flow model for flows in karstaquifersShuai LuFudan University, Chinaslu@fudan.edu.cnIn this talk, we investigate a coupled continuum pipe flow(CCPF) model which describes the fluid flows in karstaquifers. After generalizing the well-posedness of the forwardproblem to the anisotropic exchange rate case whichis a space-dependent variable, we present the uniquenessof this parameter by measuring the Cauchy data. Finally,some regularization schemes are provided to solveone proposed inverse problem.A heat source reconstruction formula fromsingle internal measurements using a familyof null controlsAxel OssesUniversidad de Chile, Chileaxosses@dim.uchile.clWe consider the inverse problem of determining the spatialdependence f(x) of the source term in a heat equationu t − γ∆u = f(x)σ(t) in Ω × (0, T ) assuming σ(t) known,