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PD09 Abstracts 45 PD09 Abstracts 45 IP1 functions. Free Boundary Regularity Yann Brenier We discuss level surfaces of solutions u to singular semilin- CNRS ear elliptic equations such as Δu = δ(u). The level surface Universit´ede Nice Sophia-Antipolis u = 0 can be interpreted as a solution to a free boundary [email protected] problem, the problem of finding the optimal shape of in- sulating material. A variant of this equation, the Prandtl- Batchelor equation, describes the wake created by a boat. IP4 We will prove that stable free boundaries are smooth in The Importance of PDEs for Modeling and Simu- three-dimensional space. Numerical methods are needed lation on Peta and Exascale Systems to explore counterexamples in higher dimensions and to establish a more robust regularity theory with numerically As scientists and engineers tackle more complex problems effective bounds. involving multiphysics and multiscales and then seek to solve these problems through computation, the partial dif- David Jerison ferential equation formulations which incorporate and more Massachusetts Institute of Technology closely approximate the complex physical or biological phe- [email protected] nomena being modeled will also be important to unlocking the approach taken on Peta and Exascale systems. Scien- tists and engineers working on problems in such areas as IP2 aerospace, biology, climate modeling, energy and other ar- Fluid Biomembranes: Modeling and Computation eas demand ever increasing compute power for their prob- lems. For Petascale and Exascale systems to be useful mas- We study two models for biomembranes. The first one is sive parallelism at the chip level is not sufficient. I will de- purely geometric since the equilibrium shapes are the min- scribe some of the challenges that will need to be considered imizers of the Willmore energy under area and volume con- in designing Petascale and eventually Exascale systems. straints. We present a novel method based on ideas from Through the combination of High Performance Comput- shape differential calculus. The second model incorporates ing (HPC) hardware coupled with novel mathematical and the effect of the inside (bulk) viscous incompressible fluid algorithmic approaches emerging from the original PDE and leads to more physical dynamics. We use a parametric formulations some efforts toward breakthroughs in science approach, which gives rise to fourth order highly nonlinear and engineering are described. While progress is being PDEs on surfaces and involves large domain deformations. made, there remain many challenges for the mathematical We discretize these PDEs in space with an adaptive fi- and computational science community to apply ultra-scale, nite element method (AFEM), with either piecewise linear multi-core systems to Big science problems with impact on or quadratic polynomials, and a semi-implicit time step- society. In conclusion, some discussion not only on the ping scheme. We employ the Taylor-Hood element for the most obvious way to use ultra-scale, multi-core HPC sys- Navier-Stokes equations together with iso-parametric ele- tems will be given but also some thoughts on incorporat- ments, the latter being crucial for the correct approxima- ing more physics in the algorithms derived from the PDE tion of curvature. We discuss several computational tools models which might allow us to better use such systems to such as space-time adaptivity and mesh smoothing. We tackle previously intractable problems. also discuss a method to execute refinement, coarsening, and smoothing of meshes on manifolds with incomplete in- Kirk E. Jordan formation about their geometry and yet preserve position IBM T.J. Watson Research and curvature accuracy. This is a new paradigm in adap- [email protected] tivity. Ricardo Nochetto IP5 Department of Mathematics Singular Limits in Thermodynamics of Viscous Flu- University of Maryland, College Park ids [email protected] We discuss some recent results concerning singular limits of the full Navier-Stokes-Fourier system, where one or several IP3 characteristic numbers become small or tend to infinity. Rearrangement, Convection and Competition The main ingredients of our approach read as follows: Rearrangement theory is about reorganizing a given func- • a general existence theory of global-in-time weak so- tion (or map) in some specific order (monotonicity, cycle lutions for any finite energy data; monotonicity etc...). This is somewhat similar to the con- vection phenomenon in fluid mechanics, where fluid parcels are continuously reorganized in a stabler way (heavy fluid • uniform bounds independent of the value of the sin- at bottom and light fluid at top). This can also be related gular parameters based on total dissipation balance to some competition models in economy, where agents act (Second law of thermodynamics); according to their rank. In our talk, we make these analo- gies more precise by analysing the Navier-Stokes equa- • analysis of propagation of acoustic waves, in particu- tions with Boussinesq approximation of the buoyancy force lar on unbounded or “large’ spatial domains. and two distinct approximations of this model (Darcy and Boussinesq). We will see how these approximations are related to the concept, well known in optimal transport theory, of rearrangement of maps as gradient of convex Eduard Feireisl Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1 Czech Republic [email protected] 46 PD09 Abstracts IP6 berger and Rosinger but it is actually based on topological Q-curvature in Conformal Geometry processes known as convergence structures. In this talk , I will survey some analytic results concerned Roumen Anguelov, Jan Harm Van Der Walt with the top order Q-curvature equation in conformal ge- University of Pretoria ometry. Q-curvature is the natural generalization of the [email protected], [email protected] Gauss curvature to even dimensional manifolds. Its close relation to the Pfaffian, the integrand in the Gauss-Bonnet formula, provides a direct relation between curvature and topology. The notion of Q-curvature arises naturally in CP1 conformal geometry in the context of conformally covari- Existence of a Unique Solution to Equations Mod- ant operators. In 1983, Paneitz gave the first construction eling Fluid Flow with Time-Dependent Density of the fourth order conformally covariant Paneitz opera- Data tor in the context of Lorentzian geometry in dimension four. The ambient metric construction, introduced by Fef- We consider a system of nonlinear equations modeling the ferman and Graham , provides a systematic construction flow of a barotropic, compressible fluid. The equations in general of conformally covariant operators. Each such consist of a hyperbolic equation for the velocity, an alge- operator gives rise to a semi-linear elliptic equation anal- braic equation (the equation of state) for the pressure, and ogous to the Yamabe equations which we shall call the a modified version of the conservation of mass equation. Q-curvature equation. These equations share a number of We prove the existence of a unique solution to the model’s common features. Among these we mention the following: equations with time-dependent density data and spatially- (i) the lack of compactness: the nonlinearity always oc- dependent velocity data, under periodic boundary condi- cur at the critical exponent, for which the Sobolev imbed- tions. ding is not compact; (ii) the lack of maximum principle: Diane Denny for example, it is not known whether the solution of the Texas A&M University - Corpus Christi fourth order Q-curvature equation on manifolds of dimen- [email protected] sions greater than four may touch zero. In spite of these difficulty, there has been significant progress on questions of existence, regularity and classification of entire solutions CP1 for these equations in the recent literature. In the talk, I Localization of Analytic Regularity Criteria on the will give a brief survey of the subject with emphasize on Vorticity and Balance Between the Vorticity Mag- applications to problems in conformal geometry and the nitude and Coherence of the Vorticity Direction in connection between the a class of conformal covariant op- the 3D NSE erators to that of the fractional Laplacian opeators. 2q 2q−3 1 Sun-Yung Alice Chang DaVaiga has shown that Duq ∈ L (0,T) is a regu- Princeton University larity class for the Navier-Stokes Equations for any 3 ≤ [email protected] q<∞. Beale-Kato-Majda proved the regularity for vor- ticity in time-space L1L∞-norm. Geometric conditions on the vorticity direction improve the regularity and the local- IP8 ized versions have been recently obtained by Grujic. The p q Title Not Available at Time of Publication scaling-invariant regularity class of weighted L L -type for vorticity magnitude with coherence factor as a weight will Abstract not available at time of publication. be demonstrated in a localized setting. Felix Otto Rafaela Guberovic University of Bonn Seminar for Applied Mathematics, ETH, Zurich, Germany Switzerland [email protected] University of Virginia, Charlottesville, USA [email protected] IP9 Zoran Grujic Title Not Available at Time of Publication Department of Mathematics University of Virginia Abstract not available at time of publication. [email protected] George C. Papanicolaou Stanford University CP1 Department of Mathematics [email protected] Fully-Coupled Aeroelastic Computations of High Reynolds Number Flows CP1 Strong (two-way) coupling of fluid and structure presents interest to vary engineering applications, particularly when Solving Nonlinear Systems of Pdes Via Real Set- the flow is turbulent and sensitive to structural motions. A Valued Maps CFD based algorithm, using large-eddy simulation (LES), The existence and regularity of solutions of nonlinear PDEs is proposed for the numerical investigation of strong aeroe- resulting from constitutive relations of fluids are problems lastic coupling. The aeroelastic response of a cranked dou- with acknowledged difficulty. We prove that the solutions ble delta wing to forced flap motion is subject of investi- of very large classes of systems of nonlinear PDE can be gation.
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