Center for Applied Mathematical Sciences
 Colloquia for the Fall 2013 Semester Anthony Suen USC Monday, September 9 KAP 414 3:30 PM - 4:30 PM Global weak solutions of the equations of compressible We prove the global-in-time existence of weak solutions of the three-dimensional isothermal, compressible magnetohydrodynamic (MHD) equations in the whole R^3 space. The initial density is strictly positive, essentially bounded and is close to a constant in L^2, and the initial velocity and magnetic field are both small in L^2 and bounded in L^n for some n>6. Here the initial data may be discontinuous across a hypersurface of R^3. Monday, September 30 KAP 414 3:30 PM - 4:30 PM Panel: Applying for jobs and grants Jacob Bedrossian Courant Institute Monday, October 7 KAP 414 3:30 PM - 4:30 PM Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Joint work with Nader Masmoudi. Alan Schumitzky USC Monday, October 14 KAP 414 3:30 PM - 4:30 PM Estimating an unknown probability distribution: a convexity approach We consider the following estimation problem. Let X1,..., XN be a sequence of N independent random vectors with common but unknown probability distribution F. The { Xi } are not observed. Let Y1,..., YN be another sequence of N independent random vectors which are observed. Assume the conditional probability P(Yi | Xi) is known for i = 1,..., N. The problem is to estimate the probability distribution F given the { Yi }. The main theoretical result is that the maximum likelihood estimate of F can be found in the space of discrete distributions with no more that N support points. An elegant proof of this will be given using results of convexity theory in a topological vector space. The practical numerical problem of actually determining the discrete maximum likelihood estimator of F is still not solved in a satisfactory manner. Examples from the area of applied pharmacokinetics will be given. Ruth Williams UCSD Monday, October 21 KAP 414 3:30 PM - 4:30 PM Resource Sharing in Stochastic Networks Stochastic models of processing networks arise in a wide variety of applications in science and engineering, e.g., in high-tech manufacturing, transportation, telecommunications, computer systems, customer service systems, and biochemical reaction networks. These "stochastic processing networks" typically have entities, such as jobs, vehicles, packets, customers or molecules, that move along paths or routes, receive processing from various resources, and that are subject to the effects of stochastic variability through such variables as arrival times, processing times and routing protocols.Networks arising in modern applications are often heterogeneous in that different entities share (i.e., compete for) common network resources. Frequently the processing capacity of resources is limited and there are bottlenecks, resulting in congestion and delay due to entities waiting for processing.The control and analysis of such networks present challenging mathematical problems.This talk will explore the effects of resource sharing in stochastic networks and describe associated mathematical analysis based on elegant fluid and diffusion approximations. Illustrative examples will be drawn from biology and telecommunications. Irena Lasiecka U of Memphis Monday, October 28 KAP 414 3:30 PM - 4:30 PM Long time behavior of solutions to flow-structure interactions arising in modeling of subsonic and supersonic flows of gas. We consider flow - structure interaction comprising of a modified wave equation coupled to a nonlinear plate equation. The model has no damping imposed neither on the structure nor the flow. The regime of the parameters considered includes supersonic flows - the latter known for depleting ellipticity from the corresponding static model. Thus, both well-posedness of finite energy solutions and long time behavior of the modelhave been open questions in the literature. The results presented include:• Existence, uniqueness and Hadamard wellposedness of finite energy solutions.• Existence of global and finite dimensional attracting sets for the structure in the absence of any mechanical dissipation. The key "hidden regularity type" inequalities, responsible for proving existence of nonlinear semigroup, are derived by using microlocal analysis.The existence of an attracting set is proved without imposing any form of dissipation on the model. This is achieved by exploiting "compensated compactness" related to the dispersive character of the flow equation. To our knowledge, this is the first complete exhibition and rigorous justification of this fact - previously known experimentally only. In order to resolve the difficulty, we follow the decoupling method of [1] which reduces the problem to a study of nonlinear plates with the delay terms.This presentation is based on a joint work with Igor Chueshov, Kharkov University and Justin Webster, Oregon State University. Michael Waterman USC Monday, November 11 KAP 414 3:30 PM - 4:30 PM Sequence Comparison Using Word Counts Recently word count statistics have received attention due to their computational efficiency. Those statistics are for entire sequences. Local alignment-free sequence comparison arises in the context of identifying similar segments of sequences that may not be alignable in the traditional sense. We propose a randomized approximation algorithm that is both accurate and efficient. Fadil Santosa U of Minnesota Monday, November 18 KAP 414 3:30 PM - 4:30 PM Resonances -- Analysis and Optimization We consider the problem of resonances for Schroedinger's equation and Helmholtz equation. These equation provide a model for an optimal design problem in which the goal is to create an optical structure that has a resonance that has low loss. In addition we will also discuss an asymptotic method for calculating low-loss resonances which is the forward problem in this inverse design problem. Zahar Hani Courant Institute, NYU Monday, November 18 KAP 414 2:00 PM - 3:00 PM Special Time Out-of-equilibrium dynamics for the nonlinear Schroedinger equation: From energy cascades to weak turbulence. Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations on bounded domains. This is partly due to the fact that dispersion does not translate into decay in this setting (in contrast to the case of unbounded domains like $R^d$). In this talk, we will take the cubic nonlinear Schroedinger equation as our model, and discuss some aspects of its out-of-equilibrium dynamics, from energy cascades (i.e. migration of energy from low to high frequencies) to weak turbulence. Guo Luo Caltech Monday, November 25 KAP 414 3:30 PM - 4:30 PM Potentially Singular Solutions of the 3D Incompressible Euler Equations Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data has been one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall.The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of thesingularity. A careful local analysis also suggests that the blowing-up solution develops a self-similar structure near the point of the singularity, as the singularity time is approached. Katya Krupchyk University of Helsinki Monday, December 2 KAP 414 3:30 PM - 4:30 PM Resolvent estimates for elliptic operators and their applications. More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise, related to the distribution of eigenvalues of such operators. Applications to inverse boundary problems for rough potentials and to the absolute continuity of spectra for periodic Schr\"odinger operators will be presented as well. This talk is based on joint works with Gunther Uhlmann. James von Brecht UCLA Wednesday, December 4 KAP 414 2:00 PM - 3:00 PM Special Time Co-Dimension One Self Assembly The interest in and study of compact, co-dimension one minimizers has at least a century-old history: in 1904, J. J. Thomson proposed minimizing the electrostatic potential over sets of particles restricted to a sphere as part of his model of the atom. Modern physical examples of these assemblies occur in the realm of interacting nanoparticles. Many species of virus rely on the formation of a hollow sphere to enclose and deliver their genetic material, for example. Inorganic polyoxometalate (POM) macroions also form into hollow spherical structures in a similar way. I will discuss recently developed mathematical theory that characterizes when spherical assemblies define energy favorable structures, as well as applications to physical models of these assemblies. Colloquia for the Spring 2013 Semester Vlad Vicol Princeton University Monday, January 28 KAP 414 3:30 PM - 4:30 PM On the inviscid limit for the stochastic Navier-Stokes equations We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. We prove that the limiting measures are supported on bounded vorticity solutions of the 2D Euler equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow.Motivated by 2D turbulence considerations we are lead to consider the problem of well-posedness for the stochastic 2D Euler equations. This is joint work with Nathan Glatt-Holtz and Vladimir Sverak. Monday, February 4 KAP 414 3:30 PM - 4:30 PM The Whiteman Lecture Daniel Tataru UC Berkeley Monday, February 11 KAP 414 3:30 PM - 4:30 PM Geometric PDE's The aim of this talk is to present a group of geometric pde's where the objects of study are maps into manifolds. Examples include the harmonic maps, the harmonic heat flow, wave maps and Schroedinger maps.Some recent results and ideas will be surveyed. Guenther Walther Stanford University Monday, February 25 KAP 414 3:30 PM - 4:30 PM Optimal and fast detection with the scan and with the average likelihood ratio Scan statistics are the standard tool for a range of detection problems, such as the detection of spatial disease clusters. There have been a number of recent claims in the literature, based on empirical findings,that scan statistics are inferior to an approach that involves the average likelihood ratio. The talk will look into this issue and present heuristics as well as mathematical optimality results. If time permits, a connection to efficient algorithms will be discussed. Ronald Graham Monday, March 4 KAP 414 2:00 PM - 3:00 PM Special Time Juggling Mathematics and Magic The mystery of magic and the art of juggling have surprising links to interesting ideas from mathematics. In this talk, I will illustrate some of these connections. Fan Chung Graham UC San Diego Monday, March 4 KAP 414 3:30 PM - 4:30 PM Can you hear the shape of a network? New directions in spectral graph theory We will discuss some recent developments in several new directions of spectral graph theory, including random walks for directed graphs, ranking algorithms, graph gauge theory, network games, graph limits and graphlets, for example. Steve Shkoller UC Davis Wednesday, March 13 KAP 414 3:30 PM - 4:30 PM Free-boundary problems in fluid dynamics Free-boundary problems in fluid dynamics involve solutions to Euler equations on domains which are evolving in time and which are a priori unknown. The boundaries of these fluid domains have evolution laws which require knowledge of the fluid flow, while solving for the fluid flow, in turn, requires information about the geometry of the boundary. Such moving boundaries arise in the study of ocean waves, shock waves, phase transitions, and many other interesting wave patterns. In this lecture, I will survey some of the recent results in this area. Maria Schonbek UC Santa Cruz Monday, April 8 KAP 414 3:30 PM - 4:30 PM L^2 asymptotic stability of mild Navier-Stokes solutions. We consider the initial value problem for the Navier-Stokes equations modeling an incompressible fluid in three dimensions. It is well-known that this problem has a unique global-in-time mild solution for a suciently small initial condition u0 and for a small external force F in suitable scaling invariant spaces. We show that these global-in-time mild solutions are asymptotically stable under every (arbitrary large) L2-perturbation of their initial conditions.The work is joint with Grsegorz Karch and Dominika Pilarczyk. John A. Burns Virginia Tech. Monday, April 15 KAP 414 3:30 PM - 4:30 PM Parabolic Boundary Control Problems with Delayed Actuator Dynamics In this talk we discuss some control, optimization and design problems for a convection diffusion equation and investigate the impact of including actuator dynamics and delays. The problem is motivated by applications to control and design of energy efficient buildings where actuation is provided by a HVAC system. To provide some indication of the scope of this problem, it is helpful to note that buildings worldwide account for a approximately 40% of global energy consumption, and the resulting greenhouse gas emissions, significantly exceed those of all transportation combined. In the United States a 50% reduction in buildings energy usage is equivalent to taking every passenger vehicle and small truck in the United States off the road and a 70% reduction in buildings energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector.As a mathematical object, a whole building system is the composition of diverse dynamic subsystems and is a complex, multi-scale, nonlinear, and uncertain dynamical system. Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques to optimize this system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. In order to control a whole building system for energy minimization one must address a variety of theoretical and computational science problems at various levels from room to complete building envelopes. We focus on a single room example where the basic model is governed by a parabolic partial differential equation which is augmented to include a model of an actuator with delays. We show that under suitable conditions, the coupled system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design. Stephen Childress Courant Institute Monday, April 22 KAP 414 3:30 PM - 4:30 PM In search of a stable jelly-fish like flying machine. Flapping-wing aircraft offer an alternative to helicopters in achieving maneuverable flight at small scales, although stabilizing such aircraft remains a key challenge. Experimental studies of the stable hovering of rigid structures in an oscillating airflow suggest a design which mimics the flapping contractions of a jelly fish. We have constructed a prototype of such a craft, and test flown it in free flight with an external power source. Our results indicate the possibility of passively stable hovering of a flapping-wing craft. Charles Doering U of Michigan Monday, April 29 KAP 414 3:30 PM - 4:30 PM Ultimate state'' of two-dimensional Rayleigh-Bénard convection Rayleigh-Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above. Heat transport by convection an important physical process for applications in engineering, atmosphere and ocean science, and astrophysics, and it serves as a fundamental paradigm of modern nonlinear dynamics, pattern formation, chaos, and turbulence.Determining the bulk transport properties of high Rayleigh number convection turbulent convection remains a grand challenge for experiment, simulation, theory, and analysis. In this talk, after a general survey of the theory and applications of Rayleigh-Bénard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes equations. These bounds challenge some popular theoretical arguments regarding the asymptotic high Rayleigh number heat transport scaling.
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