Wednesdays at 4 PM in Eckhart 202 unless otherwise
noted.
A century ago Einstein postulated that the origin of the Brownian motion was due to the light water molecules continuously bombarding the heavy pollen. This explained the Brownian motion in the frameowrk of the Newtonian mechanics. Since the discovery of quantum mechanics it has been a major challenge to verify the emergence of diffusion from the Schr\"odinger equation. In this talk I will report a rigorous derivation of a diffusion equation from a scaling limit of a random Schr\"odinger equation in a weak random potential. This is a joint work with L. Erdos and M. Salmhofer.
We study nonlinear photonic crystals as an example of a nonlinear dispersive system. The nonlinear Maxwell equations (NLM) is analyzed based on there modal decomposition, and an expansion of the exact solution of the NLM into an asymptotic series with respect to three small parameters. These parameters are introduced through the excitation current to scale respectively: (i) the magnitude of the nonlinearity; (ii) the range of wavevectors involved in the modal composition of the current scaling its spatial extension; (iii) the frequency bandwidth the current scaling its time extension. The conducted studies produce an asymptotic sequence of approximations of increasing accuracy to the electromagnetic wave. Its first, the most significant term, is governed by the Nonlinear Schrodinger equation (NLS). In particular, we show that the NLS regime is the medium response to an almost monochromatic excitation current. The developed approach not only provides rigorous estimates of the accuracy of approximations of the NLM by the NLS and its natural extensions in terms of powers of the three small parameters, but it also allows under rather general conditions to decompose the system into almost independent subsystems.
Generalized Boussinesq systems are formulated for weakly nonlinear surface water waves over arbitrary bottom topographies. These systems are derived under a curvilinear coordinate system. Our main goal is to produce disperive systems for studying pulse shaped waves in a randomly varying environment. These include solitary waves.
Although determining the mean velocity profile in steady turbulent flow of an incompressible fluid through a channel is apparently one of the ``easiest'' problems in turbulence theory, it is still unsolved. There is much lore about this problem, some of it based on arguments about overlapping scaling (inner-outer) regions. We have developed an alternative approach involving multiple-scale concepts and scaling patches. Among other things, it clarifies the hierarchical scaling structure of the profile and the existence and nature of ``logarithmic'' sections. This is joint work with J. Klewicki, P. McMurtry and T. Wei.
This talk is devoted to the proof of the hydrodynamical limit from kinetic equations (including B.G.K. like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method. Notice that no a priori knowledge on high velocities distributions for kinetic functions is needed: High energetic particules can be controlled. The case of the shallow water system with topography (where a source term is added) is included. This is joint work with F.Berthelin.
There are several motivations to study the isometric immersions with Sobolev type regularity of say an $m$-dimensional domain into a given Euclidean space. One motivation is geometrical. ` It is well known that $C^2$ isometric immersions have a good classification and enjoy strong rigidity properties while the celebrated results of Nash and Kuiper show that $C^1$ isometric immersions can be much more complicated (e.g. the image of $S^2$ can be contained in an arbitrarily small ball). One may consider now the Sobolev classes of maps which lie somewhat in between. On the other hand, spaces of this type arise in the elasticity theory of plates, first formulated by Kirchhoff, and give rise to new questions which can be formulated for higher dimensional sheets. The main example which I will consider is the space of isometric immersions $u$ from a two dimensional domain $\Omega$ to ${\mathbb R}^3$ which are in the Sobolev class $W^{2,2}$, i.e. $\nabla^2 u$ is in $L^2$.
Arnol'd in 1991 characterized ``pseudoperiodic'' flows on the 2-dimensional torus. We consider small random (dffusive) perturbations of such flows. Under appropriate scaling of time, we search for an averaged picture which describes the evolution of local ``energies''. Under certain circumstances, we identify a certain limiting Markov process with glueing conditions (as suggested by Freidlin in 1996) which characterizes energy evolution.
There has been a resurgence of interest in the homogenization theory (averaging behavior) of nonlinear pde, which arise in many applications like combustion, front propagation, percolation, phase transitions, etc.. in random media. This general environment lacks the compactness of the the periodic settings which were studied before. It is therefore necessary to develop a different approach. In the lecture I will present a number of concrete examples, I will review the methods used in the periodic setting, I will discuss why these arguments cannot be applied in the random setting, and I will explain how it is possible, using the sub-additive, to circumvent these difficulties to obatin (rigorously) the effective equations.
Imaging with passive or active sensor arrays is done usually by backpropagation of the data, which is also called Kirchhoff migration in seismic imaging. This works particularly well if there is are lot of data: large arrays and broadband probing signals. It does not work well at all, however, if the medium between the array and the region to be imaged is cluttered, that is, a random medium. I will describe Kirchhoff migration, explain why it fails to produce reasonable images when there is clutter, and introduce another imaging method, the back propagation of space-time correlations of the data, which is coherent interferometry. I will show the results of several numerical simulations as well as theoretical results that provide a new resolution theory for imaging with coherent interferometry
We present new estimates obtained jointly with J. Bourgain, and partially with P. Mironescu. Each one has a different flavour, but, in fact, they are closely related. The first one asserts that $$ \Big|\int_\Gamma f(s) \overarrow t(s) \Big|\leq C|\Gamma| \, \| \nabla f\|_{L^3}\quad \forall f $$ where $\Gamma\subset \Bbb R^3$ is a closed rectifiable curve, $|\Gamma|$ denotes the length of $\Gamma$, $C$ is a universal constant and $\overarrow t$ is the tangent to $\Gamma$. The second estimate concerns the classical system, in $\Bbb R^3$, $$ \aligned \text{div } u &= 0\\ \text{curl } u &= f.\endaligned $$ Our new estimate asserts that $$ \| u \|_{L^{3/2}}\leq C\| f \|_{L^1}. $$ A third new estimate concerns the system $$ \Delta u = f \text { in } \Bbb R^3, $$ where $f$ is a divergence-free vector-field. Our new estimate asserts that $$ \| u \|_{L^3} \leq C\|f \|_{L^1}. $$ Such inequality are unusual because it is well-known that standard elliptic estimates fail in $L^1$.
The stochastically forced Navier-Stokes equation (SNS) is both an important model of physical interest and an important testing ground to develop methods to analyze stochastic partial differential equations. Consider the two dimensional Navier-Stokes equation subject to random excitation. I will give essentially optimal conditions on the structure of the sochastic forcing under which the dynamics posses a unique statistical steady state. The conditions use information on the geometry of the forcing and are independent of the viscosity (or Reynolds number). Hence the results hold for a fixed forcing as the viscosity is made smaller. These very recent results are the culmination of a long project under taken by the speaker, his collaborators and others in the community to understand the ergodic theory for dissipative SPDEs and Hormander's "sum of squares" theorem for hypo-elliptic operators in an infinite dimentional setting. The talk will also touch on the tools from Malliavin calculus and anticipative stochastic processes used to prove the result. This is joint work with Etienne Pardoux and Martin Hairer building on earlier joint works with Ya Sinai and Weinen E. when only a few degrees of freedom are stochastically excited. The result amounts to a version of Hormander's "sum of squares" theorem for hypo-elliptic operators in an infinite dimentional setting.
Many physical systems such as fluid dynamics are governed by a coupled nonlinear evolutionary system of equations. In order to effectively compute numerically such a coupled system of nonlinear PDEs, it is necessary to linearize and decouple the equations at a certain stage. For example, explicit treatment of nonlinear and global terms in time discretizations is a practical and effective way of decoupling complicated systems. On the other hand, some of the linear operators such as the diffusion terms ought to be treated implicitly due to efficiency considerations. It is the proper combination of explicit and implicit treatment of various linear and nonlinear terms that results in an accurate and efficient numerical scheme. The art of time discretization, is therefore to find a suitable operator decomposition and estimate the spectrum of each operator in order to insure the stability of the scheme under reasonable time step size. In this talk numerical schemes are presented for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the convection and pressure terms are treated explicitly in time, with the incompressibility constraint replaced by a pressure Poisson equation. The computation of the momentum and kinematic equations are fully decoupled, resulting is a class of extremely efficient Navier-Stokes solvers. The cost of solving 3D Navier-Stokes equation is therefore comparable to solving a heat equation and a Laplace equation. These methods work well for both large and small Reynolds number flows. Moreover, the schemes are not projection-type methods and are free of numerical boundary layers resulting from time consistency issues inherent in such splitting methods. Full time accuracy is achieved for all flow variables in the L^\infty norm. In addition to efficiency, this class of schemes enjoys some remarkable stability properties. Indeed, a first order semi-implicit discretization of these explicit pressure treatment schemes has been proven to be unconditionally stable. Additionally, since the key to the schemes is the proper pressure Poisson formulation at the PDE level, any kind of spatial discretization such as finite difference, spectral (collocation and Galerkin) methods, and finite element method can be incorporated into the scheme. In particularly, standard continuous finite element spaces can be used to handle the general 3D domains. Moreover, it is proven that the so called inf-sup compatibility condition need not be imposed on the pressure and velocity spaces if the elements are at least $C^1$.
Gradient reaction-diffusion systems arise in the context of modeling the kinetics of second-order or weakly first-order phase transitions, with a broad range of applications. These systems are known to exhibit a variety of non-trivial spatio-temporal behaviors, most notably the phenomenon of propagation and traveling waves. We introduce a variational formulation for the traveling wave solutions in cylindrical geometries, which allows us to construct a certain class of special traveling wave solutions and study a number of their properties. These solutions are special in a sense that they are characterized by a non-generic fast exponential decay ahead of the wave and play an important role in propagation phenomena for the initial value problem. In particular, we show that no solution of the initial value problem that is initially sufficiently localized can propagate faster than the speed of the obtained traveling wave. We also show that only this type of traveling wave solutions can be selected as the asymptotic limit of the solution in the reference frame associated with its leading edge at long times. The considered variational formulation gives easily verifiable upper and lower bounds for propagation speeds. This is joint work with M. Lucia and M. Novaga.
The FEniCS project is working toward developing the theory and tools needed to automatically generate finite element code for the numerical solution of partial differential equations as part of a broader vision of the automation of computational mathematical modeling. Finite element methods offer us a machine where the kind of approximating functions is an input parameter, but we still must obtain practical, computable representations for a particular basis for these approximating spaces. I will present a general framework, encoded in FIAT, for the automatic generation of representations for basis functions for a wide class of finite element spaces. In the second part of the talk, I will switch gears to discussing the local computation on each element of the finite element mesh. While FIAT and a simple change of variables give us a representation for the basis functions everywhere, they do not imply particular algorithms for efficiently computing the local element matrices needed to build the global matrix. As doing these small local calculations actually represents a considerable amount of the total effort expended in a finite element computation, we present a graph-based optimization technique to determine very efficient algorithms for computing these local matrices in terms of certain quantities computed by FIAT. This is joint work with many people, especially Matt Knepley (ANL), Anders Logg (TTI-C), and Ridgway Scott (UC).
We want to provide rigorous results for experimental observations of vortices in Bose-Einstein condensates: when the trap holding the atoms is rotated, vortices are observed in the system: for intermediate velocity, there is a single bending vortex line, while for large velocity, the number of vortices increases and a lattice is formed. We investigate the behavior of the wave function minimizing the Gross Pitaevskii energy. We find the critical velocity for the nucleation of the first vortex and provide a justification for the structure of the vortex line, which relies on the analysis of a reduced energy depending on the vortex line only. For a fast rotating condensate, we investigate the structure of the lattice using wave functions in the Lowest Landau Level. We find that the minimizer has a distorted vortex lattice, determine the optimal distortion and relate it to the decay of the wave function.
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We consider evolution equations which are invariant under a group of scaling transformations. Our main examples are the nonlinear heat and Schrödinger equations, as well as the Navier-Stokes system. We show that there exist solutions which are asymptotic to different self-similar solutions along different time sequences going to infinity. In fact, the set of self-similar solutions obtainable in this fashion as an asymptotic limit from a single solution can be infinite dimensional. Furthermore, the flow operator at a fixed positive time induced by the equation, followed by an appropriate spatial dilation, generates a chaotic discrete dynamical system. This is joint work with Thierry Cazenave and Flavio Dickstein.
In this talk we will give sharp explicit estimates for the dimension of the attractor of the damped/driven 2D Navier-Stokes equations and the attractor of the Navier-Stokes equations on the elongated torus.
The notion of entropy is motivated by the second law of thermodynamics and has played an important role in the theory of hyperbolic systems of conservation laws. The term has various uses in the mathematical literature usually connected with additional conservation laws that the system satisfies. In this talk I will review various uses of entropies: (i) The notion of relative entropy and how it is used to assess the stability properties of approximating theories such as viscosity or relaxation theories. (ii) the role that nonlinear transport equations play in structural properties of polyconvex elastodynamics. (iii) The representation of entropy pairs through the kinetic formulation and their role in the study of oscillations for systems of two conservation laws, and the existence of weak solutions in the functional framework of the energy norm.
It is a general belief that systems of interacting particles (and, more generally, Hamiltonian systems) are neither ergodic nor integrable. Instead, they have KAM islands with a regular dynamics situated in chaotic sea(s). Although such structure of a phase space has been recovered in a large variety of numerical experiments, there are no general rigorous results of that type. Moreover, even exact rigorously analized visual examples of such systems were absent till recently. The major difficulty is to understand exact structure of the boundaries of islands. I'll present such examples. Their analysis leads to some new ideas and questions related to the problem of mixing of particles in a physical space (container).
In the talk, we shall discuss various local regularity issues for the 3D non-stationary Navier-Stokes equations. They include interior and boundary versions of Ladyzhenskaya-Prodi-Serrin condition, including $L_{3,\infty}$-case, and Caffarelli-Kohn-Nirenberg type theory.
In a large class of inverse problems, we seek the coefficients in a partial differential equation, inside some domain, given the Neumann to Dirichlet map. These problems are severely ill posed, so proper parameterizations are important in the inversion scheme. I will present a finite volume approach to inversion, on so-called ``optimal grids''. This approach combines classic techniques in inverse problems with ideas from model reduction, discrete inverse spectral problems and numerical PDE's to address the issue of parameterization in a rigorous fashion. I will describe the inversion algorithm, show numerical results and prove convergence in one dimensional (layered) media. Extensions to two-dimensional problems will be discussed, as well.
Click here for the abstract
Birkhoff billiard is a model problem in Hamiltonian dynamics. Periodic orbits in Hamiltonian systems are second in importance only to the rest points, which are absent in billiards. We will describe recent results on periodic orbits in convex billiards along with some applications. In particular, we will describe the construction of nonintegrable billiards possessing a continuous family of periodic orbits.
The purpose of this work is to locate localized damage in a structure with distributed sensors. Given a configuration of transducers, we assume that a full response matrix for the healthy structure is known. It is used as a basis for comparison with the response matrix that is recorded when there is damage. We have carried out a numerical experiment with the wave equation in two dimensions. The healthy structure is a domain containing many scatterers. We want to image two point-like defects with the help of 12 sensors regularly distributed. Because of the complexity of the environment, the traces have a lot of delay spread and travel time migration does not work well. Instead, the traces are back propagated numerically in the medium, assuming that we have some knowledge of the background. Since the time at which the back propagated field will focus on the targets is unknown, we compute the Shannon entropy of the image and pick the time where it is minimal. The TV norm proves also a good indicator. This works well for distributed sensors networks because the information will dramatically reduce at the time of refocusing. When there are several defects, the Singular Value Decomposition of the response matrix is performed at each frequency to resolve each of them. This optimally compensated time-reversal imaging algorithm gives good and stable results.
I will show how small stochastic perturbations of dynamical systems can lead to not random stable oscillations and equilibriums, which are not available in the system without these perturbations. If the system is generic, for any initial point and a time scale, one can point out a stable attractor (metastable state) such that the perturbed system spends most of the time (in the given time scale) near this attractor. Bifurcations in the metastable states lead to stochastic resonance. This effects are manifestations of laws of large deviations.
We consider the problem of minimizing the first eigenvalue of an elliptic operator of the type $-\Delta+v\cdot\nabla$ with bounded drifts $v$, in bounded domains with Dirichlet boundary conditions. The minimization or maximization problems over the fields $v$ in a given domain lead to some nonlinear equations. The minimization problem over the domain with given measure and over the vector field with a given bound has a unique solution and the minimizing domain is a ball. This result generalizes the usual result of Faber and Krahn for the Laplace operator. This talk is based on a joint paper with N. Nadirashvili (CNRS, Marseille) and E. Russ (Universite Paul Cezanne Aix-Marseille III)
The understanding of scale interactions for the incompressible Euler and Navier-Stokes equations has been a major challenge. Here I will present a new multiscale analysis for the 3D incompressible Euler equation with rapidly oscillating initial data. We first present a multiscale analysis based on the Lagrangian formulation. By using a Lagrangian description, we can characterize the nonlinear convection of small scales exactly and turn a convection dominated transport problem into an elliptic problem for the stream function. At the end, we derive a coupled multiscale system for the flow map and the stream function, which is well-posed. Based on our understanding in the Lagrangian formulation, we derive a similar multiscale analysis using the Eulerian formulation, which is more effective for computational purpose. Our multiscale analysis reveals some interesting structures of the Reynolds stress terms and provide a theoretical guidance in developing a systematic multiscale modeling of incompressible flow. Numerical results will be presented to demonstrate the accuracy and the robustness of the multiscale method.
We discuss phase-space formulation of the nonlinear Schr\"odinger equation with a white-noise potential in order to shed light on two problems: the rate of dispersion and the singularity formation. Our main tools are the energy laws and the variance identity. The calculations involved are elementary. For the problem of dispersion, we show that in the absence of dissipation the ensemble-average dispersion in the critical or defocusing case follows the cubic-in-time law while in the supercritical and subcritical focusing cases the cubic law becomes an upper and lower bounds respectively. In the presence of dissipation the cubic law is replaced by the linear-in-time law. We have also found that the presence of a white-noise random potential merely changs the singularity condition but does not prevent singularity formation in the critical and supercritical focusing cases. We show that the finite-time singularity in the supercritical focusing case is of the blow-up type.
It is well-known that, in the diffusive scaling, the first order expansion of the Boltzmann equation gives rise to the celebrated incompressible Navier-Stokes-Fourier system. We establish the validity of such a diffusive expansion up to any order for all time near a Maxwellian. In particular, our results lead to error estimates for the incompressible Navier-Stokes-Fourier approximation.
In a joint work with Bob Kohn, we give a new control-type interpretation on the level-set approach to motion by curvature. More precisely, we give a family of discrete-time two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. The value function of a deterministic control problem is normally a first order Hamilton-Jacobi equation, while the level-set formulation of motion by curvature is a second-order (degenerate) parabolic equation.
Over the recent years it has been observed that the linearized instabilities of the Kelvin Helmholtz equation may in some cases lead, at the level of non linear problem to the appearance of singularities. This was done both by numerical simulations, Moore (1979) and Meiron Baker and Orszag (1982) among others and by functional analysis by Caflisch and Orellana (1989).
On the other hand people have realized that these problems are in some sense elliptic. This was observed for first time for small global perturbation by Duchon and Robert (1988). Then this type of property was systematically explored (leading to local results) by Sijue Wu and by Gilles Lebeau.
With the time reversibility of the equations one can use these results to construct solutions with prescribed singularities generalizing the above examples. I intend to show (following the approach of Lebeau and Kamotski with the use of paradifferential calculus) that these results extend to a larger class of problem including for instance the Rayleigh Taylor problem and others.
In the mean time I want stress the difference between these equations (only well posed in the analytic framework) and the water waves problem, which turns out to be well posed in classical Sobolev spaces.
For the Schroedinger flow from $R^2\times R^+$ to the $2$-sphere $S^2$, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to zero. This is joint work with Stephen Gustafson and Kyungkeun Kang.
When estimating solutions of dissipative partial differential equations in $L^p$-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian $(-\Delta)^\alpha$, the approach of integration by parts no longer applies. In a recent work, we obtain a lower bound for the integral involving $(-\Delta)^\alpha$ by combining a pointwise inequality for $(-\Delta)^\alpha$ with the Bernstein inequality for fractional derivatives. As an application of this lower bound, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations refer to the resulting equations by replacing $-\Delta$ of the Navier-Stokes equations by $(-\Delta)^\alpha$.
Many problems in physics, material sciences, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well-known examples include phase transitions of condensed matter, conformational changes of biopolymers, and chemical reactions. The state of these systems is confined for long periods of time in metastable regions in configuration space and only rarely switches from one region to another. The separation of time scale is a consequence of the disparity between the effective thermal energy and typical energy barrier in these systems, and their dynamics effectively reduces to a Markov chain on the metastables regions. The analysis and computation the transition pathways and rates between the metastable states is a major computational challenge, especially when the energy landscape exhibits multiscale features. I will review recent work done by scientists from several disciplines on probing such energy landscapes, introduce concepts such as reaction coordinate and free energy, and show how these concepts can be made mathematically precise. I will then present a new method, the string method, that has proven to be very effective for some complex systems in material science and chemistry.
The notion of Relative Entropy Inequality is standard for several linear PDEs that are conservation laws, parabolic, hyperbolic, integral equations. Biological applications lead naturally to birth and death processes that can be described by zeroth order terms. They also lead to models where several balance law combines together (number of individuals, total mass of the population...) but no obvious conservation law follows. In this talk, we introduce the notion of General Relative Entropy Inequality that applies to PDEs that are not in conservation form. We show how the eigenelements come in the definition of the entropy and we give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state or attraction to periodic solutions. This last point is motivated by the question: can tumor growth be seen as a lost of circadian control? This talk is taken from papers with J. Clairambault, P. Michel, S. Mischler and L. Ryzhik
This talk will describe the simulation, design and optimization of a qubit for use in quantum communication or quantum computation. The qubit is realized as the spin of a single trapped electron in a semi-conductor quantum dot. The quantum dot and a quantum wire are formed by the combination of quantum wells and gates. The design goal for this system is a "double pinchoff", in which there is a single trapped electron in the dot and a single (or small number of) conduction states in the wire. Because of considerable experimental uncertainty in the system parameters, the optimal design should be "robust", in the sense that it is far away from unsuccessful designs. We use a Poisson-Schrodinger model for the electrostatic potential and electron wave function and a semi-analytic solution of this model. Through a Monte Carlo search, aided by an analysis of singular points on the design boundary, we find successful designs and optimize them to achieve maximal robustness.
For questions, contact Eduard Kirr at ekirr@math.uchicago.edu
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