June 21, 2018

September 18-20, 2008
Conference on Non-linear Phenomena in Mathematical Physics:
Dedicated to Cathleen Synge Morawetz
on her 85th birthday

Co-sponsored by:

Speaker Abstracts

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Claude Bardos, ( Paris VI)
Remarks on Navier Stokes and Euler Equation

In this talk I want to emphasize the following points.
1 The importance of the ``basic" properties of Euler equation (like the conservation of energy).
2 The instabilities of the solution.
3 The existence of solutions with decaying energy in connection with the fact that the zero viscosity limit of the $2d$ Navier-Stokes with no slip boundary condition is still an open problem.
4 The interpretation of the Kolmogorov inertial range in term of Wigner measure.

Lia Bronsard, (McMaster)
On the mixed state in anisotropic superconductors

In this talk, I will present an overview of the effect of anisotropy in the mathematical study of superconductors. Anisotropy is very important in the understanding of high temperature superconductors, and it presents very nice unexpected mathematical results. I will present our study on periodic minimizers of the Anisotropic Ginzburg-Landau and Lawrence-Doniach models for anisotropic superconductors, in various limiting regimes. We are particularly interested in determining the direction of the internal magnetic field (and vortex lattice) as a function of the applied external magnetic strength and its orientation with respect to the axes of anisotropy. We identify the corresponding lower critical fields, and compare the Lawrence-Doniach and anisotropic Ginzburg-Landau minimizers in the periodic setting. This talk represents joint work with S. Alama and E. Sandier.

Suncica Canic, (Houston)
Nonlinear moving-boundary problem for blood flow: analysis and computationmoving-boundary problem for blood flow: analysis and computation

The focus of the lecture will be on a benchmark fluid-structure interaction problem between pulsatile blood flow and viscoelastic arterial walls. The model equations couple the Navier-Stokes equations for an incompressible, viscous fluid with the linearly elastic/viscoelastic membrane equations.
The analysis and numerical computation of a solution to the problem is complicated due to the multi-physics and multi-scale nature of the underlying problem, and due to the exceedingly complicated nonlinear coupling between the fluid and the structure equations. Existence of strong solutions for large data is still an open problem and computational algorithms for the calculation of a solution are far from optimal.

In this talk, an analysis of the existence of a solution to the reduced, effective problem will be presented. The existence proof motivated the design of a novel computational scheme for the full problem. The new scheme, based on an implicit kinematic coupling and on a clever operator splitting approach, provides a superbly stable and efficient computational scheme to study this fluid-structure interaction problem in blood flow.

The speaker will give a brief overview of the current status in the field of fluid-structure interaction in blood flow, and outline the efforts by the group in Houston toward using nonlinear analysis and scientific computation to move evidence-based medicine closer to the future of quantitative medicine.

This work was done in collaboration with Roland Glowinski, Giovanna Guidoboni, and Taebeom Kim.

Gui-Qiang Chen, (Northwestern)
Shock Reflection-Diffraction Phenomena, Transonic Flow, and Free Boundary Problems

In this talk we will discuss a research project on shock reflection-diffraction phenomena and related topics, inspired and motivated highly by Cathleen Morawetz's fundamental works on transonic flow and related areas. We will start with various shock reflection-diffraction phenomena and their fundamental scientific issues. Then we will describe how the shock reflection-diffraction problems can be formulated into free boundary problems for nonlinear partial differential equations of mixed-composite hyperbolic-elliptic type. The problems involve two types of transonic flow: One is a continuous transition through a pseudo-sonic circle, and the other is a jump transition through the transonic shock as a free boundary. Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including some recent results on the existence, stability, and regularity of global solutions of shock reflection-diffraction by wedges. This talk is based mainly on the joint work with Mikhail Feldman.

Costas Dafermos, ( Brown)
Hyperbolic Conservation Laws with Weak Dissipation

Susan Friedlander, (Southern California)
Energy Conservation and Onsager's Conjecture for the Euler Equations

Onsager conjectured that weak solutions of the Euler equations for 3 D incompressible fluids conserve energy only if they have a certain minimal smoothness. As a consequence, in 3 D turbulent flows, energy dissipation might exist even in the limit of vanishing viscosity. We discuss some recent results where we prove that energy is conserved in a Besov space with regularity "almost" that conjectured by Onsager.
This is joint work with Alexey Cheskidov, Peter Constantin and Roman Shvydkoy.

Irene M. Gamba (Texas at Austin)
Sharp estimates for the Boltzmann Equation

We prove L1 and L8 Gaussian weighted estimates to the collisional integral and its derivatives associated to the Boltzmann equation. Such control allow us to prove the propagation and creation of L1 and L8 Gaussian weighted bounds to solutions of the homogeneous Boltzmann equation, and to any of its derivatives in n-dimensions, for realistic intra-molecular potentials leading to collisional kernels of variable hard potentials type with for unbounded, integrable angular cross sections (Grad's forms).One of the interesting developments is the sharp Povzner estimates and summability of moments to variable hard potentials and unbounded, integrable cross section which carries on to all derivatives. We will also discuss some extension Young's type estimates both in the case of elastic and inelastic collisions, by means of symmetrization and Fourier representation of the collisional operator.
This work is in collaboration with Vlad Panferov and Cedric Villani, and more recently with Ricardo Alonso and Emanuel Carneiro.

Manoussos Grillakis, (Maryland)
Correlation Estimates and Applications to Schroedinger Equations

In the present talk I will outline various methods that can be employed in order to obtain correlation type estimates for Schroedinger equations. The idea originated in the work of Lin and Strauss inspired by earlier work of C. Morawetz and W. Strauss. Recent advances due to Colliander, Keel, Stafillani, Takaoka and Tao make it a general and powerfull tool. I will outline a general method of obtaining such estimates. In two space dimensions one can obtain an apriori estimate which is the nonlinear analog of a bilinear estimate obtained by Bourgain. In higher space dimensions one can obtain global in time estimates for the density after the collapse of some internal variables. I will explain how these estimates can be used in order to prove scattering for nonlinear Schr¨odinger equations and how to obtain apriori estimates for the Bose-Einstein condensation problem. For the Bose-Einstein problem one can use the original idea of Lin, Strauss and Morawetz in order to collapse some internal variables. It turns out that this is a natural operation if we consider what happens as the number of particles tends to infinity. This work is in collaboration with a) J.Colliander, N. Tzirakis and b) D. Margetis.

Yan Guo (Brown)
Boltzmann equation in bounded domains.

In bounded domains, we establish well-posedness and exponential decay for solutions to the Boltzmann equation near Maxwellians, in the presence of in-flow, bounce back, specular, or diffuse refections boundary conditions. If the domain is strictly convex, then these solutions will remain continuous away from the grazing set at the boundary.

Tom Hou, CalTech
On the stabilizing effect of convection in 3D incompressible flows.

We investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. We demonstrate this by constructing a new 3D model which is derived from axisymmetric Navier-Stokes equations with swirl using a set of new variables. The only difference between our 3D model and the reformulated Navier-Stokes equations is that we neglect the convection term in the model. If we add the convection term back to the model, we will recover the full Navier-Stokes equations.

This model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations. In particular, the strong solution of the model satisfies an energy identity similar to that of the full 3D Navier-Stokes equations. We prove a non-blowup criterion of Beale-Kato-Majda type as well as a non-blowup criterion of Prodi-Serrin type for the model. Moreover, we we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. This partial regularity result is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.
Despite the striking similarity at the theoretical level between our model and the Navier-Stokes equations, the former has a completely different behavior from the full Navier-Stokes equations. We will present convincing numerical evidence which seems to support that the 3D model develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.

Izabella Laba, (British Columbia)
Arithmetic progressions in sets of fractional dimension

Let $E\subset {\bf R}$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions. (Joint work with Malabika Pramanik.)

Peter Lax, (Courant)
Spectral Representation and Translation Representation.

For selfadjoint operators whose spectrum is the whole real line with constant multiplicity,spectral representation and translation
representation are Fourier transforms of each other.For the automorphic wave equation the translation representation is given by integrals along one parameter families of horocycles.

Consider automorphic functions with respect to discrete subgroups of isomorphisms of two-dimensional hyperbolic space.It is known that for generic subgroups the spectrum of the wave equation has only a finite point spectrum.This raises an intriguing question about the geometry of the horocycles.

Louis Nirenberg
, (Courant)
Some remarks on nonlinear second order elliptic equations.

Some properties of such equations are presented,with some applications.

Kevin Payne, (Milano)
Weak well-posedness of the Dirichlet problem for equations of mixed elliptic-hyperbolic type

We present joint work with Daniela Lupo and Cathleen Morawetz on the question of existence and uniqueness of solutions to the Dirichlet problem for mixed type equations. While it is well known that the presence of hyperbolicity renders such a problem overdetermined for solutions with classical regularity, we show well-posedness for solutions belonging to suitably weighted Sobolev spaces. This follows from global energy estimates which are obtained by exploiting integral variants of Friedrichs’ multiplier method. Attention is paid to the problem of obtaining results with minimal restrictions on the boundary geometry and the form of the type change function in preparation for the construction of stream functions in the hodograph plane for transonic flows about profiles.

Jalal Shatah, (Courant)
The Method of Space-time Resonances

Sylvia Serfaty, (Courant & Universite Pierre et Marie Curie)
From the Ginzburg-Landau energy to vortex lattice problems

In a joint work with Etienne Sandier, we study the behavior of vortices for minimizers of the 2D Ginzburg-Landau energy of superconductivity with an applied magnetic field, in a certain asymptotic regime where the vortices become point-like. In the regime of applied fields we are interested in, it is observed that vortices are densely packed and form triangular lattices names Abrikosov lattices.
We derive rigorously from the Ginzburg-Landau energy, via methods of Gamma convergence, first a leading order ``mean field model" describing the optimal density of vortices; second a next order limiting energy which governs the position of the vortices after blow-up at their inter-distance scale. This limiting energy is a logarithmic interaction between points in R^2. By using tools from number theory (modular form), it turns out that, among lattice configurations, this energy is uniquely minimized by the triangular (or hexagonal) lattice.

Gigliola Staffilani, (MIT)
Kato's smoothing effect for solutions to the  capillary water-wave problem.

In a collaboration with Hans Cristianson and Vera Hur we proved that the solutions to the Cauchy problem for exact free-surface water waves in presence of surface tension, as t>0, gain 1/4 derivative smoothness compared to the initial profile, this is what we call the 1/4 Kato's smoothing effect. The major difficulty in proving this result is severe nonlinearity on free surface. To deal with a nonlinearity, first, we reformulate the problem as a nonlinear dispersive equation for a modified velocity on the free surface, whose linear part may be recognized as a hybrid of the wave equation and the Schroedinger or the Korteweg-de Vries equation. Our novel formulation exhibits strong dispersive property due to surface tension, and indeed, smoothing effects. Dispersion allows us to treat nonlinear terms with first or second spatial derivatives by means of techniques of oscillatory integrals. But this would not be enough.
Secondly, we view the most severe nonlinear term as a "linear component" of the equation, but with a variable coefficient which happens to depend on the solution itself. That is, we reduce the size of the nonlinear terms at the cost of making the linear part more complicated. A sophisticated microlocal analysis approach is to establish smoothing effects for this 'water-wave operator' with variable coefficient. We provide more refined analysis than classical energy estimates, which is the only estimate known so far in the analysis of the Cauchy problem for water waves. Our result requires less number of derivatives in the choice of Sobolev spaces, which is a major improvement of this project and proves some new Kato type smoothing effect for the solutions.

Walter Strauss (Brown University)
Two Problems: Nonlinear Wave Scattering and Plasma Stability

I will speak on two different problems, both of which are closely related to Cathleen's past work. Both are concerned with understanding the asymptotic behavior of waves in the absence of dissipation.
First I will consider the global scattering problem of nonlinear scalar waves in the defocusing case. After a survey of the older work on NLKG and NLS, I will present the very recent results of Benoit Pausader on fourth-order equations.
Next I will consider the Vlasov-Maxwell system that models collisionless plasmas. There are many equilibria and one wants to know which ones are stable and which ones unstable. I will describe the recent work of Zhiwu Lin and myself on this problem.

S.R.S Varadhan (Courant)
:Large deviations for Random walks in a Random Environment

We will explore some open problems regarding large deviation behavior of random walks in a random environment. They have to do with the existence of positive harmonic functions. The random environment can be viewed as a limiting case of periodic environments and the question is how much of the theory in the periodic case extends to the stationary random case.

Horng-Tzer Yau, (Harvard)
Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the $z$-axis. Suppose the solution satisfies, for some $0 \le \e \le 1$, $|v (x,t)| \le C_* r^{-1+\varepsilon } |t|^{-\varepsilon /2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.

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