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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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| December 23, 2024 |
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In this talk I want to emphasize the following points. Lia Bronsard, (McMaster) 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. 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) 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) Susan Friedlander, (Southern California) 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. Irene M. Gamba (Texas at Austin) 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. 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 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. Izabella Laba, (British Columbia) For selfadjoint operators whose spectrum is the whole real line
with constant multiplicity,spectral representation and translation
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.
Some properties of such equations are presented,with some applications. Kevin Payne, (Milano) 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. 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. 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. Walter Strauss (Brown University) 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.
S.R.S Varadhan (Courant) 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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