PDE Seminar, Fall 2008
Wednesday's from 4 to 5 PM at MIT (room 2-147).
Next seminar:
Full schedule:
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September 10: Hongjie Dong (Brown University)
``Elliptic and parabolic equations in divergence form with partially VMO coefficients.''
Abstract: We consider second order elliptic and parabolic equations in divergence form
with partially VMO leading coefficients. The solvability in Sobolev spaces is proved for
equations in the whole space, the half space and bounded Lipschitz domains. For equations
in domains, additionally we assume leading coefficients are VMO in a neighborhood of the
boundary of domains. We give a unified approach of both the Dirichlet boundary problem
and the conormal derivative problem.
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September 12: Jonathan Lenells (Cambridge)
``Global weak solutions of the Hunter-Saxton equation.''
Special date and time: Friday 3:30 PM in 2-131
Abstract: The integrable Hunter-Saxton equation models the
propagation of
orientation waves in a liquid crystal director field. Geometrically, it
describes the geodesic flow on the infinite-dimensional space Diff(S1)/S1
with respect to a certain right-invariant metric. We
show that this space has constant positive curvature and is in fact
isometric to a subset of the L2
unit sphere. This
geometric interpretation is then used to study properties of the
solutions and to
extend solutions beyond wave breaking.
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September 24: Steve Shkoller (UC Davis)
``Free Boundary Problems for incompressible and compressible 3D Euler
equations.''
Abstract:
We describe a new method for treating free boundary problems in perfect
fluids, and prove local-in-time well-posedness in Sobolev spaces for the
free-surface 3D Euler equations with arbitrary initial data, and without
any irrotationality assumption on the fluid. This is a free boundary problem
for the motion of an incompressible perfect liquid in vacuum, wherein the
motion of the fluid interacts with the motion of the free-surface at
highest-order. For the compressible problem,
the vanishing of the density at the vacuum boundary induces degenerate
hyperbolic equations that become characteristic, requiring a separate
analysis of time, normal, and tangential derivatives to handle the manifest
1/2-derivative loss. Unfortunately, the methods for incompressible flow do
not work for the degenerate compressible regime; a priori nonlinear
estimates are obtained using the geometric structure of the Euler equations,
and an existence theory is developed using a novel approximation scheme
employing an artificial phase.
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October 8: Nir Gavish (Tel Aviv University)
``New singular solutions of the critical and supercritical Nonlinear
Schrodinger equation (NLS).''
Abstract: The study of singular solutions of the
NLS goes
back to the 1960s, with applications in nonlinear optics and,
more recently, in BEC. Asymptotic and numerical studies conducted in
the 80s showed that singular solutions of
the critical NLS collapse with the Townes (R) profile at a blowup rate
known as the loglog law. Recently (2003) Merle and Raphael proved
this result rigorously for a large class of initial conditions.
Concurrently,
it was demonstrated experimentally that the profile of collapsing laser
beams is given by the Townes profile.
Thus, all the research that was carried out from the eighties until
these days leads to the belief that the Townes profile is the only
attractor of blowup solutions of the critical NLS.
In this talk I will present new families of singular solutions of the
critical and supercritical NLS that
collapse with a self-similar ring profile, and whose blowup rate is
different from the one of the "old" singular solutions. I will show,
experimentally and theoretically, that these new blowup profiles are
attractors for large super-Gaussian initial conditions.
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October 15: Tristan Roy (UCLA)
``Global existence for the defocusing cubic wave equation in dimension 3.''
Abstract:
In this talk I will discuss recent developments regarding the global existence of solutions to the defocusing
cubic wave equation in dimension 3 below the energy norm. This initial value problem is known to be globally
well-posed in Hs × Hs-1, s ≥ ¾ . We design the I
method (originally invented by the "I team" for the semilinear Schrodinger equations) for this wave equation and
give another proof of global existence for s > ¾ . Then we prove global well-posedness for
s > 7⁄10 under the additional assumption of spherical data by adding
new components to the method, such as Morawetz-type estimates, radial Sobolev inequalities and a "greedy"
algorithm. Finally we get back to the general problem and we show that this equation is globally well-posed
for s > 13⁄18 . The proof is based upon an adapted linear-nonlinear
decomposition of the solution.
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October 22: Rupert Frank (Princeton)
``The sharp constant in the Hardy-Sobolev-Maz'ya inequality.''
Abstract:
(Based on joint work with R. Benguria and M. Loss.)
We show that the sharp constant in the Hardy-Sobolev-Maz'ya
inequality on the three dimensional halfspace is given by the Sobolev
constant on the whole space. This is achieved by a duality argument
relating the problem to a Hardy-Littlewood-Sobolev-type inequality
whose sharp constant is determined as well. We discuss an equivalent
formulation in hyperbolic space and prove a similar result for the
Moser-Trudinger inequality in two dimensions.
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October 29: Ian Tice (Brown)
``Ginzburg Landau vortex dynamics driven by an applied boundary current.''
Abstract:
This talk concerns recent results on the time-dependent
Ginzburg-Landau equations on a smooth, bounded 2-D domain, subject to
both an applied magnetic field and an applied boundary current. The
boundary current is modeled with a gauge-invariant inhomogeneous Neumann
boundary condition. After a quick discussion of well-posedness, I will
present a result that controls the growth of the energy of the
solutions. I will then turn to the study of the dynamics of the vortices
of the solutions in the limit as the Ginzburg-Landau parameter vanishes.
In the original time scale, the vortices do not move, and the solutions
undergo a phase relaxation. In an accelerated time scale, the vortices
do move, and I derive their dynamical law, identifying a novel Lorentz
force term induced by the applied boundary current.
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November 5: Christian Hainzl (University of Alabama, Birmingham)
``Mathematical aspects of the Bardeen-Cooper-Schrieffer equation of superfluidity.''
Abstract:
Although the BCS-gap equation is highly non-linear, we are able to give a precise characterization of the interaction potentials
which give rise to a non-trivial solution, or in physics terms, a superfluid state. Moreover we evaluate the asymptotic behavior
of the critical temperature and the energy gap in the limit of small coupling, as well as in the small density limit. We thereby
improve the formulas from the physical literature. We achieve this by the precise spectral analysis of operators whose kinetic
symbol degenerates on a manifold of co-dimension one.
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November 12: Jill Pipher (Brown University)
``Absolute continuity of elliptic measure.''
Abstract:
In this talk we survey some methods for proving
quantitative absolute continuity of the measure
associated to elliptic second order operators in divergence
or nondivergence form. In particular, we are interested in
methods which work in the absence of L2 identities, and
yet imply solvability of some Dirichlet or Neumann boundary
value problems. I'll discuss applications to specific operators
whose coefficients are Carleson measures.
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November 19: Justin Holmer (Brown University)
``Soliton motion for the delta impurity model.''
Abstract:
We consider the nonlinear Schroedinger equation with delta potential i∂t u + ½ ∂x2 u - qδ0 u + |u|2u=0,
where the coupling constant q is small, and study the motion of soliton solutions, which takes place on a time scale q-1/2.
We also consider the dynamics of resolution of solutions toward the bound state, which takes place on a time scale O(1). Comparisons to numerics are presented.
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November 20: Bruce Driver (UCSD)
``Holomorphic functions and subelliptic heat kernels over Lie groups.''
Special date and time: Thursday 4-5 PM room 2-142
Abstract:
(Based on joint work with Laurent Saloff-Coste and Leonard Gross)
A Hermitian form q on the dual space, g*, of a Lie algebra, g,
of a Lie group, G, determines a Laplacian, Δ, on G.
Assuming Hörmander's condition for hypoellipticity, the subelliptic heat
semigroup, etΔ/4, is given by convolution by a C∞
probability density ρt. Analogous to earlier work in the strongly
elliptic case, we are able to show that if G is complex, connected, and
simply connected then the Taylor expansion defines a unitary map from the
space of holomorphic functions in L2 ( G, ρt ) onto (a
subspace of) the dual of the universal enveloping algebra in the norm induced
by q. This work is related to an extension of the bosonic Fock space to the
noncommutative Lie group setting.
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December 3: Vera Hur (MIT)
``The dispersive property of surface water waves.''
Canceled
Previous semesters:
Spring
Semester 2008
Fall
Semester 2007