Schedule:
8:30-8:45
COFFEE
MORNING SESSION Chair: Hailiang Liu
8:45-9:30 Weimin Han
Mathematical Theory and Numerical Analysis for Bioluminescence Tomography
9:30-10:15 Dong Li
On semilinear wave equations and Klein-Gordon equations with
critical nonlinearities
10:15
- 10:30 COFFEE BREAK
10:30-11:30 James Ralston
The determination of moving boundaries
by reflected waves
11:30-12:15 Gary Lieberman
New results for solutions of some degenerate elliptic equations
12:15-2:00 LUNCH
AFTERNOON SESSION Chair: Paul
Sacks
2:00-2:45 Hailiang Liu
Global recovery of high frequency wave fields
2:45-3:30 Diego Moreira
Least
super-solution method for free boundary problems of flame propagation type
3:30
- 3:45 COFFEE BREAK
3:45-4:30 Xiaoyi Zhang
Regularity of almost periodic modulo scaling solutions for mass-critical
NLS and applications
4:40-5:15 Jue Yan
Discontinuous Galerkin Method for convection
diffusion equations and Hamilton-Jacobi equations
Abstracts:
James Ralston
The determination of moving boundaries
by reflected waves
In recent work with Gregory Eskin I have been studying the (theoretical)
possibility of determining a (subsonic) moving
boundary from the Cauchy data of waves on a distant fixed surface. We
have some positive results, but there are interesting obstructions to
determining moving boundaries this way. These were discovered by Stefanov in 1992.
Weimin Han
Mathematical Theory and Numerical Analysis for Bioluminescence Tomography
Over the last couple of years
molecular imaging has been rapidly developed to study physiological and
pathological processes in vivo at the cellular and molecular levels.
Among molecular imaging modalities, optical imaging stands out for its unique
advantages, especially performance and cost-effectiveness.
Bioluminescence tomography (BLT) is an emerging optical imaging mode with
promising biomedical advantages. In this talk, we explain the biomedical
significance of BLT, present results on theoretical and numerical analysis of
BLT models.
Dong Li
On semilinear wave equations and Klein-Gordon equations with
critical nonlinearities
I will explain
recent results on the construction of wave operators for a large class of semilinear wave equations and Klein-Gordon equations with critical
nonlinearities. This is joint work with Juhi Jang and
Xiaoyi Zhang.
Gary Lieberman
New results for solutions of some degenerate elliptic equations
Hailiang Liu
Global recovery of high frequency wave
fields
Computation of high frequency
solutions to wave equations is important in many applications, and notoriously
difficult in resolving wave oscillations. Gaussian beams are asymptotically
valid high frequency solutions concentrated on a single curve through the
physical domain, and superposition of Gaussian beams provides a powerful tool
to generate more general high frequency solutions to PDEs. An alternative way
to compute Gaussian beam components such as phase, amplitude and Hessian of the
phase, is to capture them in phase space by solving Liouville
type equations on uniform grids. In this talk I shall present a systematic
construction of asymptotic high frequency wave fields from computations in phase
space for acoustic wave equations (also for the Schrödinger equation). The k-th order Gaussian beam superposition is shown to converge
to the original wave field in the energy norm, at an optimal rate in arbitrary
spatial dimension. This is in collaboration with James Ralston (UCLA).
Diego Moreira
Least super-solution method for free boundary problems of flame propagation type
Regularizing methods in free boundary problems are models for a wide spectrum of problems in nature. They are of particular interest in the theory of flame propagation to
describe the behavior of laminar flames as an asymptotic limit for high energy activation. These methods go back to Zeldovich and Frank-Kamenetski in late 30’s. A
rigorous mathematical treatment was developed in the pioneering works of Caffarelli-Beresticky-Nirenberg (1989) and Caffarelli-Vazquez (1995).
In the last two decades, several
authors studied the free boundary problem arising as the limit as $\epsilon \to 0$ of the following family of nonlinear
elliptic equations: $\Delta u = \beta_\epsilon(u)$,
where$\beta_\epsilon$ is an approximation of the Dirac delta concentrated
at zero. It is known that under certain geometrical conditions on the limit
function, this function becomes a viscosity solution of an elliptic free
boundary problem and the free boundary is a $C^{1,
\alpha}$ surface. Some counter-examples show that these assumptions are
necessary if one intends to obtain further regularity of the free boundary.
In this lecture, we will dicuss a more general class
of free boundary problems obtained as a limit as $ \epsilon \to 0$ to the
following regularizing family of semi-linear equations $\Delta u = \beta_\epsilonF(\nabla u)$, where $F$ is a Lipschitz
function bounded away from $0$ and infinity. The least super-solution approach
is used to construct solutions satisfying geometric properties of the level
surfaces that are uniform in $\epsilon$. This allows to prove that the free
boundary of the limit has the ”right” weak geometry,
in the measure theoretical sense. The classification of the global solutions
appearing in the blow-up analysis is performed by introducing a rather
geometric method, reminiscent of Alexandrov maximum
principle, where moving planes are replaced by barriers with uniformly curved
free boundaries (constructed by using Kelvin transforms in larger and larger
spheres whose centers and radii go to infinity). Finally, the limit function is
proven to be a viscosity and pointwise solution ($H^{n-1} a.e$) to an elliptic
free boundary problem and the free boundary is proven to be a $C^{1, \alpha}$-
surface around $H^{n-1} a.e.$ point.
Xiaoyi Zhang
Regularity of almost periodic modulo
scaling solutions for mass-critical NLS and applications
In this talk, I will introduce our
recent work joint with Dong Li. We consider the $L_x^2$ solution $u$ to mass
critical NLS $iu_t+\Delta u=\pm |u|^{\frac 4d} u$. We proved that in dimensions $d\ge 4$, if the solution is spherically symmetric and is \emph{almost periodic modulo scaling}, then it must lie in $H_x^{1+\eps}$
for some $\eps>0$. Moreover, the kinetic energy of
the solution is localized uniformly in time. One important application of the
theorem is a simplified proof of the scattering conjecture for mass critical
NLS without reducing to three enemies \cite{ktv:2d}, \cite{kvz:blowup}. As another important application, we establish
a Liouville type result for $L_x^2$ initial data with
ground state mass. We prove that if a radial $L_x^2$ solution to focusing mass
critical problem has the ground state mass and does not scatter in both time
directions, then it must be global and coincide with the solitary wave up to
symmetries. Here the ground state is the unique, positive, radial solution to
elliptic equation $\Delta Q-Q+Q^{1+\frac 4d}=0$. This is the first rigidity type result in
scale invariant space $L_x^2$.
Jue
Yan
Discontinuous Galerkin Method for convection
diffusion equations and Hamilton-Jacobi equations
We will first introduce the finite
element discontinous Galerkin
method: the general scheme setup, formulation and its advantages as a
numerical method solving PDEs. Then we will talk about our recent development
on a class of new DG method to directly solve convection diffusion equations.
We will discuss the design of the numerical flux formula and show the optimal
convergence of the numerical solution to a series of linear and nonlinear
problems. We will also present a new DG method solving Hamilton-Jacobi
equations. Our scheme can sharply capture the kinks with discontinuous
derivatives and converge to the entropy solution for both convex and non-convex
Hamiltonian.