The 13th Summer School and International Conference on Nonlinear Partial...
Transcript of The 13th Summer School and International Conference on Nonlinear Partial...
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The 13th Summer School and International Conference on
Nonlinear Partial Differential Equations
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The 13th Summer School and International Conference on
Nonlinear Partial Differential Equations
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Á ²§¥ì�ÆThe Ricci Flow and Its Geometric Applications
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¸�§�®�Æ!�1�ÆRemainder estimates and convergence for boundary expansionsfor Loewner-Nirenberg problem
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±Á§E��ÆStructure of Helicity and Global Solutions of IncompressibleNavier-Stokes Equations
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=7g§�®���ÆSchauder estimates for fully nonlinear nonlocal equations
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¢�§¥I<¬�Æ!�²�²á�ÆCross-diffusion models in population dynamics
6401
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o¿§¥I�Æ�Large-Time Behavior of Solutions to One-Dimensional Com-pressible Navier-Stokes System in Unbounded Domains withLarge Data
6401
15:55-16:15 � �
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�À§{Iâ��ÆConformal metrics in R2m with constant Q-curvature and ar-bitrary volume
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pl§Iá���ÆI[�ÆnØïÄ¥%Vortices of a modified Ginzburg-Landau model
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'ŧ�²�²á�ÆEstimates for fully nonlinear elliptic and parabolic equations
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¨¡§¥=Jyvaskyla�ÆOn the Euler-Lagrange equation of a functional by Polya andSzego
6401
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Ù��§�®�ÆWell-posedness and break-down criterion for the incompress-ible Euler equations with free boundary
6401
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o°f§�®���ÆGradient estimates and asymptotics for elliptic equations fromcomposite materials
6401
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Wellposedness for the non-isothermal model of the nematic Liq-
uid crystals
6401
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Úåå§oA�ÆOn impinging flows and impinging jet flows
6401
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X�§E��ÆGlobal Wellposedness of 2D Incompressible Elastodynamics
6401
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ÛU©§�l¥©�ÆNon-uniqueness of Admissible Weak Solutions to CompressibleEuler Systems with Source Terms
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Convergence to an equilibrium of nonlinear parabolic equations
on RN
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Non perturbation methods and Sobolev norms estimates for
Klein-Gordon equations
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�Sµ§þ°�Ï�ÆSteady Euler flows with physical boundaries
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Wellposedness for the non-isothermal model of the nematic
Liquid crystals
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In this talk, we establish the local and global existence and uniqueness of strong
solution for the non-isothermal model of the nematic Liquid crystals.
On impinging flows and impinging jet flows
Úåå§oA�Æ
In this talk, we will discuss some recent results on steady impinging flows and
impinging jet flows. First, we will introduce the existence and uniqueness of the com-
pressible subsonic impinging flows. And then some recent results on incompressible
oblique impinging jet flows will be also presented. Finally, we will give the existence
and uniqueness results on incompressible impinging jet in ideal rotational flows.
Convergence to an equilibrium of nonlinear parabolic
equations on RN
Ú�÷§e�|æ#=�=�Æ
We consider the Cauchy problem
ut −∆u = f(u)(x ∈ RN , t > 0), u(0, x) = u0(x)(x ∈ RN),
where u0 is a nonnegative function with compact support, and f is a smooth function
satisfying f(0) = 0. A well-known open question is whether any globally bounded
solution u of this problem must converge to a stationary solution as t goes to infinity.
If the space dimension N = 1, a positive unswer was given by Manato and the speaker
in 2010.
In this talk I will report a recent result obtained by Polacik and the speaker,
which gives a positive answer to this question for any N ≥ 2, under the following
extra condition: all the nonnegative zeros of f are nondegenerate (i.e., f(u) = 0 and
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u ≥ 0 imply f ′(u) 6= 0).
Non perturbation methods and Sobolev norms estimates for
Klein-Gordon equations
���§úô�Æ
In this talk I will consider a time periodic Klein-Gordon equation with periodic
boundary conditions. Based on the nonperturbative methods similar to those in
Bourgain and so on, we are interested in controlling the possible growth of Sobolev
norms of solution to equation. This is a joint work with Zheng Han and Wei-Min
Wang.
Vortices of a modified Ginzburg-Landau model
pl§Iá���ÆI[�ÆnØïÄ¥%
In this talk, we consider the vortices of a modified Ginzburg-Landau model,
especially the radial solution with given degree. We prove existence, quantization
result, monotonicity property and asymptotic behavior of solution as r tends to
infinity.
Estimates for fully nonlinear elliptic and parabolic equations
'ŧ�²�²á�Æ
Establishing a priori C2 estimates is a central issue in the study of fully nonlin-
ear second order equations. Such estimates are also of fundamental importance in
applications. In this talk we present an overview of techniques developed over the
past decades since the groundbreaking work of Caffarelli, Nirenberg and Spruck in
1985, followed by a report recent progresses. We shall?focus on second derivative
estimates for equations on real or complex manifolds.
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Remainder estimates and convergence for boundary
expansions for Loewner-Nirenberg problem
¸�§�®�Æ!�1�Æ
In 1974, Loewner and Nirenberg discussed a class of semilinear elliptic equations
in a bounded domain with solutions blowing up on boundary. In this talk, we discuss
a recent result of the optimal boundary expansions for its solutions in the context of
the finite regularity. We also discuss the convergence of the expansion under extra
assumptions.
Global Wellposedness of 2D Incompressible Elastodynamics
X�§E��Æ
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Gradient estimates and asymptotics for elliptic equations
from composite materials
o°f§�®���Æ
Recently there is growing interest in partial differential equations with high con-
trast coefficients arising from the study of composite materials. In this talk, we
consider the Lame system with partially infinite coefficients which models a linear
isotropic elastic body containing two adjoint inclusions with infinity elastic parame-
ters.
As the distance ε between the surfaces of discontinuity of the coefficients of the
system tends to zero, we establish upper bounds on the blow up rate of the gradients
of solutions. Especially, for strictly convex inclusions, we show the upper bound
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is ε−1/2, which is expected to be optimal, since the optimal bound for anti-plane
elasticity is known to be ε−1/2 and some numerical results support this. For the
simplified model—-equation cases, we further obtain the asymptotic expression for
the gradient of solutions. This talk is based on joint work with Professor Jiguang
Bao and Yanyan Li.
Large-Time Behavior of Solutions to One-Dimensional
Compressible Navier-Stokes System in Unbounded Domains
with Large Data
o¿§¥I�Æ�
We are concerned with the large-time behavior of solutions to the initial and
initial boundary value problems with large initial data for the compressible Navier-
Stokes system describing the one-dimensional motion of a viscous heat-conducting
perfect polytropic gas in unbounded domains. The temperature is proved to be
bounded from below and above independently of both time and space. Moreover, it
is shown that the global solution is asymptotically stable as time tends to infinity.
Note that the initial data can be arbitrarily large. This result is proved by using
elementary energy methods.
Condensate Development in a Model of Photon Scattering
4°�§OÖu²á�Æ
We study long-time dynamics in a model of the Kompaneets equation. The
Kompaneets equation describes evolution of photon energy spectrum due to Compton
scattering of photons by electrons, an important energy transport mechanism in
certain plasmas. For our model, we prove global existence for initial data with any
finite moment, convergence to equilibrium in large time, and failure to conserve
photon number for large solutions, due to formation of a shock at zero energy. This
is joint work with Dave Levermore (UMD) and Robert Pego (CMU).
u¥���Æ 1133��5 �©�§ÛÏùS�9ISÆâ¬Æ 19�
Cross-diffusion models in population dynamics
¢�§¥I<¬�Æ!�²�²á�Æ
Cross-diffusion system is an important class of reaction-diffusion problems. At
the individual level, the basic underlying assumption for cross-diffusion is that the
transition probability only depends upon departure conditions, e.g., population den-
sity and environmental condition at the departure location. We will discuss the
Shigesada-Kawasaki-Teromoto model for two competing species. This talk is based
on joint works with Wei-Ming Ni, Michael Winkler, Shoji Yotsutani.
Non-uniqueness of Admissible Weak Solutions to
Compressible Euler Systems with Source Terms
ÛU©§�l¥©�Æ
In this talk, we will present the recent results on the non-uniqueness of admissi-
ble weak solutions to the compressible Euler system with source terms, which include
rotating shallow water system and the Euler system with damping as special exam-
ples. In particular, we construct a class of finite-states admissible weak solutions
when the source term is anti-symmetric such as rotations. This is a joint work with
Prof. Chunjing Xie and Prof. Zhouping Xin.
Steady Euler flows with physical boundaries
�Sµ§þ°�Ï�Æ
In this talk, we discuss the steady Euler flows past a wall or through a nozzle.
The focus in on the subsonic flows with large vorticity.
Schauder estimates for fully nonlinear nonlocal equations
=7g§�®���Æ
The Schauder estimate is one of the most important regularity result for elliptic
partial differential equations. In 1989, L. Caffarelli introduced a nonlinear perturba-
tion method and established the Schauder estimate for fully nonlinear second order
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elliptic PDEs. In this talk, we will talk about a nonlocal analogue of Caffarelli.
s result, which answers a remainder open problem after a series of recent work of
Caffarelli-Silvestre on fully nonlinear integro-differential equations. This is joint work
with T. Jin.
Conformal metrics in R2m with constant Q-curvature and
arbitrary volume
�À§{Iâ��Æ
We study the polyharmonic problem ∆mu = ±eu in R2m, with m ≥ 2. In
particular, we prove that for any V > 0, there exist radial solutions of ∆mu = −eu
such that ∫R2m
eudx = V.
It implies that for m odd, given any Q0 > 0 and arbitrary volume V > 0, there exist
conformal metrics g on R2m with constant Q-curvature equal to Q0 and vol(g) = V .
This answers some open questions.
Well-posedness and break-down criterion for the
incompressible Euler equations with free boundary
Ù��§�®�Æ
In this talk, I will first review some recent well-posedness results on the water-
wave problem. Then I will talk about our well-posedness result in the low regularity
Sobolev space and a break-down criterion in terms of the mean curvature, the gradient
of the velocity and the Taylor sign stability condition.
On the Euler-Lagrange equation of a functional by Polya and
Szego
¨¡§¥=Jyvaskyla�Æ
I will talk about a conjeture of Polya and Szego on minimal electrostatic capac-
ity sets in convex shape optimization. The functional, associated to the conjecture,
u¥���Æ 1133��5 �©�§ÛÏùS�9ISÆâ¬Æ 111�
involves capacity and perimeter. The talk focus on the regularity properties of gen-
eralized solutions of the corresponding Euler-lagrange equation. This is joint work
with Nicola Fusco.
Structure of Helicity and Global Solutions of Incompressible
Navier-Stokes Equations
±Á§E��Æ
In this paper we construct a family of finite energy smooth solutions to the
threedimensional incompressible Navier-Stokes equations. The initial data are con-
structed so that the velocity fields are almost parallel to the corresponding vorticity
fields in a very large portion of the spatial domain, which can be viewed as pertur-
bations from steady Beltrami flows. The choices of such initial data are motivated
by an observation on the structure of the helicity defined by the solutions to the 3D
incompressible Euler and Navier-Stokes equations. The latter may be of independent
interest.
The Ricci Flow and Its Geometric Applications
Á ²§¥ì�Æ
The Ricci flow is a useful tool to understand the geometry and topology of
manifolds. In this talk, I will give a survey to the theory and report some recent ap-
plications to the classification of four-dimensional manifolds with positive curvature.
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