6,067 research outputs found
Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces
In this paper, we prove that Kato smoothing effects for magnetic
Schr\"odinger operators can yield the endpoint Strichartz estimates for linear
wave equation with magnetic potential on two dimensional hyperbolic spaces.
This result serves as a cornerstone for the author's work \cite{Lize} and
collaborative work \cite{LMZ} in the study of asymptotic stability of harmonic
maps for wave maps from to .Comment: revised and enlarge
Global Schr\"odinger map flows to K\"ahler manifolds with small data in critical Sobolev spaces: High dimensions
In this paper, we prove that the Schr\"odinger map flows from with
to compact K\"ahler manifolds with small initial data in critical
Sobolev spaces are global. This is a companion work of our previous paper [23]
where the energy critical case was solved. In the first part of this
paper, for heat flows from () to Riemannian manifolds with
small data in critical Sobolev spaces, we prove the decay estimates of moving
frame dependent quantities in the caloric gauge setting, which is of
independent interest and may be applied to other problems. In the second part,
with a key bootstrap-iteration scheme in our previous work [23], we apply these
decay estimates to the study of Schr\"odinger map flows by choosing caloric
gauge. This work with our previous work solves the open problem raised by
Tataru.Comment: slightly enlarged, submitte
Asymptotic stability of solitons to 1D Nonlinear Schrodinger Equations in subcritical case
In this paper, we prove the asymptotic stability of solitary waves to 1D
nonlinear Schr\"odinger equations in the subcritical case with symmetry and
spectrum assumptions. One of the main ideas is to use the vector fields method
developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with
respect to of the linearized equation caused by the one dimension setting
and the weak nonlinearity caused by the subcritical growth of the nonlinearity
term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of
solutions to 1D Schr\"odinger equations obtained by Staffilani to control the
high moments of the solutions emerging from the vector fields method
Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces
In this paper, we prove that the large energy harmonic maps from
to are asymptotically stable under the wave map equation.Comment: We improve the expressions, some improper notations are removed,
grammer errors and typos are corrected, more background materials are
involve
Comparisons of Reasoning Mechanisms for Computer Vision
An evidential reasoning mechanism based on the Dempster-Shafer theory of
evidence is introduced. Its performance in real-world image analysis is
compared with other mechanisms based on the Bayesian formalism and a simple
weight combination method.Comment: Appears in Proceedings of the Third Conference on Uncertainty in
Artificial Intelligence (UAI1987
Decay and scattering of solutions to nonlinear Schr\"odinger equations with regular potentials for nonlinearities of sharp growth
In this paper, we prove the decay and scattering in the energy space for
nonlinear Schr\"odinger equations with regular potentials in namely,
. We will prove
decay estimate and scattering of the solution in the small data case when
, . The index is
sharp for scattering concerning the result of W. Strauss [21].Comment: 22 page
Shooting Method with Sign-Changing Nonlinearity
In this paper, we study the existence of solution to a nonlinear system:
\begin{align}
\left\{\begin{array}{cl}
-\Delta u_{i} = f_{i}(u) & \text{in } \mathbb{R}^n,
u_{i} > 0 & \text{in } \mathbb{R}^n, \, i = 1, 2,\cdots, L
% u_{i}(x) \rightarrow 0 & \text{uniformly as } |x| \rightarrow \infty
\end{array}
\right. \end{align} for sign changing nonlinearities 's. Recently, a
degree theory approach to shooting method for this broad class of problems is
introduced in \cite{LiarXiv13} for nonnegative 's. However, many systems
of nonlinear Sch\"odinger type involve interaction with undetermined sign.
Here, based on some new dynamic estimates, we are able to extend the degree
theory approach to systems with sign-changing source terms
Asymptotic behaviors of Landau-Lifshitz flows from to K\"ahler manifolds
In this paper, we study the asymptotic behaviors of finite energy solutions
to the Landau-Lifshitz flows from into K\"ahler manifolds. First, we
prove that the solution with initial data below the critical energy converges
to a constant map in the energy space as for the compact
Riemannian surface targets. In particular, when the target is a two dimensional
sphere, we prove that the solution to the Landau-Lifshitz-Gilbert equation with
initial data having an energy below converges to some constant map in
the energy space. Second, for general compact K\"ahler manifolds and initial
data of an arbitrary finite energy, we obtain a bubbling theorem analogous to
the Struwe's results on the heat flows.Comment: This version improves the results in original Theorem 1.2 by
including the positive curvature cas
Asymptotic Stability of Solitons to Nonlinear Schrodinger Equations on Star Graphs
In this paper, we prove the asymptotic stability of nonlinear Schrodiger
equations on star graphs, which partially solves an open problem in D. Noja
\cite{DN}. The essential ingredient of our proof is the dispersive estimate for
the linearized operator around the soliton with Kirchhoff boundary condition.
In order to obtain the dispersive estimates, we use the Born's series technique
and scattering theory for the linearized operator.Comment: 2 figures, 33 page
An Extended Discrete Hardy-Littlewood-Sobolev Inequality
Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case:
\mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality
with logarithm correction for a critical case: \mu=n and p=q, by limiting the
inequality on a finite domain. The best constant in the inequality and its
corresponding solution, the optimizer, are studied. First, we obtain a sharp
estimate for the best constant. Then for the optimizer, we prove the uniqueness
and a symmetry property. This is achieved by proving that the corresponding
Euler-Lagrange equation has a unique nontrivial nonnegative critical point.
Also, by using a discrete version of maximum principle, we prove certain
monotonicity of this optimizer
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