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 $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$.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 $\Bbb R^d$ with $d\ge 3$ 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 $d=2$ was solved. In the first part of this paper, for heat flows from $\Bbb R^d$ ($d\ge 3$) 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 $t$ 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 $\Bbb H^2$ to $\Bbb H^2$ 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 $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p\le1+\frac{4}{d-2}$, $d\ge3$. The index $1+\frac{2}{d}$ 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 $f_i$'s. Recently, a degree theory approach to shooting method for this broad class of problems is introduced in \cite{LiarXiv13} for nonnegative $f_i$'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 R2\Bbb R^2 to K\"ahler manifolds

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from $\Bbb R^2$ 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 $t\to \infty$ 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 $4\pi$ 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