rough initial data
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Author(s):  
Weiyong He ◽  
Yu Zeng

Abstract In this paper, we prove that there exists a dimensional constant $\delta> 0$ such that given any background Kähler metric $\omega $, the Calabi flow with initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in L^\infty (M)\ \textrm{and}\ (1- \delta )\omega < \omega_{u_0} < (1+\delta )\omega, \end{equation*}admits a unique short-time solution, and it becomes smooth immediately, where $\omega _{u_0}: = \omega +\sqrt{-1}\partial \bar \partial u_0$. The existence time depends on initial data $u_0$ and the metric $\omega $. As a corollary, we get that the Calabi flow has short-time existence for any initial data satisfying \begin{equation*} \partial \bar \partial u_0 \in C^0(M)\ \textrm{and}\ \omega_{u_0}> 0, \end{equation*}which should be interpreted as a “continuous Kähler metric”. A main technical ingredient is a new Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time-weighted Hölder norms.





2016 ◽  
Vol 16 (3) ◽  
pp. 361-388 ◽  
Author(s):  
Sören Bartels ◽  
Marijo Milicevic

AbstractVarious iterative methods are available for the approximate solution of non-smooth minimization problems. For a popular non-smooth minimization problem arising in image processing, we discuss the suitable application of three prototypical methods and their stability. The methods are compared experimentally with a focus on choice of stopping criteria, influence of rough initial data, step sizes as well as mesh sizes. An overview of existing algorithms is given.



2016 ◽  
Vol 59 (7) ◽  
pp. 1345-1366 ◽  
Author(s):  
ChunYan Huang ◽  
BoLing Guo ◽  
DaiWen Huang ◽  
QiaoXin Li


2015 ◽  
Vol 430 (2) ◽  
pp. 777-809 ◽  
Author(s):  
Zineb Hassainia ◽  
Taoufik Hmidi


2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Chunyan Huang

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation(1/2⩽α⩽1)in any spatial dimensionn⩾1with rough initial data. For1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spacesMp,11-2α(1⩽p⩽∞). Forα=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data inB˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroupe-a+it-Δαto overcome the derivative in the nonlinear term.



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