The Calabi Flow with Rough Initial Data

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.

Stability and Experimental Comparison of Prototypical Iterative Schemes for Total Variation Regularized Problems

Various 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.

On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

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.