Analyticity for the (generalized) Navier-Stokes equations with rough initial data

We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in critical Besov spaces $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$ with $1< p<\infty, \ 1\le q\le \infty$ and the solution is global if $u_0$ is sufficiently small in $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$. In the case $p=\infty$, the analyticity for the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in modulation space $M^{-1}_{\infty,1}(\mathbb{R}^n)$ is obtained. We prove the global well-posedness for a fractional Navier-stokes equation ($\alpha=1/2$) with small data in critical Besov spaces $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p\leq\infty)$ and show the analyticity of solutions with small initial data either in $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p<\infty)$ or in $\dot{B}^0_{\infty,1} (\mathbb{R}^n)\cap {M}^0_{\infty,1}(\mathbb{R}^n)$. Similar results also hold for all $\alpha\in (1/2,1)$.Comment: 31 page

Long Trend Dynamics in Social Media

A main characteristic of social media is that its diverse content, copiously generated by both standard outlets and general users, constantly competes for the scarce attention of large audiences. Out of this flood of information some topics manage to get enough attention to become the most popular ones and thus to be prominently displayed as trends. Equally important, some of these trends persist long enough so as to shape part of the social agenda. How this happens is the focus of this paper. By introducing a stochastic dynamical model that takes into account the user's repeated involvement with given topics, we can predict the distribution of trend durations as well as the thresholds in popularity that lead to their emergence within social media. Detailed measurements of datasets from Twitter confirm the validity of the model and its predictions