On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.     

Emergent dynamics of Cucker–Smale particles under the effects of random communication and incompressible fluids

We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in ${\mathbb{R}}^d$$(d=2,3)$. Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in ${\mathbb{R}}^2$. In a large coupling regime and periodic spatial domain ${\mathbb{T}}^2:={\mathbb{R}}^2/{\mathbb{Z}}^2$, we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space $H_{\rho }^1({\mathbb{R}}^d)$. For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^k)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})$. This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.     

An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equations (NLS) on ${\mathbb{R}}^d$ with random and rough initial data. By working in the framework of $L^p({\mathbb{R}}^d)$ spaces, $p>2$, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.