P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ${ℝ}^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\psi \left(x\right)=(x,y_1\left(x\right)-y_2\left(x\right))$, $x\in [0,\alpha ]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha ]$. In other words: singularities propagate along arcs with finite turn.

In Albano-Cannarsa [1] the authors proved that, under some conditions, the singularities of the semiconcave viscosity solutions of the Hamilton-Jacobi equation propagate along generalized characteristics. In this note we will provide a simple proof of this interesting result.

This article is devoted to incompressible Euler equations (or to Navier-Stokes equations in the vanishing viscosity limit). It describes the propagation of quasi-singularities. The underlying phenomena are consistent with the notion of a cascade of energy.

For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the...

Significant information about the topology of a bounded domain $\Omega$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial \Omega }$, from the boundary of $\Omega$. We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial \Omega }$, as well as applications to homotopy equivalence.

We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $\Omega \subset {ℝ}^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m,\Theta }$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,\Theta }$. Under additional assumptions $P_m$ must be locally hyperbolic.

Existence and regularity theorems for Fuchsian type differential operators and the theory of second microlocalization are presented.

The nonlinear dissipative wave equation $u_{tt}-\Delta u+{\left|u_t\right|}^{h-1}{u}_t=0$ in dimension $d\>1$ has strong solutions with the following structure. In $0\le t\<1$ the solutions have a focusing wave of singularity on the incoming light cone $\left|x\right|=1-t$. In $\{t\ge 1\}$ that is after the focusing time, they are smoother than they were in $\{0\le t\<1\}$. The examples are radial and piecewise smooth in $\{0\le t\<1\}$

Nous construisons un calcul paradifférentiel adapté à l'équation de Schrödinger qui nous permet de montrer un théorème de propagation des singularités pour l'équation de Schrödinger non linéaire en adaptant la méthode de Bony. Nous construisons également la version tangentielle du calcul précédent qui nous permet de montrer un théorème de réflexion transverse des singularités pour l'équation de Schrödinger non linéaire. Nous utilisons alors ce théorème pour calculer l'opérateur...