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A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in 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) ψ ( x ) = ( x , y 1 ( x ) - y 2 ( x ) ) , x [ 0 , α ] , where y 1 , y 2 are convex and Lipschitz on [ 0 , α ] . In other words: singularities propagate along arcs with finite turn.

Cascade of phases in turbulent flows

Christophe Cheverry (2006)

Bulletin de la Société Mathématique de France

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.

Dynamics of wave propagation and curvature of discriminants

Victor P. Palamodov (2000)

Annales de l'institut Fourier

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...

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

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

Global controllability and stabilization for the nonlinear Schrödinger equation on an interval

Camille Laurent (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Localizations of partial differential operators and surjectivity on real analytic functions

Michael Langenbruch (2000)

Studia Mathematica

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 Ω 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 , Θ of the principal part P m is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for P m , Θ . Under additional assumptions P m must be locally hyperbolic.

Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc Joly, Guy Métivier, Jeffrey Rauch (1998/1999)

Séminaire Équations aux dérivées partielles

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

Propagation et réflexion des singularités pour l'équation de Schrödinger non linéaire

Jérémie Szeftel (2005)

Annales de l’institut Fourier

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...

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