Calcul symbolique et singularités des solutions des équations aux dérivées partielles non linéaires
Calcul symbolique non linéaire pour une onde conormale simple
On considère une solution , assez régulière, d’une équation aux dérivées partielles non linéaire. Si est conormale par rapport a une hypersurface simplement caractéristique pour l’équation linéarisée, on étudie l’équation de transport satisfaite par son symbole principal, et on en déduit la propagation de la propriété “ est conormale classique”.
Characterization of Jacobian varieties in terms of soliton equations.
Classical Solutions of the Chain Equation. I.
Colombeau's theory and shock wave solutions for systems of PDEs.
Complex methods in non-linear analysis
Conservation laws for the nonlinear Schrödinger equation
Construction de solutions singulières pour des équations aux dérivées partielles non linéaires
Construction solutions of PDE in parametric form.
Convergence of power series along vector fields and their commutators; a Cartan-Kähler type theorem
We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.
Convex integration of non-linear systems of partial differential equations
Geometrical techniques are employed to prove a global existence theorem for -solutions to underdetermined systems of non-linear order partial differential equations, , which satisfy certain convexity conditions. The solutions are not unique, but satisfy given approximations on lower order derivatives. The main result, which includes the relative case generalizes the work of M. Gromov on non-linear first order systems.
Convolution equations in the space of Laplace distributions
A formal solution of a nonlinear equation P(D)u = g(u) in 2 variables is constructed using the Laplace transformation and a convolution equation. We assume some conditions on the characteristic set Char P.