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A parametrix construction for wave equations with C 1 , 1 coefficients

Hart F. Smith (1998)

Annales de l'institut Fourier

In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions n = 2 and n = 3 .

Continuité-Sobolev de certains opérateurs paradifférentiels.

Abdellah Youssfi (1990)

Revista Matemática Iberoamericana

L'objet de ce travail est l'étude de la continuité des opérateurs d'intégrales singulières (au sens de Calderón-Zygmund) sur les espaces de Sobolev Hs. Il complète le travail fondamental de David-Journé [6], concernant le cas s = 0, et ceux de P. G. Lemarié [10] et M. Meyer [11] concernant le cas 0 < s < 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...

Remarks on Gårding inequalities for differential operators

Xavier Saint Raymond (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Classical Gårding inequalities such as those of Hörmander, Hörmander-Melin or Fefferman-Phong are proved by pseudo-differential methods which do not allow to keep a good control on the supports of the functions under study nor on the smoothness of the coefficients of the operator. In this paper, we show by very simple calculations that in certain special situations, the results that can be obtained directly are much better than those expected thanks to the general theory.

Stability of hydrodynamic model for semiconductor

Massimiliano Daniele Rosini (2005)

Archivum Mathematicum

In this paper we study the stability of transonic strong shock solutions of the steady state one-dimensional unipolar hydrodynamic model for semiconductors in the isentropic case. The approach is based on the construction of a pseudo-local symmetrizer and on the paradifferential calculus with parameters, which combines the work of Bony-Meyer and the introduction of a large parameter.

Wave fronts of solutions of some classes of non-linear partial differential equations

P. Popivanov (1992)

Banach Center Publications

1. This paper is devoted to the study of wave fronts of solutions of first order symmetric systems of non-linear partial differential equations. A short communication was published in [4]. The microlocal point of view enables us to obtain more precise information concerning the smoothness of solutions of symmetric hyperbolic systems. Our main result is a generalization to the non-linear case of Theorem 1.1 of Ivriĭ [3]. The machinery of paradifferential operators introduced by Bony [1] together...

Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity

Boris Haspot (2012)

Annales de l’institut Fourier

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ℝ N with N ≥ 2 . We address the question of well-posedness for large and small initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where u 0 ∈ B p , r N p - 1 with 1 ≤ p < + ∞ , 1 ≤ r ≤ + ∞ ). This improves the classical analysis where u 0 is considered belonging in B p , 1 N p - 1 such that the velocity u remains...

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