Phase Space Analysis of Partial Differential Equations

Members:
Jean-Michel Bony, Ecole Polytechnique, Paris
Jean-Yves Chemin, Université de Paris 6
Ferruccio Colombini, Università di Pisa
Isabelle Gallagher, Université de Paris 7
Herbert Koch, Universität Bonn
Gilles Lebeau, Université de Nice Sophia-Antipolis
Guy Métivier, Université de Bordeaux 1
Ludovico Pernazza, Università di Pavia
Daniel Tataru, University of California, Berkeley
Enrique Zuazua, Universidad Autónoma de Madrid

Research interests:
Partial Differential Equations, PDE, is a well established field of research in Mathematical Analysis. It grew over a couple of centuries to be one of the richest domains of Mathematics. Major progresses have been accomplished in the analysis of PDE's during the last twenty years, both from theoretical and applied points of view. Many of them are based on the development of powerful tools microlocal analysis. The idea, at the crossing point of harmonic analysis, functional analysis, quantum mechanics and algebraic analysis, is that many phenomena depend jointly on position and frequency and therefore must be understood and described in the phase space. The introduction of phase space analysis in the seventies put in evidence a number of unifying factors between traditionally diverse sectors, like e.g. hyperbolic and (degenerate) elliptic equations; moreover the theory has been geometrized, putting in evidence its invariance character. The basic tools that have been developed, with important contributions of the applicants, concern the theories of pseudodifferential calculus, Fourier-Bros-Iagolnizter transform, Fourier integral operators, Weyl's calculus and other quantizations and refinements based on Littlewood-Paley decompositions; extension to non smooth coefficients and nonlinear equations have been made through para-differential calculi. Many problems which are central to this proposal originate from physics and other applications. Just to cite a few, where members of the group are very active, we may mention Euler, Navier-Stokes and related equations from fluid mechanics, both in the incompressible case (global solutions, singularities, rotating fluids, water waves, etc) and for compressible fluids (low Mach number flows, shocks, etc). Maxwell's equations from electromagnetism and field equations are basic models for the theory of hyperbolic equations, including now nonlinear problems (global solutions, decay, stability, shock formation, etc). Nonlinear optics leads to various models (geometric optics, non linear Schrödinger equations, Zakharov systems and other dispersive equations). The control of systems governed by PDE's (observability, stabilization, exact controllability) is an important and very active domain; strictly related is the study of fine properties of solutions of PDE with non-smooth coefficients (existence of solutions to the Cauchy problem, unique continuation, observability), a topic with important and immediate applications in inverse problems and control theory. Local solvability and its dual problem, unique continuation, strongly profits from a microlocal point of view, which actually has made possible the solution of Nirenberg-Treves conjecture. Furthermore its microlocal aspect is made quite evident when local solvability for operators of nonprincipal type is considered. A particularly interesting problem in Microlocal Analysis and in the applications to PDE's is the one of finding lower bounds for pseudodifferential operators whose symbol is non-negative.

Activities:
Regular seminars on the previous subjects
2-days workshops, tentatively 1 each year, on some particular aspect of the general theme
2-weeks to 2-month research stays of one or more member(s) of the group or of other researchers in the field
Tutoring activity of post-doc students at the Centro De Giorgi working on the subject

Some related publications:
1) Dehman, Belhassen; Lebeau, Gilles; Zuazua, Enrique, Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. (4) 36 (2003), 525-551.
2) Lebeau, Gilles, Régularit du problème de Kelvin-Helmholtz pour l'équation d'Euler 2d. A tribute to J. L. Lions. ESAIM Control Optim. Calc. Var. 8 (2002), 801-825.
3) Croisille, Jean-Pierre; Lebeau, Gilles, Diffraction by an immersed elastic wedge. Lecture Notes in Mathematics, 1723. Springer-Verlag, Berlin, 1999. vi+134 pp.
4) Métivier, Guy; Zumbrun, Kevin, Large Viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175 (2005), vi+107 pp.
5) Guès, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin, Multidimensional viscous shocks II: the small viscosity limit. Comm. Pure and Armander. Bull. Soc. Math. France 122 (1994), 77-118. 6) Guès, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal. 175 (2005), 151-244.
7) Bony, Jean-Michel, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. école Norm. Sup. (4) 14 (1981), 209-246. 8) Bony, Jean-Michel; Chemin, Jean-Yves, Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122 (1994), 77-118.
9) Bahouri, Hajer; Chemin, Jean-Yves, Microlocal analysis, bilinear estimates and cubic quasilinear wave equation; in: Autour de l'analyse microlocale. Astérisque 230 (1995), 93-141.
10) Chemin, Jean-Yves, Fluides parfaits incompressibles. Astérisque 230 (1995), 177 pp.
11) Koch, Herbert; Tataru, Daniel, Sharp counterexamples in unique continuation for second order elliptic equations. J. Reine Angew. Math. 542 (2002), 133-146.
12) Koch, Herbert; Tataru, Daniel, Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58 (2005), 217-284.
13) Koch, Herbert; Tzvetkov, Nicolay, Nonlinear wave interactions for the Benjamin-Ono equation. Int. Math. Res. Not. 2005, 1833-1847.
14) Castro, Carlos; Zuazua, Enrique, Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal. 164 (2002), 39-72.
15) Castro, Carlos; Zuazua, Enrique, Unique continuation and control for the heat equation from an oscillating lower dimensional manifold. SIAM J. Control Optim. 43 (2004/05), 1400-1434.
16) Colombini, Ferruccio; Pernazza, Ludovico; Treves, Francois, Solvability and nonsolvability of second-order evolution equations. Hyperbolic problems and related topics, 111-120, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003.
17) Colombini, Ferruccio; Lerner, Nicolas, Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77 (1995), 657-698.
18) Bony, Jean-Michel; Broglia Fabrizio; Colombini, Ferruccio; Pernazza, Ludovico, Nonnegative Functions as Squares or Sums of Squares, to appear in J. Funct. Anal.
19) Castro, Carlos; Zuazua, Enrique, Unique continuation and control for the heat equation from an oscillating lower dimensional manifold. SIAM J. Control Optim. 43 (2004/05), 1400-1434.
20) Gallagher, Isabelle; Saint-Raymond, Laure, Mathematical study of the betaplane model: equatorial waves and convergence results. Mém. Soc. Math. Fr. 107 (2006), v+116 pp.
21) Gallagher, Isabelle; Gallay, Thierry; Nier, Francis, Special asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. IMRN (2009), 2147–2199.
22) Bahouri, Hajer; Fermanian-Kammerer, Clotilde; Gallagher, Isabelle, Phase-space analysis and pseudodifferential calculus on the Heisenberg group, to appear in Astérisque.
23) Chemin, Jean-Yves; Gallagher, Isabelle; Paicu Marius, Global regularity for some classes of large solutions to the Navier-Stokes equations, to appear in Ann. of Math.
24) Colombini, Ferruccio; Pernazza, Ludovico; Treves, Francois, Solvability and nonsolvability of second-order evolution equations. Hyperbolic problems and related topics, 111-120, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003.
25) Colombini, Ferruccio; Lerner, Nicolas, Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77 (1995), 657-698.
26) Bony, Jean-Michel; Broglia Fabrizio; Colombini, Ferruccio; Pernazza, Ludovico, Nonnegative Functions as Squares or Sums of Squares, J. Funct. Anal. 232 (2006), 137–147.
27) Colombini, Ferruccio; Métivier, Guy, The Cauchy problem for wave equations with non Lipschitz coefficients; application to continuation of solutions of some nonlinear wave equations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 177–220.
28) Colombini, Ferruccio; Petkov, Vesselin; Rauch, Jeffrey Exponential growth for the wave equation with compact time-periodic positive potential. Comm. Pure Appl. Math. 62 (2009), 565–582.
29) Colombini, Ferruccio; Koch, Herbert, Strong unique continuation for products of elliptic operators of second order. Trans. Amer. Math. Soc. 362 (2010), 345–355.
30) Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico, On square roots of class C^m of nonnegative functions of one variable, Ann. Scuola Norm. Sup. Pisa (5) 9 (2010), 635-644.