Pseudodifferential operators, applications and dynamics

Aim of the course is to introduce the basic tools of pseudodifferential calculus, and to show applications of such techniques in the analysis of dispersive PDEs, spectral theory, or other areas. A particular emphasis will be given to the problem of growth of Sobolev norms.

Sissa page of the course link



Timetable

On Monday and Wednesday from 14:00 to 16:00, room 134



Program

  1. 9/10: Introduction to the course. The need of pseudodifferential calculus. Water waves equation and their linearization at the zero solution (see also [1,2,3] in additional material)
  2. 16/10: Crash course in Fourier analysis: Schwartz class and tempered distributions; examples. Fourier transform on Schwartz function and the basic properties. [H, K]
  3. 21/10: Crash course in Fourier analysis II: Fourier inversion theorem, Plancherel and Parseval, regularity in Fourier. Fourier transform of tempered distributions. [H, K]
  4. 23/10: Sobolev spaces and their properties. Fourier multipliers and multiplier calculus. [K]
  5. 28/10: Littlewood-Paley decomposition: basic properties, square function theorem, Bernstein inequalities. [MS chap. 8, AG chap. II.A] (see also [4,5] in additional material)
  6. 30/10: Applications of Littlewood-Paley theory: characterization of Sobolev spaces, Sobolev inequalities, tame estimates [AG, chap. II.A ] (see also [6] in additional material)
  7. 4/11: Linear constant coefficients dispersive equation: solution via Fourier transform. Schrödinger and wave equation, analysis of the solutions. Fundamental solution of Schrödinger. [Tao chap 2.1--2.2, LP chap. 1]
  8. 6/11: Strichartz estimates for Schrödinger equation. Cauchy theory of NLS semilinear: LWP in $H^s$, $s>d/2$ [Tao chap 2.3] (see also [7,8,9] in additional material)
  9. 11/11: LWP of pure power NLS in $H^1$. Conservation laws. GWP in $H^1$ in the subcritical case. Blow up through virial argument [LP chap 5-6, Tao chap 3.4 (in particular Prop. 3.23)] (see also [10] in additional material)
  10. 13/11: Towards pseudodifferential operators: symbols and their properties. Asymptotic sums [SR chap 2]. Notes
  11. 18/11: Pseudodifferential operators: action on the Schwartz class, kernel representation [SR chap 3]. Notes
  12. 20/11: Oscillatory integrals: rapid decay, stationary phase, definition of oscillatory integrals [SR chap 2]. Notes
  13. 25/11: Oscillatory integrals: properties and manipulations. Approximation of symbols.[SR chap 2]. Notes (same as above)
  14. 27/11: Symbolic calculus: adjoint and composition. Proof of the composition theorem. [SR chap 2]. Notes
  15. 2/12: Applications of symbolic calculus: commutators, symbol recovery, parametrix, change of quantization. Weyl quantization. [AG, R]. Notes
  16. 4/12: $L^2$ continuity of pseudodifferential operators. Calderon-Vaillancourt theorem. [AG chap 5.1, MS chap 9.2]. Notes
  17. 9/12: Garding-like inequalities. Flow of hyperbolic PDEs. [AG chap 5.3 and chap. C.1]. Notes Garding, Notes flow
  18. 11/12: Flow of elliptic pseudodifferential operators. Conjugation of pseudodifferential operators by flow.Notes conjugation
  19. 16/12: Egorov theorem. Application: near to identity diffeomorphism. Notes conjugation
  20. 18/12: Cauchy problem for some non-selfadjoint perturbation of Schrödinger operator. (see [12,13] in additional material)Notes Doi
  21. 15/01: Introduction to the problem of growth of Sobolev norms for linear time dependent Schrödinger equation. (see [14] in additional material).
  22. 23/01: How to improve the upper bounds. Introduction to pseudodifferential normal form (see [16]). Periodic pseudodifferential operators (see [15]).
  23. 24/01: Upper bounds on the growth of Sobolev norm for linear Schrödinger equation on the torus (see [17]).
  24. 27/01: Lower bounds on the growth of Sobolev norm for linear Harmonic oscillators (see [16]). Lower bounds for the growth of Sobolev norms (see [18]).
  25. 3/02: Introduction to paradifferential calculus. Bony quantization. Continuity of paralinear operator.
  26. 5/02: Paradifferential symbolic calculus: paraproducts and composition
  27. 7/02: Paralinearizations of products and composition operator


Textbooks

  • [AG] S. Alinhac and P. Gérard, Pseudo-differential Operators and the Nash-Moser Theorem, (AMS, Graduate Studies in Mathematics, vol. 82, 2007)
  • [H] L. Hörmander, The Analysis of Linear Partial Differential Operators I-III (Springer)
  • [K] H. Kumano-go. Pseudo-differential operators (Cambridge, Massachusetts: MIT press, 1982)
  • [LP] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations (Springer)
  • [MS] C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis, Vol I ( Cambridge University Press)
  • [R] D. Robert, Autour de l'Approximation Semi-Classique (Boston etc., Birkhauser 1987)
  • [SR] X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators}(Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.)
  • [S] M. Shubin, Pseudodifferential Operators and Spectral Theory (Springer)
  • [Tao] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis
  • [Tay] M. Taylor, Pseudo Differential Operators (Princeton Univ. Press, Princeton, N.J., 1981)




Additional material

  1. Zakharov - Hamiltonian formulation of water waves link
  2. Craig-Sulem formulation of water waves link
  3. Alazard, Burq, Zuily - The Water-Wave Equations: From Zakharov to Euler link
  4. Monica Visan, Lectures in Harmonic Analysis link
  5. T. Tao, Lectures on Littlewood - Paley decomposition link
  6. J.M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires link
  7. Dispersive Equations and Nonlinear Waves link
  8. Keel-Tao: Endpoint Strichartz estimates link
  9. Raphael: Analyse nonlineaire, sur la stabilite des ondes solitaires, link
  10. Raphael: Stability and blow up for the nonlinear Schrodinger equation, Notes of the 2008 Clay Summer School in Zurich, link
  11. Fefferman-Phong inequality, link
  12. Doi - On the Cauchy problem for Schrödinger type equations and the regularity of the solutions link
  13. Doi - Remarks on the Cauchy problem for Schrödinger-type equations link
  14. A. Maspero, D. Robert: On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms. J. Funct. Anal., 273(2):721–781, 2017.Paper
  15. Sarannen, Vainniko: Periodic pseudodifferential operators chapter
  16. D. Bambusi, B. Grebert, A. Maspero, D. Robert: Growth of Sobolev norms for abstract linear Schrödinger Equations. J. Eur. Math. Soc. (JEMS),2017.Paper
  17. Montalto - On the growth of Sobolev norms for a class of linear Schrödinger equations on the torus with superlinear dispersion .link
  18. A. Maspero: Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations. Math. Res. Lett, 2019, Paper
© Tetiana Savitska 2017