Applied Partial Differential Equations (4/4)

by Paolo Antonelli (GSSI), Sara Daneri (GSSI)

Monday 1 Feb 2021, 08:00 → Sunday 28 Feb 2021, 10:00 Europe/Rome

GSSI

Lecturers
Paolo Antonelli, paolo.antonelli@gssi.it,
Sara Daneri, sara.daneri@gssi.it

Timetable and workload
Total number of hours: 60 (30 hours for each part)
Within those hours TA sessions will be organized.

Course description
This course presents the main techniques and tools developed for the study of applied PDEs,by reviewing some results and problems in fluid dynamics and dispersive equations. The first part, focused on the general theory for nonlinear Schrödinger equations, discusses the existence of solutions and their asymptotic behavior or possible formation of singularities. In the second part the focus will be on PDEs such as the Euler and Navier-Stokes equations and in particular on the questions of existence, uniqueness and regularity of solutions in different settings. In this way the student will get acquainted with the fundamental tools exploited in this field, such as semi-group theory, fixed point arguments, a-priori estimates and compactness arguments.

Course requirements
Basic knowledge of functional analysis, notions of Lp spaces, measure theory and Fourier spaces.Also the knowledge of Sobolev spaces is strongly advised, eventually to be covered in a parallel series of tutorial lectures.

Course content
The course will be divided in two parts, the first one related to the analysis on nonlinear Schrödinger equations and the second one focused on incompressible fluid dynamics.

• Part 1: Nonlinear Schrödinger equations.
  Review of basic tools from harmonic analysis: real and complex interpolation. Derivation of effective equations for nonlinear dispersive waves. Invariances and conserved quantities: the Noether’s theorem. Existence of local regular solutions: the energy method. Local and global smoothing estimates associated to the linear propagator: dispersive estimates, Strichartz estimates, Kato smoothing estimates. The local Cauchy problem for the nonlinear Schrödinger equation in H1 and L2. Global existence and asymptotic behavior for repulsive nonlinearities; scattering theory. Formation of singularities at finite times: blow-up results based on virial arguments.Stability of solitary waves: concentration-compactness. Instability of solitary waves in the mass-critical case, universality of the blow-up profile with minimal mass.
• Part 2: Incompressible fluid flows
  Derivation of the Euler and Navier-Stokes equations from conservation principles in the continuum hypothesis; Conserved quantities and special solutions; Local existence of solutions for regular initial data via energy methods; Yudovich theorem on existence and uniqueness of two-dimensional solutions with bounded vorticity; Leray-Hopf solutions of the Navier-Stokes equations; Strong solutions and weak-strong uniqueness; Serrin’s regularity result

Examination and grading
The students will be evaluated on the basis (a) a reading seminar on a research paper related to modern developments of the topics handled during the course and (b) a written exam to assess the skills developed during the course

Books of reference

• T. Cazenave,Semilinear Schrödinger equations, Courant Lecture Notes.
• A.J. Majda, A.L. Bertozzi,Vorticity and incompressible flow, Cambridge University Press.
• C. Marchioro, M. Pulvirenti,Mathematical theory of incompressible nonviscous fluids, Springer.
• J.C. Robinson, J.L. Rodrigo, W. Sadowski The three-dimensional Navier-Stokes equations,Cambridge University Press.
• C. Sulem, P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Springer.