An introduction to the mathematical theory of quantum fluids
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Europe/Rome
Ex-ISEF/Building-Room A (GSSI)
Ex-ISEF/Building-Room A
GSSI
20
Description
The course is structured in two parts.
Part 1 - Fundamental concepts and basic results (three lectures, see schedule below).
We discuss the physical background and introduce the main mathematical models. We study the associated Cauchy problem by focusing on the mathematical analysis of global-in-time, finite energy weak solutions.
- Physical motivations: description of the main physical phenomena and features
- Mathematical models: the Gross-Pitaevskii equation, the Madelung transform and the quantum hydrodynamics (QHD) system
- Review of theory for nonlinear Schrödinger equations: dispersive estimates and well-posedness results
- The Cauchy problem for the QHD system: the polar factorization approach and finite energy weak solutions
- Dispersive properties of solutions
- The wave function lifting and stability for the one-dimensional QHD system
Part 2 - Selected topics (three lectures, to be scheduled in September/October).
This part is flexible and will be agreed upon with the audience, according to specific interests. It will present some particular problems at the frontier of the current research in the field. Below is a list of some possible topics.
- Quantized vortices.
- Initial boundary value problems: analysis of the model on domains with boundary.
- Singular limits and reduced models.
- The Landau two-fluid model and other effective models for superfluid Helium II.
- The effect of non-trivial magnetic fields: nonlinear Maxwell-Schrödinger and the quantum Euler-Maxwell system.
Preliminary Schedule (Part 1)
Lecture I: May 7, 2 pm - 4 pm, Room A (ex-ISEF building)
Lecture II: May 9, 2 pm - 4 pm, Room A (ex-ISEF building)
Lecture III: May 15, 2 pm - 4 pm, Room A (ex-ISEF building)
