Math Courses

Introduction to QHD (Quantum Hydrodynamics) (8/9)

by Prof. Pierangelo Marcati (Gran Sasso Science Institute)

→ Europe/Rome
GSSI

GSSI
Description

TOPICS

  1. Review of physical problems:  superfluidity, superconductivity, carrier transport in semiconductors.      E-K (Euler-Korteweg system)
  2. Macroscopic picture vs Schrödinger
  3. Generalized irrotationality conditions
  4. Weak solutions, mathematical difficulties, Conservation Laws
  5. Polar factorization, stability
  6. Review of the theory of Schrodinger equations
  7. Existence of finite energy weak solutions
  8. A digression about non-uniqueness via Convex integration
  9. Some extension to QMHD and simple 2 fluids models
  10. Hamiltonian structure of QHD and E-K, GCP (Generalized Chemical Potential) states
  11. Wave function Lifting
  12. 1- D genuine hydrodynamic theory and extension to 2-D
  13. From Maxwell Equations to Nonlinear Schrödinger Equations and their QHD counterpart, physical derivations
  14. Solitons and vortices, mathematical problems
  15. NLS of Gross - Pitaevskii type

 

REFERENCES

  1. Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in Quantum Fluid Dynamics. Commun. Math. Phys. 287(2), 657–686 (2009)
  2. Antonelli, P., Marcati, P.: The Quantum Hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203, 499–527 (2012)
  3. Antonelli, P., Marcati, P. & Zheng, H. Genuine Hydrodynamic Analysis to the 1-D QHD System: Existence, Dispersion and Stability. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-03998-z
  4. Antonelli P., Hientzsch L-E., Marcati P., Hao Zheng (2018). On some results for quantum hydrodynamical models. RIMS Kokyuroku: 2070 Mathematical Fluids and Gas dynamics.
  5. Antonelli, P., Marcati, P.  An introduction to the mathematical theory of quantum fluids, (review)  to appear on the volume "Fluid Dynamics, Dispersive Perturbations and Quantum Fluids", Fabio Ancona, Stefano Bianchini, Andrea Marson editors, Springer-Umi Lecture Notes of the Unione Matematica Italiana.
  6. Benzoni-Gavage, S.: Propagating phase boundaries and capillary fluids. http://math.univ-lyon1.fr/~benzoni/Levico.pdf
  7. Donatelli, D., Feireisl, E., Marcati, P.: Well/ill posedness for the Euler–Korteweg–Poisson system and related problems. Commun. Part. Differ. Equ. 40, 1314–1335 (2015)
  8. Sulem, C.; Sulem, P.-L. The nonlinear Schrödinger equation: Self-focusing and wave collapse, Applied Mathematical Sciences, 139, (1999), Springer-Verlag, New York.

 

 

 

 

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