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2018 NCTS Summer Course: Fluid, Water Wave and Phase Transition Problems
6/26(Tue), 6/27(Wed), 7/2(Mon), 7/3(Tue), 7/4(Wed) 10:00 - 12:00, 14:00 - 16:00, June 26 - July 4, 2018
Room 440, Astronomy-Mathematics Building, NTU

Chun-Hsiung Hsia (National Taiwan University)
Jerry L. Bona (University of Illinois at Chicago)
Hongqiu Chen (University of Memphis)
Shouhong Wang (Indiana University)

Chun-Hsiung Hsia (National Taiwan University)

The goal of this course is multi-fold. First, we try to give students motivation of the study of fluid problems by introducing the modelling. Second, we introduce a very useful machinery in the research of phase transitions and give several important application. On the other hand, we also teach a very delicate mathematical technique concerning the regularity of compressible fluid equations newly introduced by Didier Bresch and Pierre-Emmanuel Jabin. Over all, we hope to give both motivation and teach delicate techniques in this course. 
1. Jerry Bona and Hongqiu Chen:
Title : Model Equations for Water Wave Propagation
This set of lectures will begin with an introduction to the derivation of models for the propagation of surface and internal water waves. These take the form of partial differential and pseudo-differential equations. Once the equations are in hand, we embark upon an examination of their properties. This will include fundamental issues such as local and global well posedness, or in some cases, lack thereof. More subtle aspects of the models will then be investigated.  This will include the role of solitary waves in the evolution of solutions and comparisons between different models. The course will finish with an indication of some applications of these models to real world phenomena.
2. Shouhong Wang:
Title: Fundamental Principles of Statistical Physics and Topological Phase Transitions
Abstract: These lectures cover the following recent advances of statistical physics:
1. We postulate a potential-descending principle (PDP), and show that PDP, together with the classical equal probability principle (PEP), leads to 1) the first and second laws of thermodynamics, 2) irreversibility, and 3) the three fundamental distributions(Boltzmann-Maxwell, Bose-Einstein, and Fermi-Dirac);
2. We present a new interpretation of quantum wave functions, and introduce a new  quantum mechanism of superfluidity and High Tc
3. We briefly examine basic notions and theories of both dynamic transition and topological phase transition. A related connection to the geometric theory of 2D incompressible flows is explored. Applications to quantum phase transition and boundary layer separation of fluid flows will be explored as well.
3. Chun-Hsiung Hsia:
Title: Quantitative Regularity Estimates for Compressible Transport Equations
Abstract : In this lecture series, we introduce a new method of quantitative regularity estimates for compressible transport equations developed by Didier Bresch and Pierre-Emmanuel Jabin. This is a very useful theory for a wide class of equations. We shall most follow the accepted annals paper titled " Global Existence of Weak Solutions for Compresssible Navier–Stokes Equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor " of  Didier Bresch and Pierre-Emmanuel Jabin.

National Center for Theoretical Sciences
Mathematics Research Promotion Center






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