Some Mathematical Theories for High Reynolds Number Limit

Project: Research

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To justify the high Reynolds number limit for fluid systems in rigorous mathematics has a very long history. However, it remains a lot of challenging problems, in particular with physical boundary conditions, even though there are fruitful results on the classical Navier-Stokes equations.This project aims to study the dynamics of electrically conducting fluid in some physical regimes with boundary when the boundary behaviour has suitable regularity, that is, before the boundary separation and turbulence occur. In this case, to justify the high Reynolds number limit in rigorous mathematics is related to the study on boundary layer and multi-scale analysis. One approach to study this problem is to investigate the behaviour of the Prandtl-type boundary layer introduced by Prandtl in 1904 that sets the foundation of boundary layer theory. In fact, how to justify the Prandtl ansatz for incompressible Navier-Stokes equations with no-slip boundary condition has been an active research topic with progress limited to two space dimensional setting with structure condition except the classical work by Sammartino-Caflisch in analytic function space.For electrically conducting fluid, magnetohydrodynamic system (MHD) is a fundamental system for describing the fluid behaviour. When the Reynold and the magnetic Reynolds numbers tend to infinity at the same rate by keeping the Hartmann number being finite, by fully using the stabilizing effect of the magnetic field on the fluid, recently we succeed in justifying the Prandtl ansatz in Sobolev space in 2D when the tangential magnetic field on the boundary is away from zero. Note that the corresponding result does not hold for incompressible Navier-Stokes equations. And it remains basically unsolved in the more physical 3D setting.With the development on the Prandtl equations and the justification on the Prandtl ansatz on the MHD system, the project focuses on the high Reynolds number limit in different physical regimes, such as the Prandtl-Hartmann and the Prandtl-Shercliff regimes. For example, when the magnetic field is transverse to the boundary, Hartmann layer is a typical boundary layer pro le observed by Hartmann in 1937 in the study of electromagnetic pump. In this regime, as shown formally in [11], a mixed Prandtl- Hartmann boundary layer equations can be derived when the Reynolds number, magnetic Reynolds number and the Hartmann number satisfy some scaling condition. To justify the validity of this ansatz is then one of the problems to be considered.


Project number9042862
Effective start/end date1/09/19 → …