| 主講者: | 尤釋賢 (National University of Singapore) |
| 講題: | Green's functions and Well-posedness of Compressible Navier-Stokes equation |
| 時間: | 2018-10-26 (Fri.) 11:00 - 12:00 |
| 地點: | 數學所 演講廳(6F) (台大院區) |
| Abstract: | 1.A class of decomposition of Green's functions for the compressilbe Navier-Stokes
linearized on a constant state is introduced. The singular structures of the Green's function
was developed as an essential device to effectively use the nonlinearity to covert the
2nd order quasi-linear PDE into a system of zero-th order integral equation with a nonlinear
integrable kernel. The system of integrable equation allows a wider class of functions.
We have shown global existence and well-posedness of the compressible Navier-Stokes
equation for isentropic gas with the gas constant $\gamma \in (0,e)$ in the Lagrangian
coordinate for the class of the BV functions and $L^\infty$ around a constant state; and the
underline pointwise structure of the solutions is constructed. It is a shown that for the class
of BV solution the solution is at most piecewise $C^2$-solution even though the initial data
is piecewise $C^\infty$.
2.Mathematical models of axially symmetric rotating gaseous stars are governed by the Euler-Poisson equations. About them there is a long history of research by astrophysicists from 1920's. But they lacked mathematical rigourousness. Mathematicaly rigorous treatment was begun by J.F. G. Auchmuty and R. Beals in 1971 and many mathematical works have sucseeded it. They have been using the variational problem setting. But the speaker has developed a direct rigorous justification of the original methods adopted by astrophysicists, as joint works with Dr Juhi Jang, for a wider class of equations of state of the gas composing the stars. Moreover this result can be used as the basis of the discussion of the axisymmetric stationary metric under the Einstein-Euler equations in the framework of General Relativity. But there remains an interesting open problem in this study. |
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