Mathematics Division, National Center for Theoretical Sciences  (Taipei Office)
*** NCTS and NTU PDE Seminar ***
 
Speaker:
Shih-Hsien Yu
National University of Singapore
 
Title:
Wave Propagator and Determinant of Dirichlet-Neumann Map
 
Abstract:
In this talk we use the Laplace-Fourier transformation to derive the matrix of Dirichlet-Neumann maps for the linear compressible Navier-Stokes equations in the space-time domain R_+^2×R_+ in terms of roots of the characteristic polynomials of the differential equations for the whole space-time domain R^2×R_+.
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We classify the types of roots into characteristic, global non-characteristic, etc., in terms of their analytic properties in the transform variables together with the notion ``Laplace-Fourier path'' introduced by Liu-Yu.  The classification of the roots gives a decomposition of the matrix of kernel functions into a matrix of polynomials in characteristic roots over the ring spanned by rational functions in the transform variables and the global non-characteristic roots. We call the decomposition as the characteristic - noncharacteristic decomposition. The common denominator in the rational functions of the decomposition is named as the determinant of the Dirichlet-Neumann map; and the determinant gives the spectrum for the purely tangential wave propagation derived from the Dirichlet-Neumann map.
 
The algebraic classification for the roots are so refined that one can apply complex analysis and the Laplace-Fourier path together to invert the decomposition into a composition of wave propagators in space-time variables with explicit exponentially sharp pointwise structures.  From the structure of the wave propagators one can realize the essence of the wave propagations on the boundary due to two directions: The normal direction and the tangential direction. We also call the decomposition as the Normal-Tangential decomposition for the Dirichlet-Neumann map with respect to the wave propagators.
 
Time:
2012年06月06日 (星期三) 15:30-16:30
June 06, 2012 (Wednesday) 15:30-16:30
 
Place:
臺灣大學天文數學館440室
Room 440, Astro-Math. Building, National Taiwan University
 
Organizer:
陳俊全 教授 (臺灣大學數學系)
Prof. Chiun-Chuan Chen (National Taiwan University)