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SUMMARY:Boundary kernels for dissipative systems - Shih-Hsien Yu (Singapor
 e)
DTSTART:20120423T150000Z
DTEND:20120423T160000Z
UID:TALK36073@talks.cam.ac.uk
CONTACT:Jonathan Ben-Artzi
DESCRIPTION:In this talk we will present a study on the kernel functions o
 f the Dirichlet-­‐Neumann maps for dissipative systems in a half space.
  We start from the consideration of the Green’s function for an initial-
 ­‐boundary value problems for linear dissipative systems. With the fund
 amental solutions of the dissipative systems\, one can reduce the initial-
 ­‐boundary value problems into boundary value problems so that the well
 -­‐posedness of the system gives linear algebraic systems over the poly
 nomials in the Fourier and Laplace variables for the Dirichlet-­‐Neuman
 n datum at the boundary\, where Fourier variables are in the directions of
  boundary\, and the Laplace is for the time variable.\n\nIn order to inver
 t the Dirichlet-­‐Neumann map from the transformation variables to the 
 space-­‐time variables we introduce a path\, which contains the spectra
 l information of the systems\, in the complex plan for the time Laplace va
 riable. On this path\, the Laplace-­‐Fourier variables can be recombine
 d\, through the Cauchy’s complex contour integral\, into a form resemble
  to that for a whole space problem. Thus\, the classical results for the w
 hole space problem can be used to obtain the pointwise spae-­‐time stru
 cture for long wave components of the kernel function of the Dirichlet-­
 ‐Neumann map for points within a finite Mach region. We also apply direc
 t energy estimates to yield the pointwise structure of the kernel function
 s in any high Mach number region. Finally\, we have obtained exponentially
  sharp estimates for the kernel function in the space-­‐time variables.
  For example\, the kernel functions for both D’Alermbert wave equation w
 ith dissipation and a linearized compressible Navier-­‐Stokes equation 
 can be expressed explicitly in space-­‐time variables with errors which
  decay exponentially in both space-­‐time variables. This gives a globa
 lly quantitative and qualitative wave propagations at boundary.
LOCATION:CMS\, MR15
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