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https://hdl.handle.net/2142/86949
Description
Title
The Initial Value Problem for the Zakharov System
Author(s)
Colliander, James Ellis
Issue Date
1997
Doctoral Committee Chair(s)
Jean Bourgain
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, Fluid and Plasma
Language
eng
Abstract
The method of proof is an application of techniques developed by Bourgain and Kenig, Ponce and Vega. The equivalent fixed point problem is solved via the contraction principle in the $X\sb{s,b}$ spaces introduced by Bourgain. A detailed analysis of the nonlinear terms, exploiting the arithmetical properties of the $X\sb{s,b}$ denominators combined with the Strichartz estimates for the paraboloid, gives the d = 1, 2 results. For the more difficult d = 3 problem, Strichartz estimate for the cone and a refined convolution estimate for measures supported on the sphere are exploited to give the local result.
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