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Type of Document Dissertation Author Gorsky, Jennifer Author's Email Address jgorsky@nd.edu URN etd-04022004-171911 Title On the Cauchy Problem for a KdV-Type Equation on the Circle Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Professor Alex Himonas (advisor) Committee Member Professor Bei Hu Committee Member Professor David Nicholls Committee Member Professor Gerard Misiolek Committee Member Keywords
- KdV equation
- Camassa-Holm equation
- well-posedness
- analyticity
- Sobolev space
- initial value problem
Date of Defense 2004-04-01 Availability unrestricted Abstract We shall consider the periodic Cauchy problem for a modifiedCamassa-Holm (mCH) equation. We begin by proving well-posedness
in Bourgain spaces for sufficiently small size initial data in the
Sobolev space $H^s(mathbb{T})$, $s=1/2$, by using appropriate
bilinear estimates. Also we show that these bilinear estimates do not
hold if $s<1/2$. Well-posedness of the mCH for $s>1/2$ has been established by
Himonas and Misiol ek in [HM1]. These results indicate that
$s=1/2$ may be the critical Sobolev exponent for well-posedness.
In the second part of this work we show that the periodic Cauchy
problem for the
mCH equation with analytic initial data is analytic in the space
variable $x$
for time near zero. By differentiating the equation and the initial
condition
with respect to $x$ we obtain a sequence of initial value problems of
KdV-type equations. These, written in the form of integral equations,
define
a mapping on a Banach space whose elements are sequences of
functions equipped with a norm expressing the Cauchy estimates
in terms of the KdV norms of the components introduced in the works
of Bourgain, Kenig, Ponce, Vega and others. By proving appropriate
bilinear estimates we show that this mapping is a contraction, and
therefore we obtain a solution whose derivatives in the space variable
satisfy the Cauchy estimates.
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