578 research outputs found
Well-posedness in energy space for the periodic modified Benjamin-Ono equation
We prove that the periodic modified Benjamin-Ono equation is locally
well-posed in the energy space . This ensures the global
well-posedness in the defocusing case. The proof is based on an
analysis of the system after gauge transform.Comment: 26 pages, 0 figure. Title changed, introduction modifie
A note on ill-posedness for the KdV equation
We prove that the solution-map associated with the KdV
equation cannot be continuously extended in for . The main
ingredients are the well-known Kato smoothing effect for the mKdV equation as
well as the Miura transform.Comment: This preprint is an improved version of the previous preprint : "A
remark on the ill-posedness issues for KdV and mKdV
On ill-posedness for the one-dimensional periodic cubic Schrodinger equation
We prove the ill-posedness in H^s(\T) , , of the periodic cubic
Schr\"odinger equation in the sense that the flow-map is not continuous from
H^s(\T) into itself for any fixed . This result is slightly
stronger than the one obtained by Christ-Colliander-Tao where the discontinuity
of the solution map is established. Moreover our proof is different and
clarifies the ill-posedness phenomena. Our approach relies on a new result on
the behavior of the associated flow-map with respect to the weak topology of
L^2(\T) .Comment: To appear in Mathematical Research Letter
Global well-posedness in L^2 for the periodic Benjamin-Ono equation
We prove that the Benjamin-Ono equation is globally well-posed in H^s(\T)
for . Moreover we show that the associated flow-map is Lipschitz on
every bounded set of {\dot H}^s(\T) , , and even real-analytic in
this space for small times. This result is sharp in the sense that the flow-map
(if it can be defined and coincides with the standard flow-map on
H^\infty(\T) ) cannot be of class , , from {\dot
H}^s(\T) into {\dot H}^s(\T) as soon as .Comment: 47 page
Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation
In this paper, we prove that the Cauchy problem associated to the following
higher-order Benjamin-Ono equation is globally
well-posed in the energy space . Moreover, we study the limit
behavior when the small positive parameter tends to zero and show
that, under a condition on the coefficients , , and , the solution
to this equation converges to the corresponding solution of the
Benjamin-Ono equation
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