Well-posedness and ill-posedness of KdV equation with higher dispersion
نویسندگان
چکیده
منابع مشابه
Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case
We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...
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We consider the Cauchy problem for the KdV equation with low regularity initial data given in the space Hs,a(R), which is defined by the norm ‖φ‖Hs,a = ‖〈ξ〉s−a|ξ|a b φ‖L2 ξ . We obtain the local well-posedness in Hs,a with s ≥ max{−3/4,−a − 3/2}, −3/2 < a ≤ 0 and (s, a) 6= (−3/4,−3/4). The proof is based on Kishimoto’s work [12] which proved the sharp well-posedness in the Sobolev space H−3/4(R...
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We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H−1(R) with a solution-map that is analytic from H−1(R) to C([0, T ];H−1(R)) whereas it is ill-posed in Hs(R), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(R) to even D′(R) at any fixed t > 0 small enough....
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The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2014
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2014.01.035