The local ill-posedness of the modified KdV equation
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...
متن کاملSharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
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....
متن کاملLOCAL WELL-POSEDNESS FOR THE MODIFIED KDV EQUATION IN ALMOST CRITICAL Ĥr
We study the Cauchy problem for the modified KdV equation ut + uxxx + (u )x = 0, u(0) = u0 for data u0 in the space Ĥr s defined by the norm ‖u0‖Ĥr s := ‖〈ξ〉 sû0‖Lr′ ξ . Local well-posedness of this problem is established in the parameter range 2 ≥ r > 1, s ≥ 1 2 − 1 2r , so the case (s, r) = (0, 1), which is critical in view of scaling considerations, is almost reached. To show this result, we...
متن کاملRemark on Well-posedness and Ill-posedness for the Kdv Equation
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...
متن کاملAn Improved Local Wellposedness Result for the Modified Kdv-equation
The Cauchy problem for the modified KdV-equation ut + uxxx = (u 3)x, u(0) = u0 is shown to be locally wellposed for data u0 in the space Ĥr s (R) defined by the norm ‖u0‖ Ĥr s := ‖〈ξ〉sû0‖Lr′ ξ , provided 4 3 < r ≤ 2, s ≥ 1 2 − 1 2r . For r = 2 this coincides with the best possible result on the H-scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restrictio...
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ژورنال
عنوان ژورنال: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
سال: 1996
ISSN: 0294-1449
DOI: 10.1016/s0294-1449(16)30112-3