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The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem ut + uxxx + (u 4)x = 0, u(0) = u0 ∈ H s x(R), s > − 1 6 for the generalized Korteweg-deVries equation of order three, for short gKdV3. For real valued data u0 ∈ L 2 x(R) global wellposedness follows by the conservation of the L-norm. The main new tool is a… (More)
The Cauchy-problem for the generalized Kadomtsev-PetviashviliII equation ut + uxxx + ∂ −1 x uyy = (u )x, l ≥ 3, is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in… (More)
The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u0 in the space b H r (T), defined by the norms ‖u0‖ b Hs r (T) = ‖〈ξ〉 s b u0‖lr′ ξ , is shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the… (More)
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… (More)
The Fourier transforms of the products of two respectively three solutions of the free Schrödinger equation in one space dimension are estimated in mixed and, in the first case, weighted L norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the… (More)
The Cauchyand periodic boundary value problem for the nonlinear Schrödinger equations in n space dimensions ut − i∆u = (∇u) β , |β| = m ≥ 2, u(0) = u0 ∈ H s+1 x is shown to be locally well posed for s > sc := n 2 − 1 m−1 , s ≥ 0. In the special case of space dimension n = 1 a global L-result is obtained for NLS with the nonlinearity N(u) = ∂x(u ). The proof… (More)
The I-method in its first version as developed by Colliander et al. in  is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H(R → R), provided s > − 1 42 .
In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on T × R and T × R. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear… (More)
In this paper we prove some local (in time) wellposedness results for nonlinear Schrödinger equations ut − i∆u = N(u, u), u(0) = u0 with rough data, that is, the initial value u0 belongs to some Sobolev space of negative index. We obtain positive results for the following nonlinearities and data: • N(u, u) = u, u0 ∈ H s x(T ), s > − 1 2 , • N(u, u) = u, u0… (More)
It is shown that the spatial Sobolev norms of regular global solutions of the (2+1), (3+1) and (4+1)-dimensional Klein-Gordon-Schrödinger system and the (2+1) and (3+1)-dimensional Zakharov system grow at most polynomially with a bound depending on the regularity class of the data. The proof uses the Fourier restriction norm method.