Asymptotics for critical nonconvective type equations.

Hayashi, Nakao; Kaikina, Elena I.; Naumkin, Pavel I.

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On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock (2010)

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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces ${\hat{H}}_{s}^{r} (ℝ)$ defined by the norm ${∥v_{0}∥}_{{\hat{H}}_{s}^{r} (ℝ)} : = {∥{〈ξ〉}^{s} \hat{v_{0}}∥}_{L_{ξ}^{r^{'}}}, 〈ξ〉 = {(1 + ξ^{2})}^{\frac{1}{2}}, \frac{1}{r} + \frac{1}{r^{'}} = 1$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $\frac{2 j - 1}{2 r^{'}}$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $\frac{2 j - 1}{2 r^{'}}$ . The results for r =...