Blow-Up for the 1D Cubic NLS
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We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier–Lebesgue spaces and in modulation spaces. We show the jth equation in the hierarchy is locally well-posed for initial data in $${{\hat{H}}}^s_r({\mathbb {R}})$$ H ^ r s ( R ) for $$s \ge \frac{j-1}{r'}$$ s ≥ j - 1 r ′ and $$1 < r \le 2$$ 1 < r ≤ 2 and also in $$M^s_{2, p}({\mathbb {R}})$$ M 2 , p s ( R ) for $$s = \frac{j-1}{2}$$ s = j - 1 2 and $$2 \le p < \infty $$ 2 ≤ p < ∞ . Supplementing our results with corresponding ill-posedness results in Fourier–Lebesgue and modulation spaces shows optimality. Using the conserved quantities derived in Koch and Tataru (Duke Math J 167(17), 3207–3313, 2018), we argue that the hierarchy equations are globally well-posed for data in $$H^s({\mathbb {R}})$$ H s ( R ) for $$s \ge \frac{j-1}{2}$$ s ≥ j - 1 2 . Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- and trilinear refinements of Strichartz estimates.
We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier–Lebesgue spaces and in modulation spaces. We show the jth equation in the hierarchy is locally well-posed for initial data in $${{\hat{H}}}^s_r({\mathbb {R}})$$ H ^ r s ( R ) for $$s \ge \frac{j-1}{r'}$$ s ≥ j - 1 r ′ and $$1 < r \le 2$$ 1 < r ≤ 2 and also in $$M^s_{2, p}({\mathbb {R}})$$ M 2 , p s ( R ) for $$s = \frac{j-1}{2}$$ s = j - 1 2 and $$2 \le p < \infty $$ 2 ≤ p < ∞ . Supplementing our results with corresponding ill-posedness results in Fourier–Lebesgue and modulation spaces shows optimality. Using the conserved quantities derived in Koch and Tataru (Duke Math J 167(17), 3207–3313, 2018), we argue that the hierarchy equations are globally well-posed for data in $$H^s({\mathbb {R}})$$ H s ( R ) for $$s \ge \frac{j-1}{2}$$ s ≥ j - 1 2 . Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- and trilinear refinements of Strichartz estimates.
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