Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Molinet, Luc Tanaka, Tomoyuki |
| author_facet | Molinet, Luc Tanaka, Tomoyuki |
| contents | We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |ξ|^αξ$ with $ 1 \le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R}) $ for $ s\ge \frac{5-2α}{4} $ whenever $ 1\le α<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $α\in [\frac{3}{2},2] $. This result is optimal in the case $α\ge \frac{3}{2}$ in view of the restriction $ s>\frac{1}{2} $ required for the continuous embedding $ H^s(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R}) $. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R} $. Our local well-posedness results enable us to derive global existence of solutions for $ α\in [\frac{5}{4},2] $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$ Molinet, Luc Tanaka, Tomoyuki Analysis of PDEs We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |ξ|^αξ$ with $ 1 \le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R}) $ for $ s\ge \frac{5-2α}{4} $ whenever $ 1\le α<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $α\in [\frac{3}{2},2] $. This result is optimal in the case $α\ge \frac{3}{2}$ in view of the restriction $ s>\frac{1}{2} $ required for the continuous embedding $ H^s(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R}) $. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R} $. Our local well-posedness results enable us to derive global existence of solutions for $ α\in [\frac{5}{4},2] $. |
| title | Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.27461 |