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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.06951 |
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| _version_ | 1866909895973928960 |
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| author | Cavalcante, Márcio Nascimento, Aílton C. |
| author_facet | Cavalcante, Márcio Nascimento, Aílton C. |
| contents | We study special regularity properties of solutions to the initial-boundary value problem associated with the Korteweg-de Vries equations posed on the positive half-line. In particular, for initial data $u_0 \in H^{\frac{3}{4}^{+}}(\mathbb{R}^+)$ and boundary data $f\in H^{\frac32^+}(\R^+)$, where the restriction of $u_0$ to some subset of $(b,\infty)$ has an extra regularity for any $b>0$, we prove that the regularity of solutions $u$ moves with infinite speed to its left as time evolves until a certain time $T^*$. The existence of a stopping time $T^{*}$ appears because of the effect of the boundary function $f$. Also, as a consequence of our proof, we prove a gain in the regularity of the trace derivatives of the solutions for the Korteweg-de Vries on the half-line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06951 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Propagation of Regularity of Solutions to the KdV Equation on the positive Half-line Cavalcante, Márcio Nascimento, Aílton C. Analysis of PDEs 35Q53 We study special regularity properties of solutions to the initial-boundary value problem associated with the Korteweg-de Vries equations posed on the positive half-line. In particular, for initial data $u_0 \in H^{\frac{3}{4}^{+}}(\mathbb{R}^+)$ and boundary data $f\in H^{\frac32^+}(\R^+)$, where the restriction of $u_0$ to some subset of $(b,\infty)$ has an extra regularity for any $b>0$, we prove that the regularity of solutions $u$ moves with infinite speed to its left as time evolves until a certain time $T^*$. The existence of a stopping time $T^{*}$ appears because of the effect of the boundary function $f$. Also, as a consequence of our proof, we prove a gain in the regularity of the trace derivatives of the solutions for the Korteweg-de Vries on the half-line. |
| title | On the Propagation of Regularity of Solutions to the KdV Equation on the positive Half-line |
| topic | Analysis of PDEs 35Q53 |
| url | https://arxiv.org/abs/2511.06951 |