Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917870138556416 |
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| author | Correia, Simão Leite, Pedro |
| author_facet | Correia, Simão Leite, Pedro |
| contents | We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order $k_0$ and in dimension $d_0$, we prove a sharp local well-posedness result in $H^s(\mathbb{R}^d)$ for any $k\ge k_0$ and $d\ge d_0$. Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in [CorreiaOliveiraSilva24] (doi.org/10.1137/23M156923X). The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schrödinger equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities Correia, Simão Leite, Pedro Analysis of PDEs We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order $k_0$ and in dimension $d_0$, we prove a sharp local well-posedness result in $H^s(\mathbb{R}^d)$ for any $k\ge k_0$ and $d\ge d_0$. Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in [CorreiaOliveiraSilva24] (doi.org/10.1137/23M156923X). The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schrödinger equations. |
| title | Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.11808 |