An optimal decay estimate for the linearized water wave equation in 2D
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arXiv
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| Formato: | Preprint |
| Publicado: |
2014
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| _version_ | 1866909202133286912 |
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| author | Bulut, Aynur |
| author_facet | Bulut, Aynur |
| contents | We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-Δ)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-Δ)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=φ$, assuming only bounds on $\lVert φ\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_xφ\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-Δ)^{α/2}u=0$ for a wider range of $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_0963 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | An optimal decay estimate for the linearized water wave equation in 2D Bulut, Aynur Analysis of PDEs We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-Δ)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-Δ)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=φ$, assuming only bounds on $\lVert φ\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_xφ\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-Δ)^{α/2}u=0$ for a wider range of $α$. |
| title | An optimal decay estimate for the linearized water wave equation in 2D |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1411.0963 |