On a higher dimensional version of the Benjamin--Ono equation
| Autor: | Linares, Felipe, Riaño, Oscar G., Rogers, Keith M., Wright, James, Hickman, Jonathan |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1\Delta u+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish sharp space--time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in $L^2$-Sobolev spaces $H^{s}(\mathbb{R}^d)$, with $s>5/3$ if $d=2$ and $s>d/2+1/2$ if $d\ge 3$. We also provide ill-posedness results. Comment: We also show that in dimension 2 our results are sharp |
| Databáze: | arXiv |
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