On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space
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| Format: | Preprint |
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2020
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| _version_ | 1866910405493784576 |
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| author | Álvarez-Samaniego, Borys Álvarez-Samaniego, Wilson P. Fernández-Dalgo, Pedro G. |
| author_facet | Álvarez-Samaniego, Borys Álvarez-Samaniego, Wilson P. Fernández-Dalgo, Pedro G. |
| contents | We give some conditions under which the pressure term in the incompressible Navier-Stokes equations on the entire $d$-dimensional Euclidean space is determined by the formula $\displaystyle \nabla p = \nabla \left(\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j - F_{i,j}) \right)$, where $d \in \{2, 3\}$, ${\textbf{u}} := (u_1, \ldots, u_d)$ is the fluid velocity, $\mathbb{F}:= (F_{i,j})_{1\le i,j\le d}$ is the forcing tensor, and for all $k \in \{1, \ldots, d\}$, $\mathcal{R}_k$ is the $k$-th Riesz transform. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_02588 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space Álvarez-Samaniego, Borys Álvarez-Samaniego, Wilson P. Fernández-Dalgo, Pedro G. Analysis of PDEs 35Q30, 76D05 We give some conditions under which the pressure term in the incompressible Navier-Stokes equations on the entire $d$-dimensional Euclidean space is determined by the formula $\displaystyle \nabla p = \nabla \left(\sum_{i,j=1}^d \mathcal{R}_i \mathcal{R}_j (u_i u_j - F_{i,j}) \right)$, where $d \in \{2, 3\}$, ${\textbf{u}} := (u_1, \ldots, u_d)$ is the fluid velocity, $\mathbb{F}:= (F_{i,j})_{1\le i,j\le d}$ is the forcing tensor, and for all $k \in \{1, \ldots, d\}$, $\mathcal{R}_k$ is the $k$-th Riesz transform. |
| title | On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space |
| topic | Analysis of PDEs 35Q30, 76D05 |
| url | https://arxiv.org/abs/2004.02588 |