Diffusion-free boundary conditions for the Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2025
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| author | Dormy, Emmanuel Gerard-Varet, David |
| author_facet | Dormy, Emmanuel Gerard-Varet, David |
| contents | We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain $Ω$ of $\mathbb{R}^d$, $d=2$ or $3$. We show that diffusion-free boundary conditions $$ Δu \cdot τ\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=2, $$ $$ Δu \times n\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=3, $$
allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for $d=2$, local for $d=3$). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity $ν\rightarrow 0$. We establish that the amplitude of the boundary layer flow is in this case of order $ν$, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffusion-free boundary conditions for the Navier-Stokes equations Dormy, Emmanuel Gerard-Varet, David Analysis of PDEs Fluid Dynamics We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain $Ω$ of $\mathbb{R}^d$, $d=2$ or $3$. We show that diffusion-free boundary conditions $$ Δu \cdot τ\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=2, $$ $$ Δu \times n\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=3, $$ allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for $d=2$, local for $d=3$). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity $ν\rightarrow 0$. We establish that the amplitude of the boundary layer flow is in this case of order $ν$, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions. |
| title | Diffusion-free boundary conditions for the Navier-Stokes equations |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2506.17749 |