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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2307.03434 |
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| _version_ | 1866909758591598592 |
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| author | Miller, Evan |
| author_facet | Miller, Evan |
| contents | In this paper, we introduce the Fourier-restricted Euler and hypodissipative Navier--Stokes equations. These equations are analogous to the Euler and hypodissipative Navier--Stokes equations respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space; the $(u\cdot\nabla)u$ nonlinearity is otherwise unchanged. The constraint space restricts the divergence-free velocity to specific Fourier modes, which have a dyadic shell structure, and are constructed iteratively using permutations.
In the inviscid case -- and in the hypo-viscous case when $α<\frac{\log(3)}{6\log(2)} \approx .264$ -- we prove finite-time blowup for a set of solutions with a discrete group of symmetries. Our blowup Ansatz is odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$. The Fourier-restricted Euler and hypodissipative Navier--Stokes equations respect both the energy equality and the identity for enstrophy growth from the full Euler and hypodissipative Navier--Stokes equations respectively, which is a substantial advance over the previous literature on Euler and Navier--Stokes model equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03434 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite-time blowup for the Fourier-restricted Euler and hypodissipative Navier-Stokes model equations Miller, Evan Analysis of PDEs 35Q30, 35Q31 In this paper, we introduce the Fourier-restricted Euler and hypodissipative Navier--Stokes equations. These equations are analogous to the Euler and hypodissipative Navier--Stokes equations respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space; the $(u\cdot\nabla)u$ nonlinearity is otherwise unchanged. The constraint space restricts the divergence-free velocity to specific Fourier modes, which have a dyadic shell structure, and are constructed iteratively using permutations. In the inviscid case -- and in the hypo-viscous case when $α<\frac{\log(3)}{6\log(2)} \approx .264$ -- we prove finite-time blowup for a set of solutions with a discrete group of symmetries. Our blowup Ansatz is odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$. The Fourier-restricted Euler and hypodissipative Navier--Stokes equations respect both the energy equality and the identity for enstrophy growth from the full Euler and hypodissipative Navier--Stokes equations respectively, which is a substantial advance over the previous literature on Euler and Navier--Stokes model equations. |
| title | Finite-time blowup for the Fourier-restricted Euler and hypodissipative Navier-Stokes model equations |
| topic | Analysis of PDEs 35Q30, 35Q31 |
| url | https://arxiv.org/abs/2307.03434 |