Phase transitions in the fractional three-dimensional Navier-Stokes equations
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866914829676052480 |
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| author | Boutros, Daniel W. Gibbon, John D. |
| author_facet | Boutros, Daniel W. Gibbon, John D. |
| contents | The fractional Navier-Stokes equations on a periodic domain $[0,\,L]^{3}$ differ from their conventional counterpart by the replacement of the $-νΔ\mathbf{u}$ Laplacian term by $ν_{s}A^{s}\mathbf{u}$, where $A= - Δ$ is the Stokes operator and $ν_{s} = νL^{2(s-1)}$ is the viscosity parameter. Four critical values of the exponent $s\geq 0$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are: $s=\frac{1}{3}$; $s=\frac{3}{4}$; $s=\frac{5}{6}$ and $s=\frac{5}{4}$. In particular: i) for $s > \frac{1}{3}$ we prove an analogue of one of the Prodi-Serrin regularity criteria; ii) for $s \geq \frac{3}{4}$ we find an equation of local energy balance and; iii) for $s > \frac{5}{6}$ we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{1}{3}$ suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range $0 < s < \frac{1}{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07780 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Phase transitions in the fractional three-dimensional Navier-Stokes equations Boutros, Daniel W. Gibbon, John D. Analysis of PDEs Chaotic Dynamics Fluid Dynamics 35B33 (primary), 35B65, 35B60, 35B45, 35Q35, 35R11, 76D99 (secondary) The fractional Navier-Stokes equations on a periodic domain $[0,\,L]^{3}$ differ from their conventional counterpart by the replacement of the $-νΔ\mathbf{u}$ Laplacian term by $ν_{s}A^{s}\mathbf{u}$, where $A= - Δ$ is the Stokes operator and $ν_{s} = νL^{2(s-1)}$ is the viscosity parameter. Four critical values of the exponent $s\geq 0$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are: $s=\frac{1}{3}$; $s=\frac{3}{4}$; $s=\frac{5}{6}$ and $s=\frac{5}{4}$. In particular: i) for $s > \frac{1}{3}$ we prove an analogue of one of the Prodi-Serrin regularity criteria; ii) for $s \geq \frac{3}{4}$ we find an equation of local energy balance and; iii) for $s > \frac{5}{6}$ we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{1}{3}$ suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range $0 < s < \frac{1}{3}$. |
| title | Phase transitions in the fractional three-dimensional Navier-Stokes equations |
| topic | Analysis of PDEs Chaotic Dynamics Fluid Dynamics 35B33 (primary), 35B65, 35B60, 35B45, 35Q35, 35R11, 76D99 (secondary) |
| url | https://arxiv.org/abs/2303.07780 |