Anomalous diffusion by fractal homogenization
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910642604081152 |
|---|---|
| author | Armstrong, Scott Vicol, Vlad |
| author_facet | Armstrong, Scott Vicol, Vlad |
| contents | For every $α< \frac13$, we construct an explicit divergence-free vector field $\mathbf{b}(t,x)$ which is periodic in space and time and belongs to $C^0_t C^α_x \cap C^α_t C^0_x$ such that the corresponding scalar advection-diffusion equation $$\partial_t θ^κ+ \mathbf{b} \cdot \nabla θ^κ- κΔθ^κ= 0$$ exhibits anomalous dissipation of scalar variance for arbitrary $H^1$ initial data: $$\limsup_{κ\to 0} \int_0^{1} \int_{\mathbb{T}^d} κ\| \nabla θ^κ(t,x) \|^2 \,dx\,dt >0.$$ The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small $κ$ is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05048 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anomalous diffusion by fractal homogenization Armstrong, Scott Vicol, Vlad Analysis of PDEs Mathematical Physics Fluid Dynamics 76F30, 76F25, 35B27, 35Q35 For every $α< \frac13$, we construct an explicit divergence-free vector field $\mathbf{b}(t,x)$ which is periodic in space and time and belongs to $C^0_t C^α_x \cap C^α_t C^0_x$ such that the corresponding scalar advection-diffusion equation $$\partial_t θ^κ+ \mathbf{b} \cdot \nabla θ^κ- κΔθ^κ= 0$$ exhibits anomalous dissipation of scalar variance for arbitrary $H^1$ initial data: $$\limsup_{κ\to 0} \int_0^{1} \int_{\mathbb{T}^d} κ\| \nabla θ^κ(t,x) \|^2 \,dx\,dt >0.$$ The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small $κ$ is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one. |
| title | Anomalous diffusion by fractal homogenization |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics 76F30, 76F25, 35B27, 35Q35 |
| url | https://arxiv.org/abs/2305.05048 |