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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2604.13912 |
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| _version_ | 1866914475695669248 |
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| author | Burczak, Jan Székelyhidi, Jr., László Wu, Bian |
| author_facet | Burczak, Jan Székelyhidi, Jr., László Wu, Bian |
| contents | For any $β_0<1/3$ we construct divergence free vector fields in $ C_{x,t}^{β_0}$ and a sequence of diffusivities $κ_q \searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $ρ_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $κ_q$. At the same time $ρ_q$ remains uniformly bounded in $C_t^{0} C_x^{α_0}$, where $β_0 + 2α_0<1$. Our result confirms a conjecture of Armstrong and Vicol \cite{ArmstrongVicol} and shows sharpness of the Obukhov-Corrsin threshold within the context of iterated homogenization. Our construction confirms time-homogeneity of the dissipation anomaly, as required in turbulence theory, and as a consequence we also obtain better time regularity for the scalar $ρ_q$ than the classical prediction of Yaglom. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scalar anomalous dissipation and optimal regularity via iterated homogenization Burczak, Jan Székelyhidi, Jr., László Wu, Bian Analysis of PDEs 76B03, 35Q31 For any $β_0<1/3$ we construct divergence free vector fields in $ C_{x,t}^{β_0}$ and a sequence of diffusivities $κ_q \searrow 0$ such that, for an arbitrary initial datum from a low regularity class, the classical solution $ρ_q$ to the advection-diffusion equation exhibits anomalous dissipation along the sequence $κ_q$. At the same time $ρ_q$ remains uniformly bounded in $C_t^{0} C_x^{α_0}$, where $β_0 + 2α_0<1$. Our result confirms a conjecture of Armstrong and Vicol \cite{ArmstrongVicol} and shows sharpness of the Obukhov-Corrsin threshold within the context of iterated homogenization. Our construction confirms time-homogeneity of the dissipation anomaly, as required in turbulence theory, and as a consequence we also obtain better time regularity for the scalar $ρ_q$ than the classical prediction of Yaglom. |
| title | Scalar anomalous dissipation and optimal regularity via iterated homogenization |
| topic | Analysis of PDEs 76B03, 35Q31 |
| url | https://arxiv.org/abs/2604.13912 |