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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2107.03659 |
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| _version_ | 1866929241258459136 |
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| author | Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca |
| author_facet | Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca |
| contents | We deal with the vanishing viscosity scheme for the transport/continuity equation $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$ drifted by a divergence-free vector field $\boldsymbol{b}$. Under general Sobolev assumptions on $\boldsymbol{b}$, we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8, 28, 29, 30], and rules out the possibility of anomalous dissipation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_03659 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity Bonicatto, Paolo Ciampa, Gennaro Crippa, Gianluca Analysis of PDEs 35F10, 35K15, 35Q35 We deal with the vanishing viscosity scheme for the transport/continuity equation $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$ drifted by a divergence-free vector field $\boldsymbol{b}$. Under general Sobolev assumptions on $\boldsymbol{b}$, we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8, 28, 29, 30], and rules out the possibility of anomalous dissipation. |
| title | On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity |
| topic | Analysis of PDEs 35F10, 35K15, 35Q35 |
| url | https://arxiv.org/abs/2107.03659 |