Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field
We study anomalous dissipation in the context of passive scalars and we construct a two-dimensional autonomous divergence-free velocity field in $C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits anomalous dissipation. Our proof employs the fluctuation-dissipation formula, which...
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| creator | Johansson, Carl Johan Peter Sorella, Massimo |
| description | We study anomalous dissipation in the context of passive scalars and we
construct a two-dimensional autonomous divergence-free velocity field in
$C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits
anomalous dissipation. Our proof employs the fluctuation-dissipation formula,
which links spontaneous stochasticity with anomalous dissipation. Therefore, we
address the issue of anomalous dissipation by showing that the variance of
stochastic trajectories, in the zero noise limit, remains positive. Based on
this result, we answer Question 2.2 and Question 2.3 in [Bru\`{e} & De Lellis
'22] regarding anomalous dissipation for the forced three-dimensional
Navier-Stokes equations. |
| doi_str_mv | 10.48550/arxiv.2409.03599 |
| format | Article |
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construct a two-dimensional autonomous divergence-free velocity field in
$C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits
anomalous dissipation. Our proof employs the fluctuation-dissipation formula,
which links spontaneous stochasticity with anomalous dissipation. Therefore, we
address the issue of anomalous dissipation by showing that the variance of
stochastic trajectories, in the zero noise limit, remains positive. Based on
this result, we answer Question 2.2 and Question 2.3 in [Bru\`{e} & De Lellis
'22] regarding anomalous dissipation for the forced three-dimensional
Navier-Stokes equations.</description><identifier>DOI: 10.48550/arxiv.2409.03599</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs</subject><creationdate>2024-09</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2409.03599$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2409.03599$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Johansson, Carl Johan Peter</creatorcontrib><creatorcontrib>Sorella, Massimo</creatorcontrib><title>Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field</title><description>We study anomalous dissipation in the context of passive scalars and we
construct a two-dimensional autonomous divergence-free velocity field in
$C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits
anomalous dissipation. Our proof employs the fluctuation-dissipation formula,
which links spontaneous stochasticity with anomalous dissipation. Therefore, we
address the issue of anomalous dissipation by showing that the variance of
stochastic trajectories, in the zero noise limit, remains positive. Based on
this result, we answer Question 2.2 and Question 2.3 in [Bru\`{e} & De Lellis
'22] regarding anomalous dissipation for the forced three-dimensional
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construct a two-dimensional autonomous divergence-free velocity field in
$C^\alpha$ (with $\alpha \in (0,1)$ arbitrary but fixed) which exhibits
anomalous dissipation. Our proof employs the fluctuation-dissipation formula,
which links spontaneous stochasticity with anomalous dissipation. Therefore, we
address the issue of anomalous dissipation by showing that the variance of
stochastic trajectories, in the zero noise limit, remains positive. Based on
this result, we answer Question 2.2 and Question 2.3 in [Bru\`{e} & De Lellis
'22] regarding anomalous dissipation for the forced three-dimensional
Navier-Stokes equations.</abstract><doi>10.48550/arxiv.2409.03599</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Analysis of PDEs |
| title | Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field |
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