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Heat-content and diffusive leakage from material sets in the low-diffusivity limit
We generalize leading-order asymptotics of a form of the heat content of a submanifold (van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed i...
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| Published in: | Nonlinearity 2021-10, Vol.34 (10), p.7303-7321 |
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| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c322t-f7d474ef416ac2d6733c95ec85fe55439b4ee5a1f7fb9e9941cee4bf8189be3 |
| container_end_page | 7321 |
| container_issue | 10 |
| container_start_page | 7303 |
| container_title | Nonlinearity |
| container_volume | 34 |
| creator | Schilling, Nathanael Karrasch, Daniel Junge, Oliver |
| description | We generalize leading-order asymptotics of a form of the
heat content of a submanifold
(van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity
ɛ
goes to zero, the diffusive transport out of a material set
S
under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function
1
S
, is
ε
/
π
d
A
¯
(
∂
S
)
+
o
(
ε
)
. The surface measure
d
A
¯
is that of the so-called
geometry of mixing
, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems. |
| doi_str_mv | 10.1088/1361-6544/ac18b1 |
| format | article |
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heat content of a submanifold
(van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity
ɛ
goes to zero, the diffusive transport out of a material set
S
under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function
1
S
, is
ε
/
π
d
A
¯
(
∂
S
)
+
o
(
ε
)
. The surface measure
d
A
¯
is that of the so-called
geometry of mixing
, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.</description><identifier>ISSN: 0951-7715</identifier><identifier>EISSN: 1361-6544</identifier><identifier>DOI: 10.1088/1361-6544/ac18b1</identifier><identifier>CODEN: NONLE5</identifier><language>eng</language><publisher>IOP Publishing</publisher><subject>advection diffusion equation ; finite time coherent sets ; heat content</subject><ispartof>Nonlinearity, 2021-10, Vol.34 (10), p.7303-7321</ispartof><rights>2021 IOP Publishing Ltd & London Mathematical Society</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c322t-f7d474ef416ac2d6733c95ec85fe55439b4ee5a1f7fb9e9941cee4bf8189be3</citedby><cites>FETCH-LOGICAL-c322t-f7d474ef416ac2d6733c95ec85fe55439b4ee5a1f7fb9e9941cee4bf8189be3</cites><orcidid>0000-0001-9403-6511 ; 0000-0002-3435-307X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Schilling, Nathanael</creatorcontrib><creatorcontrib>Karrasch, Daniel</creatorcontrib><creatorcontrib>Junge, Oliver</creatorcontrib><title>Heat-content and diffusive leakage from material sets in the low-diffusivity limit</title><title>Nonlinearity</title><addtitle>Non</addtitle><addtitle>Nonlinearity</addtitle><description>We generalize leading-order asymptotics of a form of the
heat content of a submanifold
(van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity
ɛ
goes to zero, the diffusive transport out of a material set
S
under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function
1
S
, is
ε
/
π
d
A
¯
(
∂
S
)
+
o
(
ε
)
. The surface measure
d
A
¯
is that of the so-called
geometry of mixing
, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.</description><subject>advection diffusion equation</subject><subject>finite time coherent sets</subject><subject>heat content</subject><issn>0951-7715</issn><issn>1361-6544</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp1kE9LAzEUxIMoWKt3j_kAxuZtkt3NUYpaoSCo95DNvmjq_ilJqvTb29LqzdPA4zePmSHkGvgt8LqegSiBlUrKmXVQN3BCJn-nUzLhWgGrKlDn5CKlFecAdSEm5GWBNjM3DhmHTO3Q0jZ4v0nhC2mH9tO-I_Vx7GlvM8ZgO5owJxoGmj92xPjNfvmQt7QLfciX5MzbLuHVUafk9eH-bb5gy-fHp_ndkjlRFJn5qpWVRC-htK5oy0oIpxW6WnlUSgrdSERlwVe-0ai1BIcoG19DrRsUU8IPX10cU4rozTqG3satAW72i5h9fbOvbw6L7Cw3B0sY12Y1buKwi_c__gPkSWPD</recordid><startdate>202110</startdate><enddate>202110</enddate><creator>Schilling, Nathanael</creator><creator>Karrasch, Daniel</creator><creator>Junge, Oliver</creator><general>IOP Publishing</general><scope>O3W</scope><scope>TSCCA</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-9403-6511</orcidid><orcidid>https://orcid.org/0000-0002-3435-307X</orcidid></search><sort><creationdate>202110</creationdate><title>Heat-content and diffusive leakage from material sets in the low-diffusivity limit</title><author>Schilling, Nathanael ; Karrasch, Daniel ; Junge, Oliver</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c322t-f7d474ef416ac2d6733c95ec85fe55439b4ee5a1f7fb9e9941cee4bf8189be3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>advection diffusion equation</topic><topic>finite time coherent sets</topic><topic>heat content</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Schilling, Nathanael</creatorcontrib><creatorcontrib>Karrasch, Daniel</creatorcontrib><creatorcontrib>Junge, Oliver</creatorcontrib><collection>Institute of Physics Open Access Journal Titles</collection><collection>IOPscience (Open Access)</collection><collection>CrossRef</collection><jtitle>Nonlinearity</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Schilling, Nathanael</au><au>Karrasch, Daniel</au><au>Junge, Oliver</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Heat-content and diffusive leakage from material sets in the low-diffusivity limit</atitle><jtitle>Nonlinearity</jtitle><stitle>Non</stitle><addtitle>Nonlinearity</addtitle><date>2021-10</date><risdate>2021</risdate><volume>34</volume><issue>10</issue><spage>7303</spage><epage>7321</epage><pages>7303-7321</pages><issn>0951-7715</issn><eissn>1361-6544</eissn><coden>NONLE5</coden><abstract>We generalize leading-order asymptotics of a form of the
heat content of a submanifold
(van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity
ɛ
goes to zero, the diffusive transport out of a material set
S
under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function
1
S
, is
ε
/
π
d
A
¯
(
∂
S
)
+
o
(
ε
)
. The surface measure
d
A
¯
is that of the so-called
geometry of mixing
, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.</abstract><pub>IOP Publishing</pub><doi>10.1088/1361-6544/ac18b1</doi><tpages>19</tpages><orcidid>https://orcid.org/0000-0001-9403-6511</orcidid><orcidid>https://orcid.org/0000-0002-3435-307X</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0951-7715 |
| ispartof | Nonlinearity, 2021-10, Vol.34 (10), p.7303-7321 |
| issn | 0951-7715 1361-6544 |
| language | eng |
| recordid | cdi_iop_journals_10_1088_1361_6544_ac18b1 |
| source | Institute of Physics:Jisc Collections:IOP Publishing Read and Publish 2024-2025 (Reading List) |
| subjects | advection diffusion equation finite time coherent sets heat content |
| title | Heat-content and diffusive leakage from material sets in the low-diffusivity limit |
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