On a transport equation with nonlocal drift
In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions ∂ t θ + u ∂ x θ = 0 , u = H θ , where H is the Hilbert transform. We provide four essentially different proofs of this f...
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| Veröffentlicht in: | Transactions of the American Mathematical Society 2016-09, Vol.368 (9), p.6159-6188 |
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| creator | Silvestre, Luis Vicol, Vlad |
| description | In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions
∂
t
θ
+
u
∂
x
θ
=
0
,
u
=
H
θ
,
where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion
∂
t
θ
+
u
∂
x
θ
+
Λ
γ
θ
=
0
,
u
=
H
θ
,
where Λ = (−Δ)1/2, and 1/2 ≤ γ |
| doi_str_mv | 10.1090/tran6651 |
| format | Article |
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∂
t
θ
+
u
∂
x
θ
=
0
,
u
=
H
θ
,
where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion
∂
t
θ
+
u
∂
x
θ
+
Λ
γ
θ
=
0
,
u
=
H
θ
,
where Λ = (−Δ)1/2, and 1/2 ≤ γ <1. Our results also apply to the model with velocity field u = Λ
sHθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C
(s+1)/2, for all positive time.
2010 Mathematics Subject Classification. Primary 35Q35.</description><identifier>ISSN: 0002-9947</identifier><identifier>EISSN: 1088-6850</identifier><identifier>DOI: 10.1090/tran6651</identifier><language>eng</language><publisher>American Mathematical Society</publisher><ispartof>Transactions of the American Mathematical Society, 2016-09, Vol.368 (9), p.6159-6188</ispartof><rights>Copyright 2015, American Mathematical Society</rights><rights>2016 American Mathematical Society</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-a330t-efd61f0f785e911951c75fef1fd0bf442fa043a8338c29b5a0f82c45396eb1683</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttp://www.ams.org/tran/2016-368-09/S0002-9947-2015-06651-3/S0002-9947-2015-06651-3.pdf$$EPDF$$P50$$Gams$$H</linktopdf><linktohtml>$$Uhttp://www.ams.org/tran/2016-368-09/S0002-9947-2015-06651-3/$$EHTML$$P50$$Gams$$H</linktohtml><link.rule.ids>68,69,315,781,785,804,833,23329,23333,27929,27930,58022,58026,58255,58259,77841,77843,77851,77853</link.rule.ids></links><search><creatorcontrib>Silvestre, Luis</creatorcontrib><creatorcontrib>Vicol, Vlad</creatorcontrib><title>On a transport equation with nonlocal drift</title><title>Transactions of the American Mathematical Society</title><description>In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions
∂
t
θ
+
u
∂
x
θ
=
0
,
u
=
H
θ
,
where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion
∂
t
θ
+
u
∂
x
θ
+
Λ
γ
θ
=
0
,
u
=
H
θ
,
where Λ = (−Δ)1/2, and 1/2 ≤ γ <1. Our results also apply to the model with velocity field u = Λ
sHθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C
(s+1)/2, for all positive time.
2010 Mathematics Subject Classification. Primary 35Q35.</description><issn>0002-9947</issn><issn>1088-6850</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1j01LxDAURYMoWEfBn1BwI0j1pWnSZCmDXzAwG12H1zRhOrTNmETEf29L1Z2rx-Wdc-EScknhloKCuxRwFILTI5JRkLIQksMxyQCgLJSq6lNyFuN-ilBJkZGb7ZhjPkvx4EPK7fsHps6P-WeXdvnox94b7PM2dC6dkxOHfbQXP3dF3h4fXtfPxWb79LK-3xTIGKTCulZQB66W3CpKFaem5s466lpoXFWVDqFiKBmTplQNR3CyNBVnStiGCslW5HrpNcHHGKzTh9ANGL40BT2P1L8jJxQWdB-TD3_c_MfBhgHTLnrTaSakVlpQriblalFwiP8XfwOGw2FV</recordid><startdate>20160901</startdate><enddate>20160901</enddate><creator>Silvestre, Luis</creator><creator>Vicol, Vlad</creator><general>American Mathematical Society</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160901</creationdate><title>On a transport equation with nonlocal drift</title><author>Silvestre, Luis ; Vicol, Vlad</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a330t-efd61f0f785e911951c75fef1fd0bf442fa043a8338c29b5a0f82c45396eb1683</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Silvestre, Luis</creatorcontrib><creatorcontrib>Vicol, Vlad</creatorcontrib><collection>CrossRef</collection><jtitle>Transactions of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Silvestre, Luis</au><au>Vicol, Vlad</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On a transport equation with nonlocal drift</atitle><jtitle>Transactions of the American Mathematical Society</jtitle><date>2016-09-01</date><risdate>2016</risdate><volume>368</volume><issue>9</issue><spage>6159</spage><epage>6188</epage><pages>6159-6188</pages><issn>0002-9947</issn><eissn>1088-6850</eissn><abstract>In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions
∂
t
θ
+
u
∂
x
θ
=
0
,
u
=
H
θ
,
where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion
∂
t
θ
+
u
∂
x
θ
+
Λ
γ
θ
=
0
,
u
=
H
θ
,
where Λ = (−Δ)1/2, and 1/2 ≤ γ <1. Our results also apply to the model with velocity field u = Λ
sHθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C
(s+1)/2, for all positive time.
2010 Mathematics Subject Classification. Primary 35Q35.</abstract><pub>American Mathematical Society</pub><doi>10.1090/tran6651</doi><tpages>30</tpages><oa>free_for_read</oa></addata></record> |
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| issn | 0002-9947 1088-6850 |
| language | eng |
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| source | American Mathematical Society Publications (Freely Accessible); Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; American Mathematical Society Publications |
| title | On a transport equation with nonlocal drift |
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