Well-Posedness for the Extended Schrödinger–Benjamin–Ono System
In this work we prove that the initial value problem associated to the Schrödinger–Benjamin–Ono type system i ∂ t u + ∂ x 2 u = u v + β u | u | 2 , ∂ t v - H x ∂ x 2 v + ρ v ∂ x v = ∂ x ( | u | 2 ) u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , with β , ρ ∈ R is locally well-posed for initial d...
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| Veröffentlicht in: | Vietnam journal of mathematics 2024-10, Vol.52 (4), p.1043-1066 |
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| container_title | Vietnam journal of mathematics |
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| creator | Linares, Felipe Mendez, Argenis J. Pilod, Didier |
| description | In this work we prove that the initial value problem associated to the Schrödinger–Benjamin–Ono type system
i
∂
t
u
+
∂
x
2
u
=
u
v
+
β
u
|
u
|
2
,
∂
t
v
-
H
x
∂
x
2
v
+
ρ
v
∂
x
v
=
∂
x
(
|
u
|
2
)
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
with
β
,
ρ
∈
R
is locally well-posed for initial data
(
u
0
,
v
0
)
∈
H
s
+
1
2
(
R
)
×
H
s
(
R
)
for
s
>
5
4
. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold
s
=
3
2
, we employ a refined Strichartz estimate introduced in the Benjamin–Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig. |
| doi_str_mv | 10.1007/s10013-023-00664-w |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3079869797</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3079869797</sourcerecordid><originalsourceid>FETCH-LOGICAL-c314t-ea0a92ace0fea3f75031007fc264d89ca30dcc3562ec0d27aafe16f1739694083</originalsourceid><addsrcrecordid>eNp9UDtOAzEQtRBIhMAFqFaiXhjbG3tdQvhKkYIUEHSWZY9JomQ32BuFdNyBu3ABbsJJcFgEHcXMvOK9eTOPkEMKxxRAnsTUKc-BpQIhiny1RTqMQy9njJXbv5g-7pK9GKeQWKWQHXL-gLNZfltHdBXGmPk6ZM0Ys4uXBiuHLhvZcfh4d5PqCcPn69sZVlMzn1QJDqs6G61jg_N9suPNLOLBz-yS-8uLu_51Phhe3fRPB7nltGhyNGAUMxbBo-Fe9oBvjveWicKVyhoOzlreEwwtOCaN8UiFp5IroQooeZcctXsXoX5eYmz0tF6GKllqDlKVQkklE4u1LBvqGAN6vQiTuQlrTUFvDHWblk5p6e-09CqJeCuKibz59W_1P6ovL6FwAw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3079869797</pqid></control><display><type>article</type><title>Well-Posedness for the Extended Schrödinger–Benjamin–Ono System</title><source>SpringerLink Journals - AutoHoldings</source><creator>Linares, Felipe ; Mendez, Argenis J. ; Pilod, Didier</creator><creatorcontrib>Linares, Felipe ; Mendez, Argenis J. ; Pilod, Didier</creatorcontrib><description>In this work we prove that the initial value problem associated to the Schrödinger–Benjamin–Ono type system
i
∂
t
u
+
∂
x
2
u
=
u
v
+
β
u
|
u
|
2
,
∂
t
v
-
H
x
∂
x
2
v
+
ρ
v
∂
x
v
=
∂
x
(
|
u
|
2
)
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
with
β
,
ρ
∈
R
is locally well-posed for initial data
(
u
0
,
v
0
)
∈
H
s
+
1
2
(
R
)
×
H
s
(
R
)
for
s
>
5
4
. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold
s
=
3
2
, we employ a refined Strichartz estimate introduced in the Benjamin–Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.</description><identifier>ISSN: 2305-221X</identifier><identifier>EISSN: 2305-2228</identifier><identifier>DOI: 10.1007/s10013-023-00664-w</identifier><language>eng</language><publisher>Singapore: Springer Nature Singapore</publisher><subject>Boundary value problems ; Energy methods ; Mathematics ; Mathematics and Statistics ; Original Article ; Well posed problems</subject><ispartof>Vietnam journal of mathematics, 2024-10, Vol.52 (4), p.1043-1066</ispartof><rights>The Author(s) 2023</rights><rights>The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-ea0a92ace0fea3f75031007fc264d89ca30dcc3562ec0d27aafe16f1739694083</cites><orcidid>0000-0002-2391-381X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10013-023-00664-w$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10013-023-00664-w$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Linares, Felipe</creatorcontrib><creatorcontrib>Mendez, Argenis J.</creatorcontrib><creatorcontrib>Pilod, Didier</creatorcontrib><title>Well-Posedness for the Extended Schrödinger–Benjamin–Ono System</title><title>Vietnam journal of mathematics</title><addtitle>Vietnam J. Math</addtitle><description>In this work we prove that the initial value problem associated to the Schrödinger–Benjamin–Ono type system
i
∂
t
u
+
∂
x
2
u
=
u
v
+
β
u
|
u
|
2
,
∂
t
v
-
H
x
∂
x
2
v
+
ρ
v
∂
x
v
=
∂
x
(
|
u
|
2
)
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
with
β
,
ρ
∈
R
is locally well-posed for initial data
(
u
0
,
v
0
)
∈
H
s
+
1
2
(
R
)
×
H
s
(
R
)
for
s
>
5
4
. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold
s
=
3
2
, we employ a refined Strichartz estimate introduced in the Benjamin–Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.</description><subject>Boundary value problems</subject><subject>Energy methods</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Article</subject><subject>Well posed problems</subject><issn>2305-221X</issn><issn>2305-2228</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9UDtOAzEQtRBIhMAFqFaiXhjbG3tdQvhKkYIUEHSWZY9JomQ32BuFdNyBu3ABbsJJcFgEHcXMvOK9eTOPkEMKxxRAnsTUKc-BpQIhiny1RTqMQy9njJXbv5g-7pK9GKeQWKWQHXL-gLNZfltHdBXGmPk6ZM0Ys4uXBiuHLhvZcfh4d5PqCcPn69sZVlMzn1QJDqs6G61jg_N9suPNLOLBz-yS-8uLu_51Phhe3fRPB7nltGhyNGAUMxbBo-Fe9oBvjveWicKVyhoOzlreEwwtOCaN8UiFp5IroQooeZcctXsXoX5eYmz0tF6GKllqDlKVQkklE4u1LBvqGAN6vQiTuQlrTUFvDHWblk5p6e-09CqJeCuKibz59W_1P6ovL6FwAw</recordid><startdate>20241001</startdate><enddate>20241001</enddate><creator>Linares, Felipe</creator><creator>Mendez, Argenis J.</creator><creator>Pilod, Didier</creator><general>Springer Nature Singapore</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2391-381X</orcidid></search><sort><creationdate>20241001</creationdate><title>Well-Posedness for the Extended Schrödinger–Benjamin–Ono System</title><author>Linares, Felipe ; Mendez, Argenis J. ; Pilod, Didier</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-ea0a92ace0fea3f75031007fc264d89ca30dcc3562ec0d27aafe16f1739694083</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Boundary value problems</topic><topic>Energy methods</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Article</topic><topic>Well posed problems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Linares, Felipe</creatorcontrib><creatorcontrib>Mendez, Argenis J.</creatorcontrib><creatorcontrib>Pilod, Didier</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Vietnam journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Linares, Felipe</au><au>Mendez, Argenis J.</au><au>Pilod, Didier</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Well-Posedness for the Extended Schrödinger–Benjamin–Ono System</atitle><jtitle>Vietnam journal of mathematics</jtitle><stitle>Vietnam J. Math</stitle><date>2024-10-01</date><risdate>2024</risdate><volume>52</volume><issue>4</issue><spage>1043</spage><epage>1066</epage><pages>1043-1066</pages><issn>2305-221X</issn><eissn>2305-2228</eissn><abstract>In this work we prove that the initial value problem associated to the Schrödinger–Benjamin–Ono type system
i
∂
t
u
+
∂
x
2
u
=
u
v
+
β
u
|
u
|
2
,
∂
t
v
-
H
x
∂
x
2
v
+
ρ
v
∂
x
v
=
∂
x
(
|
u
|
2
)
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
with
β
,
ρ
∈
R
is locally well-posed for initial data
(
u
0
,
v
0
)
∈
H
s
+
1
2
(
R
)
×
H
s
(
R
)
for
s
>
5
4
. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold
s
=
3
2
, we employ a refined Strichartz estimate introduced in the Benjamin–Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.</abstract><cop>Singapore</cop><pub>Springer Nature Singapore</pub><doi>10.1007/s10013-023-00664-w</doi><tpages>24</tpages><orcidid>https://orcid.org/0000-0002-2391-381X</orcidid><oa>free_for_read</oa></addata></record> |
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| language | eng |
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| subjects | Boundary value problems Energy methods Mathematics Mathematics and Statistics Original Article Well posed problems |
| title | Well-Posedness for the Extended Schrödinger–Benjamin–Ono System |
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