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Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds
Consider a noncompact Einstein manifold ( M , g ) with negative Ricci curvature tensor ( Ric ij = r g ij for a curvature constant r < 0 ). Denoting by Γ ( T M ) the set of all vector fields on M , we study the Navier–Stokes equations ∂ t u + ∇ u u + grad π = div ( ∇ u + ∇ u t ) ♯ , div u = 0 , u...
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Published in: | Archiv der Mathematik 2024, Vol.122 (1), p.83-93 |
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creator | Nguyen, Thieu Huy Vu, Thi Ngoc Ha |
description | Consider a noncompact Einstein manifold (
M
,
g
) with negative Ricci curvature tensor (
Ric
ij
=
r
g
ij
for a curvature constant
r
<
0
). Denoting by
Γ
(
T
M
)
the set of all vector fields on
M
, we study the Navier–Stokes equations
∂
t
u
+
∇
u
u
+
grad
π
=
div
(
∇
u
+
∇
u
t
)
♯
,
div
u
=
0
,
u
|
t
=
0
=
u
0
∈
Γ
(
T
M
)
,
div
u
0
=
0
,
for the vector field
u
∈
Γ
(
T
M
)
. Given any initial datum
u
0
∈
Γ
(
T
M
)
, we prove that if the curvature constant
r
is large enough, then the Navier–Stokes equations on the Einstein manifold (
M
,
g
) always have a unique solution
u
(
·
,
t
)
∈
Γ
(
T
M
)
which is defined for all
t
≥
0
with
u
(
·
,
0
)
=
u
0
. We also prove the exponential decay of solutions under appropriate conditions. |
doi_str_mv | 10.1007/s00013-023-01923-5 |
format | article |
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M
,
g
) with negative Ricci curvature tensor (
Ric
ij
=
r
g
ij
for a curvature constant
r
<
0
). Denoting by
Γ
(
T
M
)
the set of all vector fields on
M
, we study the Navier–Stokes equations
∂
t
u
+
∇
u
u
+
grad
π
=
div
(
∇
u
+
∇
u
t
)
♯
,
div
u
=
0
,
u
|
t
=
0
=
u
0
∈
Γ
(
T
M
)
,
div
u
0
=
0
,
for the vector field
u
∈
Γ
(
T
M
)
. Given any initial datum
u
0
∈
Γ
(
T
M
)
, we prove that if the curvature constant
r
is large enough, then the Navier–Stokes equations on the Einstein manifold (
M
,
g
) always have a unique solution
u
(
·
,
t
)
∈
Γ
(
T
M
)
which is defined for all
t
≥
0
with
u
(
·
,
0
)
=
u
0
. We also prove the exponential decay of solutions under appropriate conditions.</description><identifier>ISSN: 0003-889X</identifier><identifier>EISSN: 1420-8938</identifier><identifier>DOI: 10.1007/s00013-023-01923-5</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Curvature ; Fields (mathematics) ; Fluid flow ; Manifolds ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Navier-Stokes equations ; Tensors</subject><ispartof>Archiv der Mathematik, 2024, Vol.122 (1), p.83-93</ispartof><rights>Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-d58c0e3e947f716f56b6821a31a71dfbca6c8423e2517663f26c76b3022eb083</cites><orcidid>0000-0003-3790-8592</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>Nguyen, Thieu Huy</creatorcontrib><creatorcontrib>Vu, Thi Ngoc Ha</creatorcontrib><title>Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds</title><title>Archiv der Mathematik</title><addtitle>Arch. Math</addtitle><description>Consider a noncompact Einstein manifold (
M
,
g
) with negative Ricci curvature tensor (
Ric
ij
=
r
g
ij
for a curvature constant
r
<
0
). Denoting by
Γ
(
T
M
)
the set of all vector fields on
M
, we study the Navier–Stokes equations
∂
t
u
+
∇
u
u
+
grad
π
=
div
(
∇
u
+
∇
u
t
)
♯
,
div
u
=
0
,
u
|
t
=
0
=
u
0
∈
Γ
(
T
M
)
,
div
u
0
=
0
,
for the vector field
u
∈
Γ
(
T
M
)
. Given any initial datum
u
0
∈
Γ
(
T
M
)
, we prove that if the curvature constant
r
is large enough, then the Navier–Stokes equations on the Einstein manifold (
M
,
g
) always have a unique solution
u
(
·
,
t
)
∈
Γ
(
T
M
)
which is defined for all
t
≥
0
with
u
(
·
,
0
)
=
u
0
. We also prove the exponential decay of solutions under appropriate conditions.</description><subject>Curvature</subject><subject>Fields (mathematics)</subject><subject>Fluid flow</subject><subject>Manifolds</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Navier-Stokes equations</subject><subject>Tensors</subject><issn>0003-889X</issn><issn>1420-8938</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKAzEUhoMoWKsv4CrgejSXmUxmKaVeoChoF-5CJpNoapu0SaagK9_BN_RJjB3BnYtzzuK_HPgAOMXoHCNUX0SEEKYFInlwk3e1B0a4JKjgDeX7YJR1WnDePB2CoxgX2U143YxA92CVslD1YStTHzSUroPpRcNo3zX0Blpnk5VL2MkkofFhJ97JrdXh6-PzMflXHaHe9DJZ7yL0Dk6ti0lbB1fSWeOXXTwGB0Yuoz75vWMwv5rOJzfF7P76dnI5KxSpUSq6iiukqW7K2tSYmYq1jBMsKZY17kyrJFO8JFSTCteMUUOYqllLESG6RZyOwdlQuw5-0-uYxML3weWPgjSYMI5wibKLDC4VfIxBG7EOdiXDm8BI_MAUA0yRYYodTFHlEB1CMZvdsw5_1f-kvgELLHhO</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Nguyen, Thieu Huy</creator><creator>Vu, Thi Ngoc Ha</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-3790-8592</orcidid></search><sort><creationdate>2024</creationdate><title>Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds</title><author>Nguyen, Thieu Huy ; Vu, Thi Ngoc Ha</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-d58c0e3e947f716f56b6821a31a71dfbca6c8423e2517663f26c76b3022eb083</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Curvature</topic><topic>Fields (mathematics)</topic><topic>Fluid flow</topic><topic>Manifolds</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Navier-Stokes equations</topic><topic>Tensors</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Nguyen, Thieu Huy</creatorcontrib><creatorcontrib>Vu, Thi Ngoc Ha</creatorcontrib><collection>CrossRef</collection><jtitle>Archiv der Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Nguyen, Thieu Huy</au><au>Vu, Thi Ngoc Ha</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds</atitle><jtitle>Archiv der Mathematik</jtitle><stitle>Arch. Math</stitle><date>2024</date><risdate>2024</risdate><volume>122</volume><issue>1</issue><spage>83</spage><epage>93</epage><pages>83-93</pages><issn>0003-889X</issn><eissn>1420-8938</eissn><abstract>Consider a noncompact Einstein manifold (
M
,
g
) with negative Ricci curvature tensor (
Ric
ij
=
r
g
ij
for a curvature constant
r
<
0
). Denoting by
Γ
(
T
M
)
the set of all vector fields on
M
, we study the Navier–Stokes equations
∂
t
u
+
∇
u
u
+
grad
π
=
div
(
∇
u
+
∇
u
t
)
♯
,
div
u
=
0
,
u
|
t
=
0
=
u
0
∈
Γ
(
T
M
)
,
div
u
0
=
0
,
for the vector field
u
∈
Γ
(
T
M
)
. Given any initial datum
u
0
∈
Γ
(
T
M
)
, we prove that if the curvature constant
r
is large enough, then the Navier–Stokes equations on the Einstein manifold (
M
,
g
) always have a unique solution
u
(
·
,
t
)
∈
Γ
(
T
M
)
which is defined for all
t
≥
0
with
u
(
·
,
0
)
=
u
0
. We also prove the exponential decay of solutions under appropriate conditions.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00013-023-01923-5</doi><tpages>11</tpages><orcidid>https://orcid.org/0000-0003-3790-8592</orcidid></addata></record> |
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issn | 0003-889X 1420-8938 |
language | eng |
recordid | cdi_proquest_journals_2912680140 |
source | Springer Link |
subjects | Curvature Fields (mathematics) Fluid flow Manifolds Mathematical analysis Mathematics Mathematics and Statistics Navier-Stokes equations Tensors |
title | Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds |
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