ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations
In this paper, we establish some ε -regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows: 0.1 lim sup ϱ → 0 ϱ 1 - 2 p - ∑ j = 1 3 1 q j ‖ u ‖ L t p L x q → ( Q ( ϱ ) ) ≤ ε , 2 p + ∑ j = 1 3 1 q j ≤ 2 with q j > 1 ; sup - 1 ≤ t...
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| description | In this paper, we establish some
ε
-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows:
0.1
lim sup
ϱ
→
0
ϱ
1
-
2
p
-
∑
j
=
1
3
1
q
j
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
ϱ
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
≤
2
with
q
j
>
1
;
sup
-
1
≤
t
≤
0
‖
u
‖
L
q
→
(
B
(
1
)
)
≤
ε
,
1
q
1
+
1
q
2
+
1
q
3
<
2
with
1
<
q
j
<
∞
;
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
1
)
)
+
‖
Π
‖
L
1
(
Q
(
1
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
<
2
with
1
<
q
j
<
∞
,
which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in
L
p
→
(
R
3
)
with
1
p
1
+
1
p
2
+
1
p
3
<
2
. This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998). |
| doi_str_mv | 10.1007/s00033-020-01400-x |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_sprin</sourceid><recordid>TN_cdi_proquest_journals_2450345829</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2450345829</sourcerecordid><originalsourceid>FETCH-LOGICAL-p227t-8d1aee409d5ec368f85f5ba1e8d5fa31fbf9a088c466609b9cdbeb6b92eaba7f3</originalsourceid><addsrcrecordid>eNpFkEtOAzEMhiMEEqVwAVaRWAecyTyXqDylCiQe61Ey45SUMhmSGdRuUO_AioNwDQ7RkxBaJDa2bH_-bf2EHHI45gDZiQcAIRhEwIDHAGy-RQY8DmUBotgmA4A4ZlGUJbtkz_tpwDMOYkDev7_YHU76mXSmW9AqRHRGUtNQ2RhvO2dbU9ExKvSTHqlvZYU-zOrQc3KxWn566nGmmTcvJqhQb2d9Z2zjaWdp94RUnNEb-WbQrZYf9519Duv42ss1s092tJx5PPjLQ_J4cf4wumLj28vr0emYteHnjuU1l4gxFHWClUhznSc6UZJjXidaCq6VLiTkeRWnaQqFKqpaoUpVEaFUMtNiSI42uq2zrz36rpza3jXhZBnFCYg4yaMiUGJD-daZZoLun-JQ_hpdbowug9Hl2uhyLn4AN4x3OQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2450345829</pqid></control><display><type>article</type><title>ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations</title><source>SpringerLink Journals - AutoHoldings</source><creator>Wang, Yanqing ; Wu, Gang ; Zhou, Daoguo</creator><creatorcontrib>Wang, Yanqing ; Wu, Gang ; Zhou, Daoguo</creatorcontrib><description><![CDATA[In this paper, we establish some
ε
-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows:
0.1
lim sup
ϱ
→
0
ϱ
1
-
2
p
-
∑
j
=
1
3
1
q
j
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
ϱ
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
≤
2
with
q
j
>
1
;
sup
-
1
≤
t
≤
0
‖
u
‖
L
q
→
(
B
(
1
)
)
≤
ε
,
1
q
1
+
1
q
2
+
1
q
3
<
2
with
1
<
q
j
<
∞
;
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
1
)
)
+
‖
Π
‖
L
1
(
Q
(
1
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
<
2
with
1
<
q
j
<
∞
,
which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in
L
p
→
(
R
3
)
with
1
p
1
+
1
p
2
+
1
p
3
<
2
. This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).]]></description><identifier>ISSN: 0044-2275</identifier><identifier>EISSN: 1420-9039</identifier><identifier>DOI: 10.1007/s00033-020-01400-x</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Arches ; Engineering ; Fluid dynamics ; Fluid flow ; Mathematical analysis ; Mathematical Methods in Physics ; Navier-Stokes equations ; Regularity ; Self-similarity ; Theoretical and Applied Mechanics</subject><ispartof>Zeitschrift für angewandte Mathematik und Physik, 2020, Vol.71 (5)</ispartof><rights>Springer Nature Switzerland AG 2020</rights><rights>Springer Nature Switzerland AG 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0001-6576-5934</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/s00033-020-01400-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00033-020-01400-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Wang, Yanqing</creatorcontrib><creatorcontrib>Wu, Gang</creatorcontrib><creatorcontrib>Zhou, Daoguo</creatorcontrib><title>ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations</title><title>Zeitschrift für angewandte Mathematik und Physik</title><addtitle>Z. Angew. Math. Phys</addtitle><description><![CDATA[In this paper, we establish some
ε
-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows:
0.1
lim sup
ϱ
→
0
ϱ
1
-
2
p
-
∑
j
=
1
3
1
q
j
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
ϱ
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
≤
2
with
q
j
>
1
;
sup
-
1
≤
t
≤
0
‖
u
‖
L
q
→
(
B
(
1
)
)
≤
ε
,
1
q
1
+
1
q
2
+
1
q
3
<
2
with
1
<
q
j
<
∞
;
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
1
)
)
+
‖
Π
‖
L
1
(
Q
(
1
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
<
2
with
1
<
q
j
<
∞
,
which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in
L
p
→
(
R
3
)
with
1
p
1
+
1
p
2
+
1
p
3
<
2
. This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).]]></description><subject>Arches</subject><subject>Engineering</subject><subject>Fluid dynamics</subject><subject>Fluid flow</subject><subject>Mathematical analysis</subject><subject>Mathematical Methods in Physics</subject><subject>Navier-Stokes equations</subject><subject>Regularity</subject><subject>Self-similarity</subject><subject>Theoretical and Applied Mechanics</subject><issn>0044-2275</issn><issn>1420-9039</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNpFkEtOAzEMhiMEEqVwAVaRWAecyTyXqDylCiQe61Ey45SUMhmSGdRuUO_AioNwDQ7RkxBaJDa2bH_-bf2EHHI45gDZiQcAIRhEwIDHAGy-RQY8DmUBotgmA4A4ZlGUJbtkz_tpwDMOYkDev7_YHU76mXSmW9AqRHRGUtNQ2RhvO2dbU9ExKvSTHqlvZYU-zOrQc3KxWn566nGmmTcvJqhQb2d9Z2zjaWdp94RUnNEb-WbQrZYf9519Duv42ss1s092tJx5PPjLQ_J4cf4wumLj28vr0emYteHnjuU1l4gxFHWClUhznSc6UZJjXidaCq6VLiTkeRWnaQqFKqpaoUpVEaFUMtNiSI42uq2zrz36rpza3jXhZBnFCYg4yaMiUGJD-daZZoLun-JQ_hpdbowug9Hl2uhyLn4AN4x3OQ</recordid><startdate>2020</startdate><enddate>2020</enddate><creator>Wang, Yanqing</creator><creator>Wu, Gang</creator><creator>Zhou, Daoguo</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope/><orcidid>https://orcid.org/0000-0001-6576-5934</orcidid></search><sort><creationdate>2020</creationdate><title>ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations</title><author>Wang, Yanqing ; Wu, Gang ; Zhou, Daoguo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p227t-8d1aee409d5ec368f85f5ba1e8d5fa31fbf9a088c466609b9cdbeb6b92eaba7f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Arches</topic><topic>Engineering</topic><topic>Fluid dynamics</topic><topic>Fluid flow</topic><topic>Mathematical analysis</topic><topic>Mathematical Methods in Physics</topic><topic>Navier-Stokes equations</topic><topic>Regularity</topic><topic>Self-similarity</topic><topic>Theoretical and Applied Mechanics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wang, Yanqing</creatorcontrib><creatorcontrib>Wu, Gang</creatorcontrib><creatorcontrib>Zhou, Daoguo</creatorcontrib><jtitle>Zeitschrift für angewandte Mathematik und Physik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wang, Yanqing</au><au>Wu, Gang</au><au>Zhou, Daoguo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations</atitle><jtitle>Zeitschrift für angewandte Mathematik und Physik</jtitle><stitle>Z. Angew. Math. Phys</stitle><date>2020</date><risdate>2020</risdate><volume>71</volume><issue>5</issue><issn>0044-2275</issn><eissn>1420-9039</eissn><abstract><![CDATA[In this paper, we establish some
ε
-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows:
0.1
lim sup
ϱ
→
0
ϱ
1
-
2
p
-
∑
j
=
1
3
1
q
j
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
ϱ
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
≤
2
with
q
j
>
1
;
sup
-
1
≤
t
≤
0
‖
u
‖
L
q
→
(
B
(
1
)
)
≤
ε
,
1
q
1
+
1
q
2
+
1
q
3
<
2
with
1
<
q
j
<
∞
;
‖
u
‖
L
t
p
L
x
q
→
(
Q
(
1
)
)
+
‖
Π
‖
L
1
(
Q
(
1
)
)
≤
ε
,
2
p
+
∑
j
=
1
3
1
q
j
<
2
with
1
<
q
j
<
∞
,
which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincaré Anal Non Linéaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in
L
p
→
(
R
3
)
with
1
p
1
+
1
p
2
+
1
p
3
<
2
. This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Nečas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).]]></abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00033-020-01400-x</doi><orcidid>https://orcid.org/0000-0001-6576-5934</orcidid></addata></record> |
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| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Arches Engineering Fluid dynamics Fluid flow Mathematical analysis Mathematical Methods in Physics Navier-Stokes equations Regularity Self-similarity Theoretical and Applied Mechanics |
| title | ε-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations |
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