On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations
In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p ( u ) = ∇ · ( | D ( u ) | p - 2 D ( u ) ) with D ( u ) = 1 2 ( ∇ u + ( ∇ u ) ⊤ ) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution...
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Published in | Journal of nonlinear science Vol. 30; no. 4; pp. 1503 - 1517 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.08.2020
Springer Nature B.V |
Subjects | |
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Abstract | In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in
R
3
, having the diffusion term
A
p
(
u
)
=
∇
·
(
|
D
(
u
)
|
p
-
2
D
(
u
)
)
with
D
(
u
)
=
1
2
(
∇
u
+
(
∇
u
)
⊤
)
,
3
/
2
<
p
<
3
. In the case
3
/
2
<
p
≤
9
/
5
, we show that a suitable weak solution
u
∈
W
1
,
p
(
R
3
)
satisfying
lim inf
R
→
∞
|
u
B
(
R
)
|
=
0
is trivial, i.e.,
u
≡
0
. On the other hand, for
9
/
5
<
p
<
3
we prove the following Liouville type theorem: if there exists a matrix valued function
V
=
{
V
ij
}
such that
∂
j
V
ij
=
u
i
(summation convention), whose
L
3
p
2
p
-
3
mean oscillation has the following growth condition at infinity,
∫
-
B
(
r
)
|
V
-
V
B
(
r
)
|
3
p
2
p
-
3
d
x
≤
C
r
9
-
4
p
2
p
-
3
∀
1
<
r
<
+
∞
,
then
u
≡
0
. |
---|---|
AbstractList | In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R3, having the diffusion term Ap(u)=∇·(|D(u)|p-2D(u)) with D(u)=12(∇u+(∇u)⊤), 3/2<p<3. In the case 3/2<p≤9/5, we show that a suitable weak solution u∈W1,p(R3) satisfying lim infR→∞|uB(R)|=0 is trivial, i.e., u≡0. On the other hand, for 9/5<p<3 we prove the following Liouville type theorem: if there exists a matrix valued function V={Vij} such that ∂jVij=ui(summation convention), whose L3p2p-3 mean oscillation has the following growth condition at infinity, ∫-B(r)|V-VB(r)|3p2p-3dx≤Cr9-4p2p-3∀1<r<+∞,then u≡0. In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p ( u ) = ∇ · ( | D ( u ) | p - 2 D ( u ) ) with D ( u ) = 1 2 ( ∇ u + ( ∇ u ) ⊤ ) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p ( R 3 ) satisfying lim inf R → ∞ | u B ( R ) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B ( r ) | V - V B ( r ) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 . |
Author | Chae, Dongho Wolf, Jörg |
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Cites_doi | 10.1088/0951-7715/29/8/2191 10.1016/j.crma.2018.12.007 10.1007/s00220-013-1868-x 10.1016/j.jfa.2016.06.019 10.1016/j.jmaa.2013.04.040 10.1137/S0036141002410988 10.1007/s00526-019-1549-5 10.1007/s11511-009-0039-6 10.1016/j.jde.2016.08.014 10.1007/s00021-015-0202-0 10.1007/978-0-387-09620-9 10.1007/978-3-0348-0551-3 10.21136/AM.1996.134320 |
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References | Koch, Nadirashvili, Seregin, Šverék (CR10) 2009; 203 Gilbarg, Weinberger (CR9) 1978; 4 Leray (CR13) 1933; 12 Sohr (CR18) 2001 Galdi (CR7) 2011 Chae (CR1) 2014; 326 Frehse, Málek, Steinhauer (CR6) 2003; 34 Wilkinson (CR19) 1960 Korobkov, Pileckas, Russo (CR11) 2015; 17 Seregin (CR15) 2016; 29 Kozono, Terasawa, Wakasugi (CR12) 2017; 272 Chae, Wolf (CR3) 2016; 261 Giaquinta (CR8) 1983 Chamorro, Jarrin, Lemarié-Rieusset (CR5) 2019; 357 Seregin (CR16) 2018; 30 Chae, Wolf (CR2) 2019; 58 Seregin, Wang (CR17) 2019; 31 Pokorný (CR14) 1996; 41 Chae, Yoneda (CR4) 2013; 405 D Chae (9615_CR2) 2019; 58 D Chae (9615_CR3) 2016; 261 M Korobkov (9615_CR11) 2015; 17 G Seregin (9615_CR17) 2019; 31 GP Galdi (9615_CR7) 2011 D Chae (9615_CR1) 2014; 326 D Chae (9615_CR4) 2013; 405 WL Wilkinson (9615_CR19) 1960 G Seregin (9615_CR15) 2016; 29 M Pokorný (9615_CR14) 1996; 41 H Sohr (9615_CR18) 2001 J Frehse (9615_CR6) 2003; 34 D Gilbarg (9615_CR9) 1978; 4 H Kozono (9615_CR12) 2017; 272 G Koch (9615_CR10) 2009; 203 M Giaquinta (9615_CR8) 1983 G Seregin (9615_CR16) 2018; 30 J Leray (9615_CR13) 1933; 12 D Chamorro (9615_CR5) 2019; 357 |
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Equ. doi: 10.1016/j.jde.2016.08.014 contributor: fullname: Wolf – volume: 17 start-page: 287 issue: 2 year: 2015 end-page: 293 ident: CR11 article-title: The Liouville theorem for the steady-state Navier–Stokes problem for axially symmetric 3D solutions in absence of swirl publication-title: J. Math. Fluid Mech. doi: 10.1007/s00021-015-0202-0 contributor: fullname: Russo – volume: 58 start-page: 111 issue: 3 year: 2019 ident: 9615_CR2 publication-title: Cal. Var. PDE doi: 10.1007/s00526-019-1549-5 contributor: fullname: D Chae – volume: 4 start-page: 381 issue: 5 year: 1978 ident: 9615_CR9 publication-title: Ann. Sc. Norm. Super. Pisa contributor: fullname: D Gilbarg – volume: 272 start-page: 804 year: 2017 ident: 9615_CR12 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2016.06.019 contributor: fullname: H Kozono – volume-title: An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems year: 2011 ident: 9615_CR7 doi: 10.1007/978-0-387-09620-9 contributor: fullname: GP Galdi – volume-title: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems year: 1983 ident: 9615_CR8 contributor: fullname: M Giaquinta – volume: 31 start-page: 269 issue: 2 year: 2019 ident: 9615_CR17 publication-title: Algebra i Analiz. contributor: fullname: G Seregin – volume: 326 start-page: 37 year: 2014 ident: 9615_CR1 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1868-x contributor: fullname: D Chae – volume: 12 start-page: 1 year: 1933 ident: 9615_CR13 publication-title: J. Math. Pures Appl. contributor: fullname: J Leray – volume-title: Non-Newtonian Fluids. Fluid Mechanics, Mixing and Heat Transfer year: 1960 ident: 9615_CR19 contributor: fullname: WL Wilkinson – volume: 34 start-page: 1064 issue: 5 year: 2003 ident: 9615_CR6 publication-title: SIAM J. Math. Anal. doi: 10.1137/S0036141002410988 contributor: fullname: J Frehse – volume: 30 start-page: 238 issue: 2 year: 2018 ident: 9615_CR16 publication-title: Algebra Anal. contributor: fullname: G Seregin – volume-title: The Navier–Stokes Equations. An Elementary Functional Analytic Approach year: 2001 ident: 9615_CR18 doi: 10.1007/978-3-0348-0551-3 contributor: fullname: H Sohr – volume: 203 start-page: 83 year: 2009 ident: 9615_CR10 publication-title: Acta Math. doi: 10.1007/s11511-009-0039-6 contributor: fullname: G Koch – volume: 29 start-page: 2191 year: 2016 ident: 9615_CR15 publication-title: Nonlinearity doi: 10.1088/0951-7715/29/8/2191 contributor: fullname: G Seregin – volume: 357 start-page: 175 issue: 2 year: 2019 ident: 9615_CR5 publication-title: Math. Acad. Sci. Paris. doi: 10.1016/j.crma.2018.12.007 contributor: fullname: D Chamorro – volume: 17 start-page: 287 issue: 2 year: 2015 ident: 9615_CR11 publication-title: J. Math. Fluid Mech. doi: 10.1007/s00021-015-0202-0 contributor: fullname: M Korobkov – volume: 261 start-page: 5541 year: 2016 ident: 9615_CR3 publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2016.08.014 contributor: fullname: D Chae – volume: 405 start-page: 706 issue: 2 year: 2013 ident: 9615_CR4 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2013.04.040 contributor: fullname: D Chae – volume: 41 start-page: 169 issue: 3 year: 1996 ident: 9615_CR14 publication-title: Appl. Math. doi: 10.21136/AM.1996.134320 contributor: fullname: M Pokorný |
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Snippet | In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in
R
3
, having the diffusion term
A
p
(
u
)
=
∇
·
(
|
D
(
u
)
|
p
-
2
D
(
u... In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R3, having the diffusion term Ap(u)=∇·(|D(u)|p-2D(u)) with... |
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SubjectTerms | Analysis Classical Mechanics Economic Theory/Quantitative Economics/Mathematical Methods Liouville theorem Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Newtonian fluids Non Newtonian fluids Theoretical |
Title | On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations |
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