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research
Global existence of weak solutions of the nematic liquid crystal flow in dimensions three
Authors
Fanghua Lin
Changyou Wang
Publication date
18 August 2014
Publisher
View
on
arXiv
Abstract
For any bounded smooth domain
Ξ©
β
R
3
\Omega\subset\mathbb R^3
Ξ©
β
R
3
, we establish the global existence of a weak solution
u
:
Ξ©
Γ
(
0
,
+
β
)
β
R
3
Γ
S
2
u:\Omega\times (0,+\infty)\to\mathbb R^3\times\mathbb S^2
u
:
Ξ©
Γ
(
0
,
+
β
)
β
R
3
Γ
S
2
of the initial-boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data
(
u
0
.
d
0
)
β
H
Γ
H
1
(
Ξ©
,
S
2
(u_0. d_0)\in {\bf H}\times H^1(\Omega,\mathbb S^2
(
u
0
β
.
d
0
β
)
β
H
Γ
H
1
(
Ξ©
,
S
2
), with
d
0
(
Ξ©
)
β
S
+
2
d_0(\Omega)\subset\mathbb S^2_+
d
0
β
(
Ξ©
)
β
S
+
2
β
(the upper hemisphere). Furthermore, (
u
,
d
u,d
u
,
d
) satisfies the global energy inequality (1.4).Comment: 24 page
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Last time updated on 30/10/2017