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The Sobolev Stability Threshold for 2D Shear Flows Near Couette
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The Sobolev Stability Threshold for 2D Shear Flows Near Couette
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Journal Article
The Sobolev Stability Threshold for 2D Shear Flows Near Couette
2018
Overview
We consider the 2D Navier–Stokes equation on
T
×
R
, with initial datum that is
ε
-close in
H
N
to a shear flow (
U
(
y
), 0), where
‖
U
(
y
)
-
y
‖
H
N
+
4
≪
1
and
N
>
1
. We prove that if
ε
≪
ν
1
/
2
, where
ν
denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains
ε
-close in
H
1
to
(
e
t
ν
∂
y
y
U
(
y
)
,
0
)
for all
t
>
0
. Moreover, the solution converges to a decaying shear flow for times
t
≫
ν
-
1
/
3
by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than
ν
1
/
2
for 2D shear flows close to the Couette flow.
Publisher
Springer US,Springer Nature B.V
