Inviscid Damping and Enhanced Dissipation of the Boundary Layer for 2D Navier–Stokes Linearized Around Couette Flow in a Channel
We study the 2D Navier–Stokes equations linearized around the Couette flow ( y , 0 ) t in the periodic channel T × [ - 1 , 1 ] with no-slip boundary conditions in the vanishing viscosity ν → 0 limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary correct...
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Veröffentlicht in: | Communications in mathematical physics 2020-10, Vol.379 (1), p.177-226 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | We study the 2D Navier–Stokes equations linearized around the Couette flow
(
y
,
0
)
t
in the periodic channel
T
×
[
-
1
,
1
]
with no-slip boundary conditions in the vanishing viscosity
ν
→
0
limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an
O
(
ν
1
/
3
)
boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of
both
the velocity
and
the vorticity associated to the boundary layer. For example, our
L
t
2
L
y
1
estimate of the boundary layer vorticity is
independent of
ν
, provided the initial data is
H
1
. For
L
2
data, the loss is only logarithmic in
ν
. Note both such estimates are false for the vorticity in the interior. To the authors’ knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected
O
(
exp
(
-
δ
ν
1
/
3
α
2
/
3
t
)
)
enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for
H
1
data that is against the boundary initially. For
L
2
data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation. |
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ISSN: | 0010-3616 1432-0916 |
DOI: | 10.1007/s00220-020-03851-9 |