regularity for the stationary navier-stokes …...regularity for the stationary navier-stokes...
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Regularity for the stationary Navier-Stokes equationsover bumpy boundaries and a local wall law
檜垣充朗 (神戸大学)
joint work with
Christophe Prange (CNRS 研究員・ボルドー大学)
The 45th Sapporo Symposium on Partial Di!erential Equations(北海道大学, 8月 17日, 2020)
Background: Boundary Roughness E!ect
We consider viscous incompressible fluids above rough bumpy boundariesx2 > !"(x1/!) with " Lipschitz and the no-slip boundary condition.
General concernThe e!ect of wall-roughness on fluid flows.! The flow may paradoxically be better behaved than flat boundaries.
The Navier wall lawThe wall law is a boundary condition on the flat boundary describing anaveraged e!ect from the O(!)-scale on large scale flows of order O(1).When the boundary is periodic, it gives a slip condition, with # = #("),
u1 = !#$2u1 , u2 = 0 on $R2+ .
Stationary: Jager ·Mikelic (’01), Gerard-Varet (’09),Gerard-Varet ·Masmoudi (’10)
Nonstationary: Mikelic ·Necasova ·Neuss-Radu (’13)IBVP: Higaki (’16)
2 / 17
Motivation
To investigate such e!ects from the point of view of the regularity theory,especially, of the mesoscopic regularity of the steady Navier-Stokes flows.
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2D steady Navier-Stokes equations
(NS!)
!"
#
"!u! +#p! = "u! ·#u! in B!1,+
# · u! = 0 in B!1,+
u! = 0 on "!1 .
u! = (u!1(x), u!2(x))
!: velocity field
p! = p!(x): pressure field
For ! $ (0, 1] and r $ (0, 1],
B!r,+ = {x $ R2 | x1 $ ("r, r) , !"(
x1!) < x2 < !"(
x1!) + r} ,
"!r = {x $ R2 | x1 $ ("r, r) , x2 = !"(
x1!)} .
" $ W 1,": boundary function, "(x1) $ ("1, 0) for all x1 $ RFor an open set # % R3 with the Lebesgue measure |#|,
"$
!|f | = 1
|#|
$
!|f | , (f)! =
1
|#|
$
!f .
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ミ飈
Regularity Theory
Small Scales (! !) ! The Schauder theoryThe small-scale regularity is determined by the regularity of data.
Ladyzenskaja (’69): Holder estimate by potential theory
Giaquinta ·Modica (’82): the Campanato spaces
Dependence on the continuity of "# when the boundary is x2 = "(x1).
Large Scales (! ! r & 1)The large-scale regularity is determined by the macroscopic properties.
Gerard-Varet (’09): C0,µ-est. uniform in ! by a mesoscopic Holder
%"$
B!r,+
|u!|2& 1
2
& C(µ)
%"$
B!1,+
|u!|2& 1
2
rµ , µ $ (0, 1) ,
combined with the classical estimates near the boundary x2 = !"(x1/!)
Kenig ·Prange (’18): linear elliptic system, mesoscopic Lipschitz
Zhuge (’20, preprint): mesoscopic Lipschitz, the quantitative method5 / 17
Main Theorems (CVPDE)
Theorem 1 (mesoscopic Lipschitz)
'M $ (0,(), ) !(1) = !(1)(*"*W 1,! ,M) $ (0, 1) s.t.' ! $ (0, !(1)], ' r $ [!/!(1), 1], any weak solution u! to (NS!) with
%"$
B!1,+
|u!|2& 1
2
& M(+)
satisfies
%"$
B!r,+
|u!|2& 1
2
& C(1)M r ,
where the constant C(1)M is independent of ! and r.
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Theorem 2 (polynomial approximation)
Fix M $ (0,(), µ $ (0, 1). Then, ) !(2) = !(2)(*"*W 1,! ,M, µ) $ (0, 1)s.t. for all weak solutions u! to (NS!) satisfying (+), the following holds.
(i) ' ! $ (0, !(2)], ' r $ [!/!(2), 1], we have
%"$
B!r,+
|u!(x)" c!rx2e1|2 dx& 1
2
& C(2)M (r1+µ + !
12 r
12 ) ,
where the coe"cient c!r = c!r(*"*W 1,! ,M, µ) is a functional of u!.
(ii) Let " be 2%-periodic in addition. Then, )# = #(*"*W 1,!) $ R s.t.' ! $ (0, !(2)], ' r $ [!/!(2), 1], we have
%"$
B!r,+
|u!(x)" c!r(x2 + !#)e1|2 dx& 1
2
& 'C(2)M (r1+µ + !
32 r$
12 ) .
RemarkThe polynomial approximation requires an analysis of the boundary layer.
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Remark (Consequences)
(i) When r = O(!), the estimates are no better than the one in Theorem1. Hence there is no improvement at this scale. On the other hand, if weconsider the case r $ [(!/!(2))", 1] with & $ (0, 1), then we see that
r1+µ + !12 r
12 & (1 + (!(2))
12 r
1""2" $µ)r1+µ .
Therefore, we call the estimates in Theorem 2 mesoscopic C1,µ-estimatesat the scales r $ [(!/!(2))", 1] with & $ (0, (2µ+ 1)$1].
(ii) A comparison between two estimates in Theorem 2 highlights theregularity improvement coming from the boundary periodicity: in fact,
!32 r$
12 < !
12 r
12 , r $ (!, 1] .
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Remark (Relation with the wall law)
(i) Let us define a polynomial P !N by
P !N (x) = (x2 + !#)e1 .
Then P !N is an explicit (shear flow) solution to
(NS!N )
!"
#
"!u!N +#p!N = "u!N ·#u!N in R2+
# · u!N = 0 in R2+
uN,1 = !#$2uN,1 , uN,2 = 0 on $R2
with a trivial pressure p!N = 0.
(ii) The second estimate reads as follows: any weak solution u! to (NS!)can be approximated, at mesoscopic scales, by the Navier polynomial P !
Nmultiplied by a constant depending on u! (a local Navier wall law).
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Strategy
We apply a compactness argument originating from the works byAvellaneda · Lin (’87, ’89) on uniform estimates in homogenization.
Compactness
The mesoscopic regularity is inherited from the limit system when ! , 0posed in a domain with a flat boundary. Here no regularity is needed forthe original boundary, beyond the boundedness of " and of its gradient.
We use such regularity in order to verify the boundary layer expansion!(("
((#
u!(x) = ($2u!1)B!r,+
)x2e1 + !v(
x
!)*+ o(r) in
%"$
B!r,+
| · |2& 1
2
,
p!(x) = ($2u!1)B!r,+
q(x
!) .
The strategies are summarized asConstruction of the boundary layer corrector (v, q)Mesoscopic regularity by compactnessIteration of the compactness argument
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Boundary Layer Corrector
The expansion u!(x) - !v(x/!) and p!(x) - q(x/!) leads to
(BL)
!"
#
"!v +#q = 0 , y $ #bl
# · v = 0 , y $ #bl
v(y#, "(y#)) = ""(y#)e1 ,
where #bl = {y $ R2 | y2 > "(y1)}.
Proposition 1
)! v $ H1loc(#
bl) to (BL) satisfying
sup#%Z
$ #+1
#
$ "
$(y#)|#v(y1, y2)|2 dy2 dy1 & C(*"*W 1,!) .
(Outlined Proof) Gerard-Varet ·Masmoudi (’10), Kenig ·Prange (’18).Equivalent problem on a strip with the Dirichlet-to-Neumann op. DN
Estimates for DN in H12uloc (Note that W 1," ', H
12uloc)
The Saint-Venant energy estimate controlling the nonlocality11 / 17
Compactness Argument
(MNS!)
!("
(#
"!U ! +#P ! = "# · (U ! . b! + b! . U !)
" (!U ! ·#U ! +# · F ! in B!1,+
# · U ! = 0 in B!1,+ , U ! = 0 on "!
1 ,
b!(x) = C!+x2e1 + !v(
x
!),, x $ B!
1,+ .
Note that # · b! = 0 in B!1,+ and b! = 0 on "!
1.
The Caccioppoli inequality
)K0 $ (0,() depending only on *"*W 1,! s.t. ' ) $ (0, 1), we have
*#U !*2L2(B!#,+) & K0
)(1" ))$2*U !*2L2(B!
1,+)
++|C!|4 + (1" ))$
43 |C!|
43,*U !*2L2(B!
1,+)
+ ((!)4(1" ))$4*U !*6L2(B!1,+) + *F !*2L2(B!
1,+)
*.
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Lemma 1
'* $ (0,(), 'M $ (0,(), 'µ $ (0, 1), ) )0 = )0(M,µ) $ (0, 18) s.t.
' " with *"*W 1,! & *, ' ) $ (0, )0], ) !µ = !µ(*,M, µ, )) $ (0, 1) s.t.
' ! $ (0, !µ], ' ((!, C!) $ ["1, 1]2, 'F ! $ L2(B!1,+)
3&3 with
*F !*L2(B!1,+) & M!µ ,
any weak solution U ! to (MNS!) with
"$
B!1,+
|U !|2 & M2(++)
satisfies
"$
B!#,+
--U !(x)" ($2U !1 )B!
#,+
+x2e1 + !v(
x
!),--2 dx & M2)2+2µ .
Remark
We can choose the scale parameter ) freely as long as ) $ (0, )0].13 / 17
Iteration
Lemma 2
Fix * $ (0,(), M $ (0,(), and µ $ (0, 1). Let )0 $ (0, 18) be theconstant in Lemma 1. Choose ) = )(M,µ) $ (0, )0] small to satisfy
4(1" ))32+C1(1" )µ)$1(6 + 28M4)
12M)
12,4 & 1
and C1(1" )µ)$2(6 + 28M4)M) & 1 ,
where C1 is a numerical constant. Moreover, let !µ = !µ()) $ (0, 1) bethe corresponding constant for ) in Lemma 1. Then, ' k $ N,' ! $ (0, )k$1()2(2+µ)!2µ)], any weak sol. u! to (NS!) with (+) satisfies
"$
B!#k,+
--u!(x)" a!k+x2e1 + !v(
x
!),--2 dx & M2)(2+2µ)k ,
|a!k| & C2)$ 3
2 (1" ))$1+6 + 26(1" ))$2M4
, 12M
k.
l=1
)µ(l$1) .
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Basic idea
Induction on k $ N using compactness (Lemma 1) at each step.
Di"culty
Nonlinearity and lack of smallness.
Let the estimates hold for k $ N and let ! $ (0, )k+2(2+µ)!2µ].
We define U !/%k = U !/%k(y) and P !/%k = P !/%k(y) by
U !/%k(y) =1
)(1+µ)k
)u!()ky)" )ka!k
+y2e1 +
!
)kv(
)ky
!),*
,
P !/%k(y) =1
)µk
)p!()ky)" a!kq(
)ky
!),*
.
Then we see that, by the induction assumption,
"$
B!/#k
1,+
|U !/%k |2 & M2
and that (U !/%k , P !/%k) is a weak solution to ...15 / 17
Modified Navier-Stokes equations!((((((((("
(((((((((#
"!yU!/%k +#yP
!/%k = "#y ·+U !/%k . ()kb!/%
k)
+()kb!/%k). U !/%k
,
")(2+µ)kU !/%k ·#yU!/%k
+#y · F !/%k in B!/%k
1,+
#y · U !/%k = 0 in B!/%k
1,+ U !/%k = 0 on "!/%k
1 ,
where
b!/%k(y) = C!
k
+y2e1 +
!
)kv(
)ky
!),, C!
k = )ka!k ,
F !/%k(y) = ")$µk+b!/%
k(y). b!/%
k(y)"
+C!ky2e1
,. (C!
ky2e1),.
Key ingredient
A suitable choice of the scale parameter ) = )(M) controlling thecoe"cients of M . This is done in the spirit of the Newton shooting.
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Proof of Theorem 2 (i)
Fix µ $ (0, 1) and set !(2) = )2(2+µ)!2µ. We take ! $ (0, !(2)].
Since every r $ [!/!(2), )] satisfies r $ ()k, )k$1] with some 2 < k $ N,%"$
B!r,+
|u!(x)" a!kx2e1|2 dx& 1
2
&%)$3"
$
B!#k"1,+
|u!(x)" a!kx2e1|2 dx& 1
2
& M)(1+µ)(k$1)$ 32 + )$
32 |a!k| !
%"$
B!#k"1,+
--v(x!)--2 dx
& 12
& M)(1+µ)(k$1)$ 32 +
))$
32 supk%N
|a!k|*!
12 ()k$1)
12 .
Then, from )k$1 $ (0, )$1r),%"$
B!r,+
|u!(x)" a!kx2e1|2 dx& 1
2
& C(2)(M,µ, ))(r1+µ + !12 r
12 ) .
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