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Page 1: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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)

Page 2: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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)

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Page 3: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 4: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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

Page 5: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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

Page 6: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 7: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 8: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 9: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 10: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 11: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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

Page 12: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 13: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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

Page 14: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 15: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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

Page 16: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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Page 17: Regularity for the stationary Navier-Stokes …...Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law 檜垣充朗(神戸大学) joint

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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