Zero-viscosity limit of the incompressible Navier-Stokes equations with sharp vorticity gradient

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Zero-viscosity limit of the incompressible Navier-Stokes equations with sharp vorticity gradient Jiajiang Liao a,b , Franck Sueur c,d , Ping Zhang e,b,∗ a Academy of Mathematics & Systems Science, The Chinese Academy of Sciences, Beijing 100190, China b School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China c Université Bordeaux, Institut de Mathématiques de Bordeaux, F-33405 Talence Cedex, France d Institut Universitaire de France, France e Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese

Academy of Sciences, Beijing 100190, China Received 2 March 2019; revised 22 October 2019; accepted 11 November 2019

Abstract It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit. Indeed we exhibit some sequences of exact solutions to the Navier-Stokes equations with vanishing viscosity which are given by multi-scale asymptotic expansions involving some characteristic boundary layers given by some linear PDEs. The convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the Euler equations. © 2019 Elsevier Inc. All rights reserved.

* Corresponding author.

E-mail addresses: [email protected] (J. Liao), [email protected] (F. Sueur), [email protected] (P. Zhang). https://doi.org/10.1016/j.jde.2019.11.018 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction and statement of the result 1.1. The equations at stake In this paper, we are concerned with the Cauchy problem in the full space R3 associated with the 3D incompressible fluid equations: 

∂t v ν + v ν · ∇v ν − νv ν + ∇p ν = 0, div v ν = 0,

(1.1)

where v ν and p ν respectively denote the fluid velocity and the scalar pressure function, and where ν ≥ 0 is the viscosity coefficient. When ν > 0 they are the Navier-Stokes equations and when ν = 0 they reduces to the Euler equations: 

∂t v 0 + v 0 · ∇v 0 + ∇p 0 = 0, div v 0 = 0.

(1.2)

1.2. Initial data with vorticity discontinuity across an hypersurface Many authors have studied the case where the initial data of (1.2) is a vortex patch, that is, when the initial vorticity is piecewise regular and discontinuous across a hypersurface of R3 . In this paper, we shall restrict for the sake of simplicity to the case where the hypersurface is C ∞ and where the vorticity is piecewise C ∞ . Let 0 be a compact connected smooth hypersurface in R3 . Then according to Jordan’s theorem, R3 \ 0 has two distinct connected components denoted by O0,± . We assume that O0,+ is the bounded one and that there exists a smooth function ϕ0 from R3 to R such that 0 = {ϕ0 = 0}, O0,± = {±ϕ0 > 0} and ∇ϕ0 = 0 in a neighborhood of 0 . We shall consider the initial velocity to be a divergence free vector field v0 in L2 (R3 ) whose vorticity def

ω0 = curl v0 is equal on O0,± to the restrictions to O0,± of a smooth compactly supported divergence free vector field on R3 . We observe that the tangential component of the vorticity can be discontinuous across the hypersurface 0 , with a well-defined jump, whereas the normal component of the vorticity is continuous across the hypersurface 0 . 1.3. A reminder on the case of the Euler equations For such vortex patch initial data, the existence and uniqueness of a local-in-time solution to the Euler equations (1.2) is well-known: there exists some T > 0 and a unique solution v 0 in L∞ ([0, T ]; Lip(R3 )) and in Lip([0, T ]; L2 (R3 )) to the Euler equations (1.2) with v0 as initial velocity, cf. for instance [8]. Moreover, for each t ∈ [0, T ], the vorticity is compactly supported and the pressure p0 in (1.2) can be chosen continuous across the hypersurface. Finally, the initial pattern is propagated in the sense that at any time in (0, T ), the vorticity ω0 is piecewise smooth and discontinuous across a time-dependent smooth hypersurface which evolves with the flow X 0 defined for (t, x) in [0, T ] × R3 by the differential equation:

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∂t X 0 (t, x) = v 0 (t, X 0 (t, x)), X 0 (0, x) = x.

def

(1.3)

def

For each t ∈ [0, T ], we set (t) = X 0 (t, 0 ) and O± (t) = X 0 (t, O0,± ). Theorem 1.1 ([2–6,8,10,18–20]). For each t ∈ [0, T ] the hypersurface (t) is C ∞ , the restrictions to O± (t) of the vorticity ω0 (t, ·) can be extended into vector fields which are C ∞ on the closure O± (t) of O± (t). We remark that the global existence for the Euler 2-D vortex patch problem was first established by Chemin [3–5], Bertozzi and Constantin [2]. Local existence for the 3-D vortex patch problem was first proved by Gamblin and Saint Raymond [8]; see also [10,19,20]. Coutand and Shkoller [6] investigated the propagation of higher Sobolev regularity for the velocity field of the vortex patch problem in both 2-D and 3-D. A very nice summary of results on vortex patch problems can be found in [18]. For the sequel it is interesting to recall that the following description of the hypersurface (t) is possible. We define ϕ 0 as the unique smooth solution for (t, x) in [0, T ] × R3 to the differential equation: 

 ∂t + v 0 · ∇ ϕ 0 = 0, ϕ 0 |t=0 = ϕ0 .

(1.4)

Then, for each t ∈ [0, T ], (t) = {ϕ(t, ·) = 0} and O± (t) = {±ϕ(t, ·) > 0}. Moreover there exists def

δ > 0 such that for 0  t  T , and x such that |ϕ 0 (t, x)| < δ, the vector n(t, x) = ∇x ϕ 0 (t, x) satisfies n(t, x) = 0. We stress that the vector n is not an unit vector even if it is so at t = 0, since it is stretched when time proceeds according to the equation: (∂t + v 0 · ∇)n = −t (∇v 0 ) · n.

(1.5)

1.4. A reminder on the inviscid limit Let us recall that the previous solution v 0 to the Euler equations on [0, T ] can be approximated by some strong solutions to the Navier-Stokes equations (1.1) on [0, T ] in the inviscid limit ν → 0+ , with a rate O(ν 3/4 ) in the energy space. Theorem 1.2 ([1,12]). There is a sequence (v ν )ν∈(0,1) of strong solutions to the Navier-Stokes equations (1.1) on [0, T ] converging to v 0 at rate O(ν 3/4 ) in L∞ ([0, T ]; L2 (R3 )) as ν → 0+ . Observe that the sequence (v ν )ν∈(0,1) in Theorem 1.2 is not required to start from the initial data v0 of the Euler solution v 0 . The optimality of the rate O(ν 3/4 ) in L∞ ([0, T ]; L2 (R3 )) is shown in [1] by considering the case of circular vortex patches. It has to be compared with the rate O(ν) associated with the case of smooth solutions to the Euler equations, see for instance [12]. The goal of this paper is to highlight how this decrease of the rate of convergence is due to some boundary layers associated with the discontinuity of the tangential component of the vorticity across the hypersurface (t).

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1.5. Multiscale asymptotic expansions and statement of the main result The main result of this paper is an extension of Theorem 1.2 where a sequence of exact solutions to the Navier-Stokes equations with vanishing viscosity is constructed by means √ of a multiscale asymptotic expansion involving an extra variable corresponding to ϕ 0(t, x)/ ν and some √ profiles V j (t, x, X) which are piecewise smooth in (t, x, X), evaluated at X = ϕ 0 (t, x)/ ν. More precisely the profiles V j (t, x, X) will be constructed as the sum of a regular part V j (t, x)  j def  and of a layer part V (t, x, X). We set O± = (t, x) : t ∈ (0, T ), x ∈ O± (t) . The regular part will be in the space H ∞ (O± ) of the functions on [0, T ] × R3 whose restrictions to O± are in the intersections of all the Sobolev spaces (built on L2 ) on O± . To describe the space in which we will take the layer parts we first introduce the function space p − S(R) of the functions f (X) whose restrictions to the half-lines R± are rapidly decreasing functions, that is for any j, k ∈ N, there is a constant Cj,k such that |X j ∂Xk f (X)| ≤ Cj,k

for ± X > 0.

def

The layer part will be in the space A∞ = H ∞ (O± ; p − S(R)). The following result establishes the existence of exact solutions to the Navier-Stokes equations with a precise description of the difference with the solution v 0 to the Euler equations on [0, T ] in the vanishing viscosity limit. Theorem 1.3. For any N ≥ 3, there are some vector fields V 1 in A∞ and (V j )2≤j ≤N in H ∞ (O± ) ⊕ A∞ and a ν-family (v ν )ν∈(0,1) of strong solutions to the Navier-Stokes equations (1.1) on [0, T ] such that, the ν-family of vector fields which map (t, x) to v ν (t, x) − v 0 (t, x) −

N  j =1

is O(ν

N− 2

 j ϕ 0 (t, x)  ν 2 V j t, x, √ ν

(1.6)

) in L∞ ([0, T ]; H (O± (t))) for 0 ≤ ≤ N as ν → 0+ .

Also we used the slightly abusive notation L∞ ([0, T ]; H (O± (t))): it describes the space of functions defined for almost all t , which associates with each such t a vector field on O± (t). There should be no ambiguity coming from this abuse of notation. This result thus establishes the existence of some strong solutions to the Navier-Stokes equations with vanishing viscosity under the form of a multiscale asymptotic expansion of arbitrary high order whose principal term is the solution to the Euler equations with a main corrector corresponding to a boundary layer, 1 1 for the velocity field, of amplitude ν 2 and of width ν 2 . This however corresponds to the reading of (1.6) in L∞ . In the energy space by using Lemma 3 of [7] which guarantees that 0 (·, ·)   j V ·, ·, ϕ √  ∞ 2 ≤ Cν 14 V j ∞ 2 1 , LT (LX (Hx )) LT (L ) ν

so that (1.6) implies that

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   ν  v − v 0 

2 L∞ T (L )

5

3 3 

1+2j 3 ≤C ν 2 + ν 4 V j L∞ (L2 (H 1 )) ≤ Cν 4 , T

X

x

i=1

so that Theorem 1.2 easily follows from Theorem 1.3. With a similar reasoning we can also deduce from Theorem 1.3 that for any 0 ≤ s < 34 , v ν converges to v 0 in L∞ ([0, T ]; H s (R3 )). Indeed the proof below will even provide more precise estimates by distinguishing the effect of normal and of tangential derivatives. Let us insist on the fact that the time T in Theorem 1.3 is the one associated with the solution v 0 to the Euler equations given in Section 1.3. Indeed in the proof below the profiles (V j ) are constructed as solutions of linear PDEs. In particular we refer to Proposition 4.1 where a profile V 1 is constructed as a solution to a linearized Prandtl-type equation. Moreover the rest (roughly speaking, the expression in (1.6)) is constructed as a solution of a variant of the Navier-Stokes equations which is weakly nonlinear in the sense that the nonlinearities are tamed by some extra powers of ν in the vanishing viscosity limit, see Section 6. Theorem 1.3 also completes the analysis in [18] where some approximate solutions of the Navier-Stokes equations with vanishing viscosity were constructed in the case where the initial data v0 is also prescribed for √ the Navier-Stokes equations. √ Accordingly the extra variable is there evaluated in ϕ 0 (t, x)/ νt, rather than in ϕ 0 (t, x)/ ν, to describe the initial set-up of the boundary layer. 2. Preliminaries Let us start this section by two general notations. • For two operators A, B, we set def

[A; B] = AB − BA, the commutator between A and B. • For a  b, we mean that there is a positive constant C such that a ≤ Cb. This constant may depend on the solution of the Euler equations given by Theorem 1.1 and may be different on different lines, but is uniform with respect to the viscosity parameter ν. 2.1. Tangential vector fields Let v 0 be the unique solution of (1.2) determined by Theorem 1.1 and X 0 (t, x) be the flow of v 0 determined by (1.3). We define the vector field: def

Dt = ∂t + v 0 · ∇.

(2.1)

We observe that Dt is tangential to the sets O± . Next we introduce a cut-off function χ(t, ·) ∈ C0∞ (R3 ) such that χ(t, x) = 0 when 0 |ϕ (t, x)| ≥ 2δ and χ(t, x) = 1 when |ϕ 0 (t, x)| < δ, and the vector fields set

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     def W0 = w01 = 0, −∂3 ϕ0 , ∂2 ϕ0 , w02 = ∂3 ϕ0 , 0, −∂1 ϕ0 ,     w03 = −∂2 ϕ0 , ∂1 ϕ0 , 0 , w04 = ∂3 (x3 (1 − χ0 )), 0, −∂1 (x3 (1 − χ0 )) ,   w05 = ∂2 (x1 (1 − χ0 )), −∂1 (x1 (1 − χ0 )), 0 , where χ0 (x) = χ(0, x). We remark that the above 5 vector fields, w0i , i = 1, · · · , 5, satisfy the admissible condition (see for instance Definition 3.1 of [8]): 

j

|w0i (x) ∧ w0 (x)|2 > 0 for any

x ∈ R3 .

1≤i
  j It is easy to observe that w0 · n0 (x) = 0 on the set x : |ϕ0 (x)| < δ . We define w j to be def

j

w j (t, X 0 (t, x)) = w0 (x) · ∇x X 0 (t, x), which verify the equations 

Dt w j = w j · ∇v 0 , j w j |t=0 = w0 .

Using (1.5) we observe that Dt (w j · n) = 0 and we deduce that (w j · n)(t, x) = 0 when |ϕ 0 (t, x)| < δ. Now we define the tangential derivatives def

def

α

Zj = w j · ∇ for 1 ≤ j ≤ 5 and Z α = Z1α1 · · · Z5 5 for α = (α1 , · · · , α5 ).

(2.2)

Observe that for any multi-index α, [Dt , Z α ] = 0.

(2.3)

On the other hand ∇Zj = Zj ∇ + ∇w j · ∇,

(2.4)

[; Zi ] = 2∇w i : ∇ 2 + w i · ∇

(2.5)

and for |α| = m ≥ 1, [; Z α ] =



∇(cβ ∇Z β ) +

|β|≤m−1



cγ ∇Z γ ,

(2.6)

|γ |≤m−1

where the functions cβ , cγ are some derivatives of w i which belong to L2 ∩ L∞ . j We observe that infR3 sup1i=j 5 |w0i × w0 | > 0 and deduce that for a 3-D vector-valued   function f , the divergence of f can be written on the set x : |ϕ 0 (t, x)| ≤ δ as div f = a −1 ∂n f · n +

 j

cj · Zj f,

(2.7)

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  where a, cj are smooth functions related to w j and a = 0 on the set x : |ϕ 0 (t, x)| ≤ δ . 2.2. Conormal norms k (R3 ) the space of functions so that Definition 2.1. We denote by Hco def

f 2k =

 |α|≤k

Z α f 2L2 (R3 ) < ∞.

We recall the generalized Sobolev-Gagliardo-Nirenberg inequality (see [9] for instance). Lemma 2.1. For u, v ∈ L∞

k (R3 ), we have Hco

  α Z 1 uZ α2 v   u L∞ v k + v L∞ u k

with |α1 | + |α2 | = k.

We also need the following parabolic interpolation inequality. Lemma 2.2. Let QT =  × (0, T ) and f ∈ L2 (QT ) ∩ Wp2,1 (QT ) for some p ∈ (5, ∞), where def 

Wp2,1 (QT ) =

 f : f, ∇x f, ∇x2 f, ∂t f ∈ Lp (QT ) .

(2.8)

Then there exists a constant C such that ∇x f L∞ (QT ) ≤ C f 1−θ (∂t f, ∇x2 f ) θLp (QT ) , L2 (Q ) T

where θ =

7p 9p−10

∈

7



9,1

.

Remark 2.3. When 5 < p < ∞, f ∈ Wp2,1 (QT ), ∇x f is actually a H older ¨ continuous function with respect to the parabolic distance. 3. Two auxiliary problems 3.1. The auxiliary layer problem Let a ∈ Cb∞ ([0, T ] × R3 ) be a positive function, which satisfies a = |n|2

  on (t, x) : |ϕ 0 (t, x)| < δ

and

inf a(t, x) ≥ c > 0

(t,x)

(3.1)

for some positive constant c. The notation Cb∞ used above stands for the set of smooth functions with bounded derivatives of any order. def

For a profile V (t, x, X), we denote by [V ]X = V |X=0+ − V |x=0− the jump across {X = 0} def

and by AV = V · ∇v 0 − 2

V ·∇v 0 ·n a

n for v 0 determined by Theorem 1.1.

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Proposition 3.1. Let g1 , g2 ∈ H ∞ (O± ), f ∈ A∞ . There is a vector field V ∈ A∞ which satisfies 

Dt V + AV − a∂X2 V = f for ± X > 0, [V ]X = g1 and [∂X V ]X = g2 for X = 0.

(3.2)

def

Proof. First we observe that the initial value V0 = V |t=0 of a vector field satisfying the conclusions of Proposition 3.1 has to satisfy a sequence of compatibility conditions on the hypersurface. For example the two first conditions are [V0 ]X (x) = g1 (0, x) and [∂X V0 ]X (x) = g2 (0, x), and the higher order compatibility conditions can be obtained by considering the jump of Dtk V on each sides of the hypersurface at the initial time. By Borel’s summation method there is V0 in H ∞ (O± (0); p − S(R)) satisfying this sequence of compatibility conditions. We refer here to [17,16] for more details on this construction in some close settings. Then we observe that the problem can be reduced to the case when g1 = g2 = 0 in (3.2) by considering V − ±g12−g2 e−|X| for ±X > 0 instead of V . Let us now present the a priori estimates in this case. The existence of a vector field satisfying the conclusions of Proposition 3.1 and these a priori estimates will follow from standard technics of linear PDEs theory. By multiplying (3.2) by V and integrating the resulting equality over def

U± (t) = O± (t) × R respectively, we find 





 Dt V · V + AV · V + a|∂X V |2 dx dX =

U± (t)

f · V dx dX.

U± (t)

For the first term, we observe that  U± (t)

1 d Dt V · V dx dX = 2 dt

 |V |2 dx dX. U± (t)

Note that |AV · V | ≤ C|V |2 for a constant C and thanks to (3.1), we find d dt



 |V |2 dx dX + U± (t)





 |V |2 + |f |2 dx dX.

|∂X V |2 dx dX ≤ C

U± (t)

U± (t)

Applying Gronwall inequality yields V 2L∞ (L2 (U

± (t)))

T

+ ∂X V 2L2 (L2 (U T

±

 V0 2L2 (U ≤ C (t)))

± (0))



+ f 2L2 (L2 (U

± (t)))

T

.

(3.3)

To gain the regularity of x, we claim that V 2L∞ (H k (L2 (U t

x

X

± (t))))

+ V 2L2 (H k (H˙ 1 (U (t)))) t x X ± ≤C V0 2H k (L2 (U (0))) + f 2L2 (H k (L2 (U x

X

±

t

x

X

± (t))))



(3.4) .

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In view of (3.3), (3.4) holds for k = 0. Inductively, let us assume that (3.4) holds for k ≤ m − 1, we are going to prove that (3.4) holds for k = m ∈ N + . To do it, we apply ∂xα with |α| = m to (3.2) to get (Dt + A − a∂X2 )∂xα V =f˜α , X ∈ R, x ∈ O± (t) with f˜α =∂xα f + [v 0 · ∇x ; ∂xα ]V − [a; ∂xα ]∂X2 V + [A; ∂xα ]V .

(3.5)

It is easy to observe that  |∂xα f · ∂xα V | dx dX ≤ U± (t)

  α 2 1  ∂ V  2 ∂ α f 2 2 + . x x L (U± (t)) L (U± (t)) 2

While it follows from Moser type inequality (see [11]) that  |[v 0 · ∇x ; ∂xα ]V · ∂xα V | dx dX ≤C V 2H k (L2 (U x

U± (t)

± (t)))

X

;

          [a; ∂xα ]∂X2 V · ∂xα V dx dX  = [a; ∂xα ]∂X V · ∂xα ∂X V dx dX   U± (t)

U± (t)

≤δ V 2H k (H˙ 1 (U x

X

± (t)))

+

C V 2 k−1 ˙ 1 ; Hx (HX (U± (t))) δ

and 



 |[A; ∂xα ]V · ∂xα V | + |A∂xα V · ∂xα V | dx dX ≤ C V 2H k (L2 (U x

U± (t)

X

± (t)))

.

Then we get, by taking L2 inner product of (3.5) with ∂xα V and inserting the above inequalities to the resulting one, that t V (t) 2H m (L2 (U (t))) x X ±

+

  V (t  )2

Hxm (H˙ X1 (U± (t  )))

0

+ f 2L2 (H m (L2 (U (t))) t x X ±

t +

  V (t  )2

dt  ≤ C V0 2H m (L2 (U

Hxm (L2X (U± (t  )))

x

X

dt  + V 2 2

± (0)))

Lt (Hxm−1 (H˙ X1 (U± (t))))

,

0

along as we choose δ to be sufficiently small. Applying Gronwall inequality and using (3.4) for k = m − 1, we conclude (3.4) for k = m. This in turn shows that (3.4) holds for any k ∈ N + . If we go back to our original case with jumps g1 , g2 and the original v 0 , V0 and f , we find that (3.2) has a unique solution V ∈ C([0, T ]; H k (O± (t), L2 (R))) ∩ L2 ([0, T ]; H k (O± (t), H˙ 1 (R± ))) for any k ≥ 1 with

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T sup

t∈[0,T ]

V (t) 2H k (O (t),L2 (R)) ±

≤ C V0 2H k (O

+

V (t) 2H k (O

± (t),H˙

1 (R

± ))

dt

0 2 ± (0),L (R))

+ f 2L2 (H k (O T

2 ± (t),L (R)))

(3.6)

+ (g1 , g2 ) H 1 (H k+1 (O± (t))) , T

where H m (R± ) denotes the space of functions f (X) which satisfies f |R+ ∈ H m (R+ ) and f |R− ∈ H m (R− ). j Inductively, by virtue of (3.2), we obtain that for any integer j ≥ 0, ∂X V satisfies ⎧ j 2 j ⎪ ⎨(Dt + A − a∂X )∂X V = ∂X f for ± X > 0, j j +1 [∂X V ]X = gj +1 and [∂X V ]X = gj +2 for X = 0, ⎪ ⎩ j j X ∈ R, x ∈ O± (0), ∂X V |t=0 = ∂X V0

(3.7)

j −3

def

where gj = a −1 (Dt gj −2 + Agj −2 − [∂X f ]X ) ∈ H ∞ (O± ) for j ≥ 3. Then we apply inequality (3.6) to (3.7) to prove by induction that for any k, m ∈ N, V ∈ C([0, T ]; H k (O± (t)), H m (R± )) ∩ L2 ([0, T ]; H k (O± (t)), H m+1 (R± )). By using the equation (3.2) we deduce that for any j, k, m ∈ N, V ∈ H j ([0, T ], H k (O± (t)), H m (R± )). Finally by induction we get for any n, j, k, m ∈ N, |X|n V ∈ H j ([0, T ], H k (O± (t)), H m (R± )), that is, V ∈ A∞ . This completes the proof of Proposition 3.1. 2 3.2. The auxiliary regular problem For a function f (t, x), we denote def

[f ] = f |ϕ 0 =0+ − f |ϕ 0 =0−

(3.8)

to be the jump across the hypersurface (t). Lemma 3.2. For any f ∈ H ∞ (O± ), g1 , g2 ∈ H ∞ (∂O± ), V0 ∈ H ∞ (O± (0)) with div V0 = 0, [V0 · n0 ] = g1,0 , where g1,0 (x) = g1 (0, x), and  g1 (t, ·)dσt = 0,

(3.9)

∂ O+ (t)

where dσt is the surface measure on ∂O+ (t). There exists a solution V ∈ H ∞ (O± ) and p ∈ (R ⊕ H ∞ )(O± ) to the equations Dt V + V · ∇v 0 + ∇p = f

for (t, x) in O± ,

(3.10)

div V = 0

for (t, x) in O± ,

(3.11)

[V · n] = g1

for (t, x) in ∂O± ,

(3.12)

[p] = g2

for (t, x) in ∂O± ,

(3.13)

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V |t=0 = V0

for x in O± (0).

11

(3.14)

The solution V is unique in H ∞ (O± ) and p is unique in H ∞ (O± ) up to a constant. Proof. For a function V ∈ H ∞ (O± ), we define (V ) ∈ (R ⊕ H ∞ )(O± (t)) via ⎧ 0 ⎪ ⎨(V ) = div f − 2∇V : ∇v , for x in O± (t), [∂n (V )] = [f · n] − Dt g1 − 2[V · ∇v 0 · n] + C(V )(t), ⎪ ⎩ [(V )] = g2 , for x in ∂O± (t),

for x in ∂O± (t),

where C(V )(t) is chosen to satisfy the compatibility condition between the first two equations. For the existence and the following properties of (V ), we refer to the reference [14]. Such a (V ) is unique up to a constant. Moreover the operator which maps V to (V ) is a pseudodifferential operator of order 0 acting on V , which satisfies the transmission property and which preserves piecewise smoothness at any order. In particular, there exists a C > 0 such that for any V ∈ H ∞ (O± ), k ≥ 0,   ∇(V ) H k (O± (t)) ≤ C 1 + V H k (O± (t)) , from which, we deduce that there exists a unique V ∈ H ∞ (O± ) satisfying Dt V + V · ∇v 0 + ∇(V ) = f,

V |t=0 = V0 .

Then equations (3.10) and (3.13) are satisfied with p = (V ). It remains to prove that (3.11) and (3.12) are satisfied as well. Indeed by taking the divergence of (3.10), we obtain 

Dt div V = 0, div V0 = 0,

which ensures that div V = 0. Taking the scalar product of (3.10) with n we find Dt ([V · n] − g1 ) = −C(V )(t), that is, [V · n] − g1 does not depend on x ∈ O± (t). While thanks to div V = 0 and (3.9), we deduce from Stokes formula that  ([V · n] − g1 )dσt = 0. ∂ O± (t)

This proves (3.12). This finishes the proof of Lemma 3.2.

2

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12

4. Construction of a sequence of approximate solutions 4.1. Construction of V 1 def

For a vector-valued function f , we define ftan = f −

f ·n a n.

We observe that ftan satisfies

  ftan · n(t, x) = 0 on (t, x) : |ϕ 0 (t, x)| < δ .

(4.1)

 Since can be uniquely parameterized by σ − sn(σ ) on the set (t, x) : |ϕ 0 (t, x)| <  δ is small, x ∞ δ . Let χ1 (s) ∈ C0 (R) is a cut-off function such that χ1 (s) = 0 when |s| ≥ 2δ and χ1 (s) = 1 when |s| < δ. Proposition 4.1. There is a profile V 1 ∈ A∞ which verifies ∂t V 1 + v 0 · ∇x V 1 + V 1 · ∇v 0 −

2V 1 · ∇v 0 · n n − a∂X2 V 1 = 0, a

(4.2)

for ±X > 0 and the jump conditions [V 1 ]X = 0

and [∂X V 1 ]X (t, x) = −

 1 [∂n v 0 ](σ )χ1 (s) tan , a

(4.3)

where x = σ − sn(σ ), σ ∈ (t). Moreover, V 1 satisfies V1 ·n=0

  for (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

(4.4)

Proof. By Proposition 3.1 that there is V 1 ∈ A∞ which satisfies the equation (4.2) for ±X > 0 and the jump condition (4.3) where x = σ − sn(σ ), σ ∈ (t). Indeed bookkeeping the proof of Proposition 3.1 we can choose the initial value V01 such that V01 · n0 = 0 for X ∈ R and |ϕ0 (x)| < δ, where n0 (x) = n(0, x) = ∇ϕ0 (x). Furthermore, in view of (1.5), V 1 · n satisfies |n|2 ), a  1 [∂X (V 1 · n)]X = − [∂n v 0 ](σ )χ1 (s) tan · n, a

Dt (V 1 · n) − a∂X2 (V 1 · n) = −2V 1 · ∇v 0 · n(1 −

(4.5)

[V 1 · n]X = 0,

(4.6)

V 1 · n|t=0 = V01 · n0 .

(4.7)

  In particular, on the set (t, x) : |ϕ 0 (t, x)| < δ , the right-hand-side of (4.5), (4.6) and (4.7) equal zero. When |ϕ0 (x)| < δ, we can use (4.5)–(4.6) and integrate by part to find that   1 d |V 1 · n|2 (t, χ 0 (t, x), X)dX + a|∂X (V 1 · n)|2 (t, χ 0 (t, x), X)dX 2 dt R R = (Dt (V 1 · n) − a∂X2 (V 1 · n))V 1 · ndX R

=0

(4.8)

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Then (4.4) follows.

13

2

4.2. Construction of p2 In this paragraph, we define the layer part p2 of the pressure profile p 2 . We set def

±∞

p (t, x, X) = 2

2V 1 · ∇v 0 · n dY, a

for ± X > 0.

(4.9)

X

It is easy to observe that p2 ∈ A∞ . We will complete the construction of p 2 below by adding another term to p2 . 4.3. Construction of V 2 and p2 Let ±∞ h2 = div x V 1 dY def

for ± X > 0.

X

Since V 1 ∈ A∞ , h2 ∈ A∞ . Moreover there holds Lemma 4.2. For any t ∈ [0, T ], s ∈ (−δ, δ), and X ∈ R, the profile h2 satisfies  h2 (t, x, X)dσt,s (x) = 0, ∂(O+ (t))s def

where ∂(O+ (t))s = {σ − sn(t, σ ) : σ ∈ ∂O+ (t)} and dσt,s is the surface measure on ∂(O+ (t))s . We refer to Section 6.1 of [18] for the proof of Lemma 4.2. Let ϕ 0 be determined by (1.4). For a profile V (t, x, X), we designate def

{V }ν (t, x) = V (t, x,

ϕ 0 (t, x) ). √ ν

(4.10)

Proposition 4.3. There exist V 2 ∈ H ∞ (O± ) and p2 ∈ H ∞ (O± ), such that Dt V 2 + V 2 · ∇v 0 + ∇p2 = v 0 

where [·] is defined by (3.8).

for (t, x) in O± ,

for (t, x) in O± , div V = 0  V 2 · n = −[{h2 }ν ] for (t, x) in ∂O± ,   p2 = −[{p 2 }ν ] for (t, x) in ∂O± ,

(4.11)

2

(4.12)

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14

Proof. There is V 20 which satisfies V 20 ∈ H ∞ (O± (0)),

div V 20 = 0,

[V 20 · n0 ] = −[{h2,0 }ν ],

(4.13)

where {h2,0 }ν (x) = {h2 }ν (0, x). We may take V 20 ≡ 0

in O− (0),

V 20 = b1 + b2 where b1 (x) = −

[{h2,0 }ν ](σ )χ1 (s)n0 (x) a0 (x)

in O+ (0),

and b2 (x) ∈ H ∞ (O+ (0)) satisfies

div b2 (x) = −div b1 (x) b2 = 0

in O+ (0),

(4.14)

on ∂O+ (0).

(4.15)

Since h2 , p 2 ∈ A∞ , we have [{h2 }ν ], [{p 2 }ν ], b1 ∈ H ∞ (∂O± ). By Lemma 4.2, [{h2 }ν ] satisfies property (3.9). Hence we have   −div b1 (x)dx = [{h2,0 }ν ](σ )dσ = 0. O+ (0)

∂ O+ (0)

Then we can find b2 ∈ H ∞ (O+ (0)) a solution to the Bogovskii equations (4.14) and (4.15). By the construction of b1 and b2 , V 20 satisfies (4.13). Then by Lemma 3.2, we have a solution V 2 ∈ H ∞ (O± ), p 2 ∈ (R ⊕ H ∞ )(O± ), and V 2 satisfies V 2 |t=0 = V 20 4.4. Construction of the tangential part of V

for x in O± (0).

2

(4.16)

2

First, we construct a profile A2 . When ±X > 0, we define def

A2 =

nh2 1 1 − ± ([V 2 ](σ )χ1 (s))tan − ([{∂n V 1 }ν ](σ )χ1 (s))tan e−|X| . a 2 a

Since h2 , V 1 ∈ A∞ , V 2 ∈ H ∞ (O± ), we deduce that A2 ∈ A∞ and A2 satisfies A2 · n = h2

  for X > 0 and (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

(4.17)

Thanks to (4.12), we find that n [V 2 · n] [{A2 }ν ] = [{h2 }ν ] − [V 2 ] + n = −[V 2 ]. a a

(4.18)

Note that ∂X h2 = −divx V 1 , we have n 1 n [{∂n V 1 }ν · n] [{∂X A2 }ν ] = − [{divx V 1 }ν ] − [{∂n V 1 }ν ] + . a a a a

(4.19)

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Whereas it follows from (2.7) that when |ϕ 0 | < δ, divx V 1 = implies [{divx V 1 }ν ] =

∂n V 1 ·n a

+

15



j cj

· Zj V 1 , which

[{∂n V 1 }ν · n]  cj · [{Zj V 1 }ν ]. + a j

While for a profile V (t, x, X), Zj {V }ν = {Zj V }ν +

w√j ·n {∂X V }ν . ν

Since [{V 1 }ν ] = 0, Zj are

tangential derivatives and w j · n = 0 when |ϕ 0 | < δ, we obtain [{Zj V 1 }ν ] = 0. As a result, it comes out 

  1 {∂X A2 }ν = − {∂n V 1 }ν . a

(4.20)

2

Proposition 4.4. There is V in A∞ such that

n 1 2 2 2 2 Dt V + AV − a∂X2 V =(Dt h2 − a∂X2 h2 ) − (V · n + h2 )∂X V 1 tan a a

(4.21) 1 2 1 1 1 − , ∇x p + V · ∇x V − (2n · ∇x ∂X V + ϕ 0 ∂X V 1 ) tan a for ±X > 0 and the jump condition 2

[V ]X = [A2 ]X ,

[∂X V 2 ]X = [∂X A2 ]X ,

(4.22)

where x = σ − sn(σ ), σ ∈ (t), and 2

V · n = h2

for X ∈ R and

  (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

(4.23)

Proof. Since V 1 , h2 , p 2 ∈ A∞ , v 0 , V 2 ∈ H ∞ (O± ) and ϕ 0 , a ∈ Cb∞ ([0, T ] × R3 ), the right2

hand-side of (4.21) is in A∞ . By Proposition 3.1, we can find a profile V ∈ A∞ satisfying 2 (4.21) and (4.22). We now prove (4.23). Indeed it follows from (1.5) and (4.1) that V · n satisfies 2

2

Dt (V · n − h2 ) − a∂X2 (V · n − h2 ) = 0, 2

[V · n − h2 ]X = 0,

2

[∂X (V · n − h2 )]X = 0,

2

(V · n − h2 )|t=0 = 0. By standard energy method as in (4.8), we deduce (4.23).

2

We set def

2

V 2 (t, x, X) = V 2 (t, x) + V (t, x, X).

(4.24)

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16

4.5. Construction of p3 Let us determine p3 via 2

2V · ∇v 0 · n 1 1 2 − (Dt h2 − a∂X2 h2 ) − (V · n + h2 )∂X V 1 · n a a a

1 − ∇x p 2 + V 1 · ∇x V 1 − (2n · ∇x ∂X V 1 + ϕ 0 ∂X V 1 ) · n. a

∂X p 3 = −

We can verify that p3 ∈ A∞ . 4.6. Construction of the higher orders i

For j > 2, we assume that the profiles, V i = V i + V ∈ H ∞ ⊕ A∞ for 1 ≤ i < j , and the profiles p i = p i + pi ∈ (R ⊕ H ∞ )(O± ) ⊕ A∞ for 2 ≤ i < j , and p j ∈ A∞ have been constructed. Let us denote ±∞ hi (t, x, X) = div x V i−1 (t, x, Y ) dY, def

for ± X > 0,

(4.25)

X

1 1 def nhi − ± ([V i ](σ )χ1 (s))tan − ([{∂n V i−1 }ν ](σ )χ1 (s))tan e−|X| . Ai = a 2 a

(4.26)

Similar to A2 , we can show that Ai satisfies 

  1 {∂X Ai }ν = − {∂n V i−1 }ν , a   for X > 0 and (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

 {Ai }ν = −[V i ],

Ai · n = hi



(4.27)

i

Assume the initial data satisfies for any i > 1, V0i = V i0 + V 0 with V i0 ∈ H ∞ (O± (0)), i V0 i

∈ A∞ 0 ,

V 0 · n0 = hi,0 (x, X)

i [V 0 ]X

divV i0 = 0, = [Ai0 ]X (x),

[V i0 · n0 ] = −[{hi,0 }ν ], i [∂X V 0 ]X



= [∂X Ai0 ](x),

 for X ∈ R and (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ ,

(4.28) (4.29) (4.30)

where hi,0 = hi |t=0 , Ai0 = Ai |t=0 . Assume that the regular part of profiles V i for 2 ≤ i < j satisfy Dt V i + V i · ∇v 0 + ∇pi = f i 

for (t, x) in O± ,

div V = 0 for (t, x) in O± ,  V i · n = −[{hi }ν ] for (t, x) in ∂O± ,   pi = −[{pi }ν ] for (t, x) in ∂O± , i

(4.31) (4.32) (4.33) (4.34)

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V i |t=0 = V i0 0

17

for x in O± (0),

(4.35)

1

where V 0 = V = v 0 , V 0 = 0, V 1 = V , V 1 = 0, and i def

f = x V

i−2

−

i−2 

V k · ∇V i−k .

(4.36)

k=2

The layer part of the pressure profile p i+1 is determined by i

2V · ∇v 0 · n 1 − Dt hi − a∂X2 hi + (V i · n + hi )∂X V 1 · n a a

1 i + −∇x p + x V i−2 + 2n · ∇x ∂X V i−1 + ϕ 0 ∂X V i−1 · n (4.37) a i−1 i−2 i−1

  1  k V · ∇x V i−k − V k · ∇V i−2 + V i · n∂X V i+1−k · n. − a

∂X p i+1 = −

k=1

k=2

k=2

Assume that the layer part of the profiles V i for 2 ≤ i < j satisfy i

i

i

i

Dt V + AV − a∂X2 V = f , i

[V ]X = [Ai ]X V

i

(4.38)

i

and [∂X V ]X = [∂X Ai ]X ,

(4.39)

i |t=0 = V 0 ,

(4.40)

where i

f =



n Dt hi − a∂X2 hi − (V i · n + hi )∂X V 1 · n + ∇x pi tan a i−1 i−2 i−1 

  − V k · ∇x V i−k − V k · ∇V i−2 + V i · n∂X V i+1−k +

k=1

k=2

i−2 i−1 0 x V + 2n · ∇x ∂X V + ϕ ∂X V i−1 . tan k=2

(4.41)

tan

i

Under the assumptions above, we can prove that V satisfies i

V · n = hi

for X ∈ R

and

  (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

(4.42)

Thus there holds i

div x V i−1 + ∂X V = 0 for X ∈ R

and

  (t, x) ∈ (t, x) : |ϕ 0 (t, x)| < δ .

(4.43)

Now we construct profile V j , p j and pj +1 . Since we hope (4.43) is satisfied for i = j . We i

define hj and f j as in (4.25) and (4.36) respectively. Since the profiles, V i = V i + V ∈ H ∞ ⊕ A∞ for 1 ≤ i < j , and the profiles, p i = p i + p i ∈ (R ⊕ H ∞ )(O± ) ⊕ A∞ for 2 ≤ i < j , and p j ∈ A∞ . We observe that

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18

hj , f j ∈ A∞ ,

  [{hj }ν ], {pj }ν ∈ H ∞ (∂O± ).

Similar to Lemma 4.2, we can prove that for any t ∈ [0, T ], s ∈ (−δ, δ) and X ∈ R, there holds  hj (t, x, X)dσt,s (x) = 0. ∂(O+ (t))s

Together with the initial assumption (4.28), we deduce from Lemma 3.2 that there exist V j ∈ A∞ and pj ∈ (R ⊕ H ∞ )(O± ). Then we can define Aj as in (4.26) and take ∂X p j +1 as in (4.37) and solve equations (4.38)–(4.40) for i = j . Notice that f ∈ A∞ , j

[Ai ]X , [∂X Ai ]X ∈ H ∞ (O± ), j

from which, and (4.29), (4.30), we deduce from Proposition 3.1 that there exists V ∈ A∞ , and accordingly pj +1 ∈ A∞ . In order to prove (4.42) for i = j , we investigate the equation for Dt (V j · n − hj ). Indeed it j

follows from (1.5) and (4.1) that V · n − hj satisfies j

j

Dt (V · n − hj ) − a∂X2 (V · n − hj ) = 0, j

[V · n − hj ]X = 0,

j

[∂X V · n − hj ]X = 0,

j

(V · n − hj )|t=0 = 0, from which and standard energy method as in (4.8), we conclude (4.42). 4.7. A last technical corrector Let def

A(t, x) = −

 1 {∂n V N }ν (σ )χ1 (s)e−|s| , 2

(4.44)

for x = σ − sn(σ ), σ ∈ ∂O± (t), χ1 is a cut-off function as above. We have [A] = 0,

  [∂n A] = − {∂n V N }ν .

(4.45)

5. A family of approximate solutions 5.1. Consistency of the ansatz Let (v 0 , p 0 ) be the solution of (1.2) determined by Theorem 1.1. Let (V j , p j ), for j = 1, · · · , N , be the profiles constructed in the previous section, and recall that A is a technical corrector defined in Section 4.7. We set

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va (t, x) = v 0 (t, x) +

√

νV 1 (t, x,

19

N N ϕ 0 (t, x) ϕ 0 (t, x) ) + · · · + ν 2 V N (t, x, √ ) + ν 2 A(t, x), (5.1) √ ν ν

def

pa (t, x) = p 0 (t, x) + νp 2 (t, x,

N ϕ 0 (t, x) ϕ 0 (t, x) ) + · · · + ν 2 p N (t, x, √ ). √ ν ν

(5.2)

By using the eikonal equation (1.4), the Euler equation (1.2) and the definitions of the profiles V j given in Section 4, we find that on both sides of the interface, (t), there holds N

∂t va + va · ∇va − νva + ∇pa = ν 2 Fa

N

and div va = ν 2 Ga ,

(5.3)

where  Fa = Dt V N + V N · ∇v 0 + V N · n∂X V 1 − |n|2 ∂X2 V N + ∇x p N − (x V N−2 + 2n · ∇x ∂X V N−1 + ϕ 0 ∂X V N−1 ) N−1   V i · ∇x V N−i + V i · n∂X V N−i+1 + i=1 √ ν(2n · ∇x ∂X V N + ϕ 0 ∂X V N ) + νx V N N N N   j  N+j +1−i  + ν2 V j · ∇x V N+j −i + V i · n∂X

−

j =1



(5.4)

i=j +1

i=j

 V · n∂X V + (a +ν N−1  N−j   j ν − 2 V 1 · n∂X V j + (a − |n|2 )∂X2 V j + (V · n − hj )∂X V 1 + 1−N 2

1

1

− |n|2 ∂X2 V 1 )

ν

j =2 N 2

+ ∂t A − νA + va · ∇A + A · ∇(va − ν A), and Ga = {divx V N }ν + divA +

N−1 

ν

k−N 2



   {divx V k }ν + ∂X V k+1 · n ν ,

(5.5)

k=0

with {·}ν being given by (4.10). Moreover since [va ] = [∂n va ] = 0

and [pa ] = 0,

(5.6)

(5.3) is satisfied in QT = [0, T ] × R3 in the sense of distributions. 5.2. Consistency estimates We claim that Fa , Ga , va , ∇va is bounded in L∞ (QT ) ∩ L1 (QT ).

(5.7)

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Let’s first investigate the term {Dt V N }. Since V N ∈ H ∞ (O± ) ⊕ A∞ and v 0 ∈ H ∞ , we have Dt V N ∈ H ∞ (O± ) ⊕ A∞ , and   {Dt V N }ν (t, x) ≤ |Dt V N (t, x)| + Dt V N (t, x, ·) L∞ (R) , which ensures that {Dt V N } is bounded in L∞ (QT ) ∩ L1 (QT ). Be careful that the derivatives in the term x V N−2 is taken in each side of (t), moreover, x V N−2 ∈ H ∞ (O± ) ⊕ A∞ . 1−N  This implies that {x V N−2 }ν is bounded in L∞ (QT ) ∩ L1 (QT ). Although the term ν 2 V 1 ·  n∂X V 1 ν looks seemingly unbounded with respect to ν, but we recall that this term equals zero   on the set (t, x) : |ϕ 0 (t, x)| < δ and we can rewrite this term as ν

1−N 2

{V 1 · n∂X V 1 }ν =

 X 

N−1

ϕ0

V 1 · n∂X V 1

ν

,

 1−N  which implies that ν 2 V 1 · n∂X V 1 ν is bounded in L∞ (QT ) ∩ L1 (QT ). Let’s mention that by definition A ∈ H ∞ (O± ) and has compact support, we can check the remaining terms in Fa , Ga and va are bounded in L∞ (QT ) ∩ L1 (QT ). Next we estimate the derivatives of Fa and Ga . For function V 1 , if we take the tangential derivatives Zj , Zj {V 1 }ν = {Zj V 1 }ν + wϕi 0·n {X∂X V 1 }ν . It is not difficult to show that w j ·n ϕ0

∈ L∞ (QT ) since w j · n = 0 on (t). Moreover the tangential derivatives of

w j ·n ϕ0

are also

in L∞ (QT ). Together with the fact that V 1 is in A∞ , we deduce that Zj {V 1 }ν is bounded in ) ∩ L∞ (QT ) for any fixed α. If we L1 (QT ) ∩ L∞ (QT ). Similarly Z α {V 1 } is bounded in L1 (QT √

take normal derivative, we have to multiply the derivative by ν in order to obtain the bounded√ ness. That is, νZ α ∇V 1 is bounded in L1 (QT ) ∩ L∞ (QT ). Along the same line, we can show that for any fixed α, √ √ Z α Fa , Z α Ga , Z α ∇va , νZ α ∇Fa , νZ α ∇Ga are bounded in L1 (QT ) ∩ L∞ (QT ). (5.8) 6. A priori estimates of the remainder R Let T  in (0, T ) where T is the maximal existence time of the solution v 0 of (1.2) determined by Theorem 1.1. The goal of this section is to establish some a priori estimates for a vector field R satisfying, in the sense of distributions, N

∂t R − νR + (va + ν 2 R) · ∇R + R · ∇va + ∇pR = −Fa ,

(6.1)

div R = −Ga ,

(6.2)

in QT  = [0, T  ] × R3 , with zero initial data and zero jumps at the interface that is [R] = [∂n R] = 0. We recall that [·] is determined by (3.8).

(6.3)

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21

6.1. Energy estimates We first present the L2 estimate of the remainder R. Proposition 6.1. There is a constant C such that for any t ∈ [0, T  ], t R(t) 2L2

  ∇R(t  )2 2 dt  ≤ C. L

+ν

(6.4)

0

Proof. Let P = I + ∇(−)−1 div . def

We define ρ = −(−)−1 div R so that (I − P )R = ∇ρ, with ρ = div R = −Ga . By (5.7) and singular integral operator theory (see [15]), ∇ 2 ρ L∞ (L2 (R3 )) ≤C Ga L∞ (L2 (R3 )) ≤ C, T

T

∇ρ L∞ (L2 (R3 )) ≤C Ga T

1

6 3 5 L∞ T (L (R ))

2

≤ C Ga L3 ∞ (L2 (R3 ))) Ga L3 ∞ (L1 (R3 )) ≤ C, T

T

for a constant C, which has changed from line to line but which is independent of ν. This leads to (I − P )R L∞ (H 1 (R3 )) ≤ C.

(6.5)

T

Next we estimate P R by taking L2 (R3 ) inner product of (6.1) with P R and by using that the decomposition R = P R + ∇ρ is orthogonal in L2 (R3 ). Since 



N

(va + ν 2 R) · ∇P R · P R dx = 0 and R3

∇pR · P R dx = 0, R3

we arrive at 1 P R(t) 2L2 + ν 2

t ∇P R(t 0



) 2L2

 t     N  dt ≤ (va + ν 2 R) · ∇ (I − P )R · P R dx dt   

0 R3 t

  +



   R · ∇va · P R + Fa · P R dx dt  .



0 R3

By (5.7) and (6.5), t  t   

   R · ∇va · P R + Fa · P R dx dt  ≤ C 1 + P R 2L2 dt  ,  0 R3

0

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22

and, since N ≥ 3,    t   t       N   2 P R L2 + ν R L4 P R L4 dt  . (va + ν R) · ∇ (I − P )R · P R dx dt  ≤ C     0 R3  0 3

1

Then, with the Sobolev inequality · L4 (R3 ) ≤ C ∇ · L4 2 (R3 ) · L4 2 (R3 ) , Young’s inequality for products and (6.5), the right hand side can be bounded by 1 ν 2

t ∇P R(t



) 2L2

t



dt + C

0



 1 + P R 2L2 dt  .

0

It is therefore sufficient to apply the Gronwall inequality to conclude the proof of (6.4).

2

6.2. Tangential derivatives estimate Recall that the iterated tangential derivatives Z α , where α is a multi-index, are defined in def

(2.2). We set ZR L∞ = max1≤i≤5 Zi R L∞ . Proposition 6.2. For t ≤ T  , t R(t) 2m + ν 0

+

t

1+ν

N−1 2

   t      2  α α  ∇R m dt  1 + Z ∇pR · Z R dx dt     |α|≤m  0 3  R N

( R L∞ + ZR L∞ ) + ν 2 ∇R L∞



(6.6)

R 2m + ν ∇R 2m−1 dt  ,

0

Proof. By (2.4) it is sufficient to prove that for any multi-index α with |α| = m ≥ 0 and t ≤ T  ,    t   t    α 2 α 2  α α  Z R(t) L2 + ν ∇Z R L2 dt  1 +  Z ∇pR · Z R dx dt     0 R3  0 (6.7) t



N−1 N 1 + ν 2 ( R L∞ + ZR L∞ ) + ν 2 ∇R L∞ R 2m + ν ∇R 2m−1 dt  . + 0

We proceed by induction. The case for m = 0 has already been done in Proposition 6.1. Let us assume (6.7) holds for |α| < m with m ≥ 1, we wish to prove that it also holds for |α| = m. To do it, we apply Z α to (6.1) and we use (2.3) to obtain that N

∂t Z α R − νZ α R + (va + ν 2 R) · ∇Z α R + Z α (R · ∇va ) + Z α ∇pR N

= −Z α Fa − ν[; Z α ]R + [(va − v 0 + ν 2 R) · ∇; Z α ]R.

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23

Then, by using a standard energy estimate, since t 

N

(va + ν 2 R) · ∇Z α R · Z α R dx dt  = 0,

0 R3

we get that 1 α Z R(t) 2L2 + ν 2

t ∇Z

α

R 2L2



dt ≤

5 

Ii ,

i=1

0

where   t    I1 =  Z α (R · ∇va ) · Z α R dx dt  , 0 R3

  t    I2 =  Z α ∇pR · Z α R dx dt  , 0 R3

  t    I3 =  Z α Fa · Z α R dx dt  , 0 R3

  t    I4 =  ν[; Z α ]R · Z α R dx dt  , 0 R3

  t  N   I5 =  [(va − v 0 + ν 2 R) · ∇; Z α ]R · Z α R dx dt  . 0 R3

Next we handle term by term above. By the estimates of Section 5.2 t I 1 + I3  1 +

R 2m dt  .

0

To handle I4 , we deduce from (2.4) and (2.5) that  t   I4  



0 R3 |β|,|γ |≤m−1

    ν cβ ∇ 2 Z β R + cγ ∇Z γ R · Z α R dx dt  ,

(6.8)

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24

where cβ , cγ are some derivatives of w i , and they are smooth functions. Then we get, by using integration by parts, that t

ν ( ∇R m−1 ∇R m + ∇R m−1 R m ) dt 

I4  0

t ≤ δν

∇R 2m dt 

+ Cν

0

t

∇R 2m−1 + R 2m dt  .

0

It remains to estimate I5 . We split it into two parts, I5 = I51 + I52 , with  t     I51 =  [(va − v 0 ) · ∇; Z α ]R · Z α R dx dt  , 0 R3

  t  N   I52 =  [ν 2 R · ∇; Z α ]R · Z α R dx dt  . 0 R3

By using Leibniz formula, we find 

[(va − v 0 ) · ∇; Z α ]R =

cα1 Z α1 (va − v 0 ) · Z α2 ∇R

α1 +α2 =α,α1 =0

+(va − v 0 ) · [∇; Z α ]R, which implies [(va − v 0 ) · ∇; Z α ]R 

√

ν ∇R m−1 ,

accordingly one has t I51 

√

ν ∇R m−1 R m dt 

0

t 

(ν ∇R 2m−1

+ R 2m ) dt 

0

t 1+

R 2m dt  ,

0

where we used inductive assumption for |α| = m − 1. Similarly we get by, using Leibniz formula, that N

N

ν 2 [R · ∇; Z α ]R = ν 2

 α1 +α2 =α,α1 =0

Applying Lemma 2.1 yields

N

cα1 Z α1 R · Z α2 ∇R + ν 2 R · [∇; Z α ]R.

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25

 N N ν 2 [R · ∇; Z α ]R ν 2 ZR L∞ ∇R m−1 + ∇R L∞ ZR m−1 + R L∞ ∇R m−1  N−1  N  ν 2 ( R L∞ + ZR L∞ ) + ν 2 ∇R L∞   √ × R m + ν∇R m−1 , which implies I52 

t ν

N−1 2

N

( R L∞ + ZR L∞ ) + ν 2 ∇R L∞



√ R 2m + ν∇R 2m−1 dt  .

0

By inserting the above estimates into (6.8) leads to (6.7).

2

6.3. Normal derivatives estimate We introduce def √

η=

(6.9)

ν∂n R.

Proposition 6.3. For t ≤ T  , t η(t) 2m−1 + 0

   t      √  ∇η 2m−1 dt   Z β ν∂n ∇pR · Z β η dx dt      |β|≤m−1  0 3  R +1+

t

(6.10)



1 + ν N−1 R 2L∞ + ν N ∇R 2L∞ R 2m + η 2m−1 dt  .

0

√ Proof. We apply Z β ν∂n , for |β| = m − 1, to (6.1) to obtain √ √ N ∂t Z β η − νZ β η + (va + ν 2 R) · ∇Z β η + Z β ν∂n (R · ∇va ) + Z β ν∂n ∇pR √ √ √ √ N = −Z β ν∂n Fa + [Dt , Z β ν∂n ]R − ν[, Z β ν∂n ]R + [(va + ν 2 R − v 0 ) · ∇, Z β ν∂n ]R. We take the L2 inner product of the above equation with Z β η, and observe that t 

N

(va + ν 2 R) · ∇Z β η · Z β η dx dt  = 0,

0 R3

to arrive at 1 β Z η(t) 2L2 + ν 2

t 0

where

∇Z β η 2L2 dt  

6  i=1

Ji ,

(6.11)

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26

  t  √   J1 =  Z β ν∂n ∇pR · Z β η dx dt  , 0 R3

 t   √   Z β ν∂n (R · ∇va ) · Z β η dx dt  , J2 =  0 R3

 t   √   Z β ν∂n Fa · Z β η dx dt  , J3 =  0 R3

  t  √   [Dt ; Z β ν∂n ]R · Z β η dx dt  , J4 =  0 R3

 t   √   ν[; Z β ν∂n ]R · Z β η dx dt  , I5 =  0 R3

 t   √ N   [(va + ν 2 R − v 0 ) · ∇; Z β ν∂n ]R · Z β η dx dt  . J6 =  0 R3

Next we deal with the estimate of Ji , 2 ≤ i ≤ 6. First, by the estimates of Section 5.2, (2.3) and (1.5)

J 2 + J3 + J4 

t

1 + R 2m−1 + η 2m−1 dt  .

0

Next, by (2.4), (2.5), Leibniz formula and integration by parts, we find t J5  0

  √ ν ∇η m−2 ( ∇η m−1 + η m−1 ) + ν∇R m−1 ( ∇η m−1 + η m−1 ) dt  t ∇η 2m−1

≤ δν



dt + Cν

0

t

∇η 2m−2 + η 2m−1 + R 2m dt  .

0

If m = 1, the term ∇η 2m−2 disappears. It remains to estimate J6 . We split it into four parts, J6 = J61 + J62 + J63 + J64 with  t     J61 =  [(va − v 0 ) · ∇; Z β ]η · Z β η dx dt  , 0 R3

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27

  t  √   J62 =  Z β [(va − v 0 ) · ∇; ν∂n ]R · Z β η dx dt  , 0 R3

 t     J63 = ν  [R · ∇; Z β ]η · Z β η dx dt  , N 2

0 R3

 t   √ N  Z β [R · ∇; ν∂n ]R · Z β η dx dt  . J64 = ν 2  0 R3

For I61 , if m = 1, it vanishes. If m > 1, we get, by using Leibniz formula as we did for I51 in Proposition 6.2 and the inductive assumption, that t J61 

√



t

ν∇η m−2 η m−1 dt  1 + 0

η 2m−1 dt  .

0

J62 can be handled as J4 . Indeed   t  √ √   J62 =  Z β ( ν(va − v 0 ) · ∇n · ∇R − ν∂n (va − v 0 ) · ∇R) · Z β η dx dt  , 0 R3

from which, we deduce t J62 

√



ν∇R m−1 η m−1 dt  0

t

R 2m + η 2m−1 dt  .

0

The estimate for J64 is similar. We write  t   √ N  J64 = ν 2  Z β ( νR · ∇n · ∇R − η · ∇R) · Z β η dx dt  . 0 R3

Applying Lemma 2.1 gives

J64  ν

N 2

t 0

As a result, we obtain



√ √ R L∞ ν∇R m−1 + R m−1 ν∇R L∞  + η L∞ ∇R m−1 + η m−1 ∇R L∞ η m−1 dt  .

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28

J64 

t ν

N−1 2

N

R L∞ + ν 2 ∇R L∞



R 2m + η 2m−1 dt  .

0

Finally, we deal with the estimate of I63 . If m = 1, this term also disappears. If m > 1, by using Leibniz formula, we have J63 ≤ J631 + J632 , where  t   J631 = ν 

  cβ1 Z β1 R · Z β2 ∇η · Z β η dx dt  ,



N 2

0 R3 β1 +β2 =β,β1 =0

 t   N  R · (Z β ∇η − ∇Z β η) · Z β η dx dt  , J632 = ν 2  0 R3

cβ1 is some derivatives of w i . We first observe that N

t

J632  ν 2

R L∞ ∇η m−2 η m−1 dt 

0

t ≤ δν

∇η 2m−2 dt 

t + Cν

0

N−1

R 2L∞ η 2m−1 dt  .

0

By integration by parts and (2.4) we arrive at J631  ν

N 2

t



( R L∞ η m−1 + R m−1 η L∞ )( ∇η m−1 + η m−1 )

0  + ( ∇R L∞ η m−1 + ∇R m−1 η L∞ ) η m−1 dt  t t



2  ≤ δν ∇η m−1 dt + C 1 + ν N−1 R 2L∞ + ν N ∇R 2L∞ R 2m + η 2m−1 dt  . 0

0

Inserting the above estimates into (6.11) leads to (6.10). This ends the proof of Proposition 6.3. 2 6.4. Pressure estimates Proposition 6.4. For m ≥ 0,     N N ∇pR m ≤ C 1 + R m 1 + ν 2 ∇R L∞ + ν 2 R L∞ ∇R m .

(6.12)

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29

Proof. We apply the operator ∇(−)−1 div to (6.1) to arrive at ∇pR = ∇(−)−1 div Fa + ∇(−)−1 div Dt R − ν∇Ga   N + ∇(−)−1 div (va + ν 2 R − v 0 ) · ∇R + R · ∇va .

(6.13)

We remark that (6.13) holds in the sense of distributions. By (6.2) and some commutations, we arrive at ∇pR = ∇(−)−1 div Fa − ∇(−)−1 Dt Ga − ν∇Ga   N N + ∇(−)−1 div R · ∇(2va + ν 2 R) − Ga (va + ν 2 R − v 0 ) .

(6.14)

When (t, x) verifies |ϕ 0 (t, x)| < δ, one has Ga = {div x V N }ν + div A. This leads to div x V N =

∂n V N · n  c j · Zj V N , + a j

i where cj is a smooth function   of w . N  N While since {Zj V }ν = Zj {V }ν = Zj [{V N }ν ] = 0, we have



{div x V }ν N





 {∂n V N }ν · n . = a

Along the same line, we have [div A] =

[∂n A] · n . a

Therefore thanks to (4.45), we deduce from (5.5) that [Ga ] = 0, and accordingly the distribution derivatives of Ga coincide with its point-wise derivatives. Consequently, we obtain √ ν∇Ga L2 ≤ C ν, for a constant C independent of t and ν. Next we claim that Dt Ga is uniformly bounded in L1 (R3 ) ∩ L∞ (R3 ). Indeed it is easy to observe from (5.5) that Ga =

N 

i

ν 2 {Gi }ν ,

i=0

for profiles Gi ∈ H ∞ (O± ) ⊕ A∞ , 0 ≤ i ≤ N . While Dt {Gi }ν = {Dt Gi }ν , we find Dt Ga =

N 

i

ν 2 {Dt Gi }ν ,

i=0

is uniformly bounded in L1 (R3 ) ∩ L∞ (R3 ). Since ∇(−)−1 is a homogeneous Fourier multiplier of degree −1, we deduce from Bessel potential theory (see [15]) that

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30

∇(−)−1 Dt Ga L2 ≤ C Dt Ga

1

6 L5

2

≤ C Dt Ga L3 2 (R3 ) Dt Ga L3 1 (R3 ) ≤ C.

Whereas it follows from singular integral theory (see [15]) that ∇(−)−1 div Fa L2 ≤ C Fa L2 ≤ C, and N

N

∇(−)−1 div (R · ∇(2va + ν 2 R)) L2 ≤C R · ∇(2va + ν 2 R) L2   N ≤C R L2 + ν 2 R L∞ ∇R L2 , N

N

∇(−)−1 div (Ga (va + ν 2 R − v 0 )) L2 ≤C Ga (va + ν 2 R − v 0 )   ≤C 1 + R L2 . Therefore, we obtain   N ∇pR L2 ≤ C 1 + R L2 + ν 2 R L∞ ∇R L2 . This proves (6.12) for m = 0. On the other hand, by taking space divergence of (6.14), we find N

pR = −div Fa + Dt Ga − νGa − div (R · ∇(2va + ν 2 R) N − Ga (va + ν 2 R − v 0 )),

(6.15)

which holds in the sense of distributions. In general, we assume that (6.12) holds for |β| < m. For |α| = m ≥ 1, we write Z α pR = Z α pR +





 ∇(cβ ∇Z β pR ) + cγ ∇Z γ pR ,

(6.16)

|β|,|γ |≤m−1

where cβ , cγ are some derivatives of w i , cβ , cγ ∈ L2 ∩ L∞ , which together with (6.15) implies Z α pR =





 ∇(cβ ∇Z β pR ) + cγ ∇Z γ pR − Z α div Fa

|β|,|γ |≤m−1

  N N + Z α (Dt Ga − νGa ) − Z α div R · ∇(2va + ν 2 R) − Ga (va + ν 2 R − v 0 ) , from which, we deduce that 1

∇Z α pR L2 ≤ C (−) 2 Z α pR L2 ≤ C

6  i=1

where

Ki ,

(6.17)

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K1 =

31

1

(−)− 2 ∇(cβ ∇Z β pR ) L2 ,

|β|≤m−1



K2 =

1

(−)− 2 (cγ ∇Z γ pR ) L2 ,

|γ |≤m−1 1

K3 = (−)− 2 Z α div Fa L2 ,  1 K4 = (−)− 2 Z α Dt Ga − νGa ) L2 ,    1 N K5 = (−)− 2 Z α div R · ∇(2va + ν 2 R) L2 ,    1 N K6 = (−)− 2 Z α div Ga (va + ν 2 R − v 0 ) L2 . Since cβ , cγ ∈ L2 ∩ L∞ , we find K1 ≤ C cβ ∇Z β pR L2 ≤ C ∇pR m−1 , K2 ≤ C (cγ ∇Z γ pR ) L2 ∩L∞ ≤ C ∇pR m−1 . While in view of (2.4), we write Z α div =





 ∇(cα1 Z α1 ) + cα2 Z α2 ,

(6.18)

|α1 |,|α2 |≤m

where cα1 , cα2 ∈ L2 ∩ L∞ . This ensures K3 ≤ C Fa m ≤ C. Along the same line, we have K4 ≤ C Z α (Dt Ga − νGa ) L2 ∩L∞ ≤ C. Finally it follows from (6.18) and Lemma 2.1 that   N N K5 ≤ C R · ∇(2va + ν 2 R) m ≤ C R · ∇va m + ν 2 R · ∇R m   N ≤ C R m + ν 2 R L∞ ∇R m + ∇R L∞ R m and     N 0 2 K6 ≤ C Ga (va + ν R − v ) m ≤ C 1 + R m . Inserting the above estimates into (6.17) and summing up the resulting inequality for |α| ≤ m leads to (6.12). This finishes the proof of Proposition 6.4. 2 Thanks to Proposition 6.4 we can deal with the estimate of the two terms involving the pressure which appears in Proposition 6.2 and Proposition 6.3. Corollary 6.5. Let α, β satisfy |α| = m, |β| = m − 1 for m ≥ 1. Then for any small δ > 0, there exist a constant C such that

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32

   t   t    α α  Z ∇pR · Z R dx dt  ≤δν ∇R 2m dt      0 R3  0 t +C

(6.19)



 1 + ν N−1 R 2L∞ + ν N ∇R 2L∞ R 2m dt  ,

0

and t  t    β√ β  Z ν∂n ∇pR · Z η dx dt  ≤ δν ( ∇R 2m + ∇η 2m−1 ) dt   0 R3

0

t +C+C

(6.20)







1 + ν N−1 R 2L∞ + ν N ∇R 2L∞ R 2m + η 2m−1 dt  .

0

Proof. First (6.19) is an immediate consequence of Proposition 6.4 and Cauchy-Schwarz inequality. To prove (6.20) we first use (2.4) and an integration by parts to obtain t  t   √  β√ β  Z ν∂n ∇pR · Z η dx dt  ≤ C ν ∇pR m−1 ( ∇η m−1 + η m−1 ) dt   0 R3

0

t ≤ δν

t ∇η 2m−1

+C

0

Then using Proposition 6.4 we obtain (6.20).

( η 2m−1 + ∇pR 2m−1 ) dt  .

0

2

By combining Proposition 6.2, Proposition 6.3 with Corollary 6.5, we obtain the following corollary. Corollary 6.6. For any m ≥ 1, there holds t R(t) 2m

+ η(t) 2m−1

+ν



 ∇R 2m + ∇η 2m−1 dt 

0 (6.21) t

   1 + ν N−1 ( R 2L∞ + ZR 2L∞ ) + ν N ∇R 2L∞ R 2m + η 2m−1 dt  . ≤C 1+ 0

6.5. L∞ estimates According to√Corollary 6.6, in order to close the estimate of R(t) m , it remains to deal with the estimate of ν R L∞ and ν ∇R L∞ . Let us denote

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Nm (t) = R(t) 2m + η(t) 2m−1

and

33

def

N m (t) = sup Nm (s).

(6.22)

s∈[0,t]

Proposition 6.7. For m0 > 1, we have   ν R(t) 2L∞ ≤ C R(t) 2m + η(t) 2m−1 ≤ CNm (t) if m ≥ m0 + 1.

(6.23)

Proof. We deduce from Proposition 20 of [13] that   R(t) 2L∞ ≤ C ∂n R(t) m0 R(t) m0 + R(t) 2m0 , which implies   ν R(t) 2L∞ ≤ C ν ∂n R(t) m0 R(t) m0 + ν R(t) 2m0   ≤ C η(t) 2m−1 + ν R(t) 2m ≤ CNm (t) for m ≥ m0 + 1, where we assume ν < 1 and η is given by (6.9). Actually it is sufficient to take m0 = 2.

2

Proposition 6.8. We have 3

ν ∇R L∞ (QT ) ≤ C + CN m2 (T ),

(6.24)

def

Proof. Recall that P = I + ∇(−)−1 div. By combining (6.1) with (6.14), we find ∂t R − νR = −P Fa + ∇(−)−1 Dt Ga + ν∇Ga N

− (va + ν 2 R) · ∇R − P (R · ∇va ) N

N

− ∇(−)−1 div (R · ∇(va + ν 2 R) − Ga (va + ν 2 R − v 0 )). We write its integral formulation as follows t R= 0

e(t−s) (−P Fa + ∇(−)−1 Dt Ga + ν∇Ga )ds t

−

N e(t−s) (va + ν 2 R) · ∇R + P (R · ∇va ) ds

(6.25)

0

t −



N N e(t−s) ∇(−)−1 div R · ∇(va + ν 2 R) − Ga (va + ν 2 R − v 0 ) ds

0

Since [R] = [∇R] = 0, the distribution derivatives ∇ 2 R coincide with its pointwise derivatives. Then by applying the maximal regularity estimate for heat semi-group, we deduce that for 2 ≤ p<∞

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(∂t R, ∇ 2 R) Lp (QT ) ≤ C P Fa Lp (QT ) + ∇(−)−1 Dt Ga Lp (QT ) + ν∇Ga Lp (QT ) N

+ (va + ν 2 R) · ∇R Lp (QT ) + R · ∇va Lp (QT )

(6.26)

+ R · ∇(va + ν R) Lp (QT ) + Ga (va + ν R − v ) Lp (QT ) . N 2

N 2

0

In view of (5.7), we have 2

1− 2

P Fa Lp (QT ) ≤ C Fa Lp (QT ) ≤ C Fa Lp 2 (Q ) Fa L∞p(QT ) ≤ C. T

Observing that Dt Ga , ν∇Ga , va and ∇va are bounded in L1 (QT ) ∩ L∞ (QT ), we infer ∇(−)−1 Dt Ga Lp (QT ) + ν∇Ga Lp (QT ) ≤ C, and N

(va + ν 2 R) · ∇R Lp (QT ) + R · ∇va Lp (QT ) N

N

+ R · ∇(va + ν 2 R) Lp (QT ) + Ga (va + ν 2 R − v 0 ) Lp (QT )   ≤C + C R Lp (QT ) + ∇R Lp (QT ) (1 + ν R L∞ (QT ) ). Inserting the above estimates into (6.23) gives rise to   (∂t R, ∇ 2 R) Lp (QT ) ≤ C + C R Lp (QT ) + ∇R Lp (QT ) (1 + ν R L∞ (QT ) ). It follows from Proposition 6.1 that R L2 (QT ) ≤ C

and

√

ν ∇R L2 (QT ) ≤ C.

Proposition 6.7 ensures that √

1

ν R L∞ (QT ) ≤ CN m2 (T ).

Then for 2 < p < ∞, we have √

2 1 1 √ 1− 2 − ν R Lp (QT ) ≤ C ν R Lp 2 (Q ) R L∞p(QT ) ≤ CN m2 p (T ) T

2 p

1− 2

1− 2

ν ∇R Lp (QT ) ≤ Cν ∇R L2 (Q ) ∇R L∞p(QT ) ≤ Cν ∇R L∞p(QT ) . T

In particular p = 6, for an arbitrary small δ ∈ (0, 1), we find 1

2

1

ν (∂t R, ∇ 2 R) L6 (QT ) ≤ C + C(N m3 (T ) + ν ∇R L3 ∞ )(1 + N m2 (T )) 3

≤ δν ∇R L∞ + Cδ −2 + Cδ −2 N m2 (T ), which together with Lemma 2.2 ensures that

(6.27)

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21

ν ∇R L∞ (QT ) ≤ Cν R L222 (Q ) (∂t R, ∇ 2 R) L226 (Q T

T)

≤ C + C (∂t R, ∇ 2 R) L6 (QT ) 3

≤ Cδν ∇R L∞ + Cδ −2 + Cδ −2 N m2 (T ). We choose δ so small that Cδ < 12 . Then we obtain (6.24) for a constant C independent of ν.

2

6.6. Bootstrapping By virtue of (6.23) and (6.24), we deduce from (6.21) that t N m (t) + ν



 ∇R 2m + ∇η 2m−1 dt 

0

t



≤C 1+

3 

 1 + ν N−2 N m (t  ) + ν N−2 N m2 (t  ) R 2m + η 2m−1 dt 



(6.28)

0

t



≤C 1+

3 

 1 + ν N−2 N m2 (t  ) R 2m + η 2m−1 dt  ,



0

where N m (t) is determined by (6.22). Let us define   def T  = sup t ≤ T : N m (t) ≤ C exp (2CT ) ,

(6.29)

where T is the lifespan of the solution of (1.2), which is determined by Theorem 1.1. We claim that as long as ν is small enough, T  = T and there holds T N m (T ) + ν



 ∇R 2m + ∇η 2m−1 dt  ≤ C exp (2CT ) .

(6.30)

0

Otherwise, for t ≤ T  , we get, by applying Gronwall’s inequality to (6.28), that t N m (t) + ν

  3

∇R 2m + ∇η 2m−1 dt  ≤ C exp Ct + ν N−2 C exp(2CT ) 2 t .



0

In particular, since N ≥ 3, if we take ν so small that  3 1 ν N−2 C exp(2CT ) 2 ≤ C, 2 we deduce that

(6.31)

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T  N m (T  ) + ν

   3 ∇R 2m + ∇η 2m−1 dt ≤ C exp CT  , 2



0

which contradicts with (6.29). This in turn shows that T  = T and there holds (6.30) if ν satisfies (6.31). 7. Proof of Theorem 1.3 We now present the proof of Theorem 1.3. First, by standard technics, from the a priori estimates of the previous section we deduce the existence, for small enough ν, of strong solutions R to (6.1)–(6.2) in [0, T ] × R3 , with zero initial data and zero jumps at the interface. def

N

Then we set v ν = va + ν 2 R. We recall that va is defined in Section 5. We deduce from (5.3), (6.1)-(6.2) and (6.30) that (v ν )ν∈(0,1) are strong solutions to the NavierStokes equations (1.1) on [0, T ]. Moreover v ν (t, x) − v 0 (t, x) −

N  √ j =1

j

ν V j (t, x,

N ϕ 0 (t, x) ) = ν 2 (A + R) √ ν

N−

is O(ν 2 ) in L∞ ([0, T ]; H (O± (t))) for 0 ≤ ≤ 1 as ν → 0+ . That is (1.6) holds for = 0 and 1. The case of (1.6) for 2 ≤ ≤ N can be proved along the same line. We skip the details here. Acknowledgments F. Sueur was supported by the Agence Nationale de la Recherche, Project IFSMACS, grant ANR-15-CE40-0010, Project BORDS, grant ANR-16-CE40-0027-01 and Project SINGFLOWS, grant ANR-18-CE40-0027-01. P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences. F. Sueur warmly thanks Beijing’s Morningside center for Mathematics for its kind hospitality during his stays in September 2017 and May 2018. References [1] H. Abidi, R. Danchin, Optimal bounds for the inviscid limit of Navier-Stokes equations, Asymptot. Anal. 38 (2004) 35–46. [2] A.L. Bertozzi, P. Constantin, Global regularity for vortex patches, Commun. Math. Phys. 152 (1993) 19–28. [3] J.-Y. Chemin, Sur le mouvement des particules d’un fluide parfait incompressible bidimensionnel, Invent. Math. 103 (1991) 599–629. [4] J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993) 517–542. [5] J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995), 177 pp. [6] D. Coutand, S. Shkoller, Regularity of the velocity field for Euler vortex patch evolution, Trans. Am. Math. Soc. 370 (2018) 3689–3720. [7] D. Iftimie, F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions, Arch. Ration. Mech. Anal. 199 (2011) 145–175. [8] P. Gamblin, X.S. Raymond, On three-dimensional vortex patches, Bull. Soc. Math. Fr. 123 (1995) 375–424.

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[9] O. Guès, Problème mixte hyperboliquebquasi-linéaire caractèriistique, Commun. Partial Differ. Equ. 15 (1990) 595–645. [10] C. Huang, Singular integral system approach to regularity of 3D vortex patches, Indiana Univ. Math. J. 50 (2001) 509–552. [11] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. [12] N. Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Commun. Math. Phys. 270 (2007) 777–788. [13] N. Masmoudi, F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal. 203 (2012) 529–575. [14] S. Rempel, B.-W. Schulze, Index Theory of Elliptic Boundary Problems, reprint of the 1982 edition, North Oxford Academic Publishing Co. Ltd., London, 1985. [15] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970. [16] F. Sueur, Couches limites semilinéaires, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006) 323–380. [17] F. Sueur, Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires, Ann. Inst. Fourier (Grenoble) 56 (2006) 183–245. [18] F. Sueur, Viscous profile of vortex patches, J. Inst. Math. Jussieu 14 (2015) 1–68. [19] P. Zhang, Q.J. Qiu, Propagation of higher-order regularities of the boundaries of 3-D vortex patches, Chin. Ann. Math., Ser. A 18 (1997) 381–390. [20] P. Zhang, Q.J. Qiu, The three-dimensional revised vortex patch problem for the system of incompressible Euler equations, Acta Math. Sin. 40 (1997) 437–448.