SOLUTIONS TO A FLUID-STRUCTURE ... - USC Dornsife

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 32, Number 4, April 2012

doi:10.3934/dcds.2012.32.1355 pp. 1355–1389

SOLUTIONS TO A FLUID-STRUCTURE INTERACTION FREE BOUNDARY PROBLEM

Igor Kukavica Department of Mathematics University of Southern California Los Angeles, CA 90089, USA

Amjad Tuffaha Department of Mathematics The Petroleum Institute Abu Dhabi, UAE

(Communicated by Irena Lasiecka) Abstract. Our main result is the existence of solutions to the free boundary fluid-structure interaction system. The system consists of a Navier-Stokes equation and a wave equation defined in two different but adjacent domains. The interaction is captured by stress and velocity matching conditions on the free moving boundary lying in between the two domains. We prove the local existence of a solution when the initial velocity of the fluid belongs to H 3 while the velocity of the elastic body is in H 2 .

1. Introduction. In this paper, we study the free boundary fluid-structure interaction system first considered by Coutand and Shkoller in [9]. The system models the motion of an elastic body in a viscous incompressible fluid. The literature contains an abundance of works on the mathematics of fluidstructure interaction. The regularity of solutions was considered for models of rigid body motion in both compressible and incompressible flows. This entailed coupling the Navier-Stokes equations with systems of ordinary differential equations [13, 28]. More recent works have appeared treating the elastic motion of a body in an incompressible flow using coupled Navier-Stokes equation with hyperbolic elasticity equations on fixed domains; for nonlinear models c.f. [22, 3, 4, 18, 19, 20], and [2, 1, 12] for linear models. Other type of structures were also considered such as plates and beams in addition to strongly damped elastic bodies, see for example [6, 11, 16]. Two notable works have treated free boundary fluid-structure interaction: The case of incompressible flow was considered in Coutand and Shkoller [9, 10] while the compressible flow case was treated more recently by Boulakia and Guerrero in [7]. 2000 Mathematics Subject Classification. Primary: 35Q30, 74F10, 76D05; Secondary: 35K15, 35K55, 35M30. Key words and phrases. Fluid-structure interaction, Navier-Stokes equations, incompressible fluids.

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IGOR KUKAVICA AND AMJAD TUFFAHA

As in [9], we consider the incompressible case in which the model consists of an incompressible Navier-Stokes equation coupled with a hyperbolic equation describing elastic motion of a body through the fluid with velocity and stress boundary conditions on the interface lying in between the fluid and the elastic body. Since the problem involves a free moving boundary, the system is better formulated in Lagrangian coordinates which allows us to consider the system on a fixed domain namely the domain at the initial time. Our first result is the regularity of solutions to the system given initial Lagrangian fluid velocity in the Sobolev space H 3 and an initial velocity of the elastic body w1 in the space H 2 satisfying appropriate compatibility conditions. We note the result in [9] provides existence of solutions and regularity given initial velocity of the fluid and the elastic body in the H 5 and the H 2 spaces respectively. In our result, the obtained regularity of the solution to the fluid velocity is v ∈ C([0, T ]; H 2 (Ωf )) ∩ L2 ([0, T ]; H 3 (Ωf )), while the regularity of the elastic displacement w and the velocity wt fall in the spaces C([0, T ]; H 11/4− (Ωe )) and C([0, T ]; H 7/4− (Ωe )) where > 0 in contrast to the result in [9] where w and wt belong to the spaces C([0, T ]; H 3 (Ωe ))) and C([0, T ]; H 2 (Ωe )). The slight loss of regularity is inherent to the coupled system and has been addressed in works on the system with fixed boundary [18, 19, 20]. We only consider domains with flat boundaries and the the case of a channel type domains for both the structure and the fluid; we impose periodic boundary conditions along the channel. The difficulties in proving our theorem are three-fold: First, the presence of variable coefficients which depend on the solution v in the Lagrangian formulation of the Navier-Stokes equation requires careful estimates. Second, since the pressure term does not disappear from the energy estimates involving tangential and time derivatives, variable coefficients Stokes type estimates are required to deal with the pressure. Most importantly, the incompatibility between the Navier-Stokes and the hyperbolic equation requires going beyond energy level type estimates of solutions and their derivatives to obtain sharper regularity results. The technique developed in [18] allows for sharper estimates of time derivatives by appealing to the hidden regularity theorem [23] for the wave equation to control boundary terms which do not vanish from the estimates due to the inherent incompatibility between the wave equation and the Navier-Stokes equation. In particular, we establish the regularity of the time derivative vtt in the L2 (Ωf × [0, T ]) space. In contrast to [18], proving the regularity result requires carefully combining all tangential and time derivative estimates with the variable coefficients Stokes type estimates for the pressure. 2. The main result. We consider the coupled system of partial differential equations modeling the fluid-structure interaction phenomenon. The problem under consideration is a free moving boundary problem involving a Navier-Stokes equation ∂t u − ∆u + (u · ∇)u + ∇p = 0

(2.1)

∇·u=0

(2.2)

and the wave equation wtt − ∆w = 0 interacting on a free interface where the velocity and stress matching conditions are prescribed. Initially, the system is defined on a three dimensional smooth domain Ω = Ωf ∪Ωe where Ωf consists of two channels extending in the directions of the x1 and the x2 axes, separated by another channel Ωe in the x3 direction. For now, we take for Γc the two disjoint boundaries between the channels Ωf and Ωe to be flat in order

FLUID-STRUCTURE INTERACTION PROBLEM

1357

to make the analysis more accessible to the reader. The case of a general domain will be treated using ideas from [20] and will be addressed subsequently. The two surfaces comprising the outer boundary of Ωf , denoted by Γf , are parametrized by (x1 , x2 , 0) and (x1 , x2 , m3 ) for m3 > 0. More precisely, Ωf = {(x1 , x2 , x3 ) : (x1 , x2 ) ∈ R2 , 0 < x3 < m1 or m2 < x3 < m3 } and Ωe = {(x1 , x2 , x3 ) : (x1 , x2 ) ∈ R2 , m1 < x3 < m2 }. In Eulerian coordinates, the system consists of a Navier-Stokes equation (2.1)– (2.2) in the variables u and p denoting the velocity and the pressure of the fluid defined on the domain Ωf (t) which evolves over time from the initial configuration according to a position function η(·, t) : Ωf → Ωf (t). Similarly, the elastic equation is also defined on a domain Ωe (t) which evolves over time according to the position function η(·, t) : Ωe → Ωe (t). The Navier-Stokes equation (2.1)–(2.2) is formulated in the Lagrangian coordinates in the variables v(x, t) and q(x, t) denoting the velocity vector and the pressure of the fluid respectively over the initial domain Ωf . In other words, we have v(x, t) = ηt (x, t) = u(η(x, t), t),

x ∈ Ωf

x ∈ Ωf .

q(x, t) = p(η(x, t), t),

(2.3) (2.4)

Similarly, the elastic equation for the displacement function w(x, t) = η(x, t) is formulated in the Lagrangian framework on the domain Ωe . We consider a linear second order hyperbolic equation with constant coefficients as an approximation to the elastic phenomenon. We thus seek solutions (η, v, w, q) to the system vti − ∂j (ajl akl ∂k v i ) + ∂k (aki q) = 0 in Ωf × (0, T )

(2.5)

aki ∂k v i = 0 in Ωf × (0, T )

(2.6)

i wtt

i

− ∆w = 0 in Ωe × (0, T ),

(2.7)

where i = 1, 2, 3 and the summation convention on repeated indices is understood throughout. Above, the coefficient aij for i, j = 1, 2, 3 is the ij entry in the 3 × 3 matrix a defined by a(x, t) = (∇η(x, t))−1 ,

(x, t) ∈ Ωf × [0, T ].

(2.8)

In the variables x1 , x2 , we impose periodic boundary conditions with period 1, i.e., the functions v, q, and w are periodic with period 1. With the outward unit normal vector with respect to Ωe denoted by N = (N 1 , N 2 , N 3 ), we impose the velocity and boundary matching conditions on the interface Γc which lies in between Ωf and Ωe as v i = wti i

v =0 i

j

∂j w N =

ajl akl ∂k v i N j

−

qaki N k

on Γc × (0, T )

(2.9)

on Γf × (0, T )

(2.10)

on Γc × (0, T ),

(2.11)

for i = 1, 2, 3. The condition (2.9) represents the matching of the velocities on the common interface, (2.10) stands for the Dirichlet boundary condition on the fluid boundary, while (2.11) represents the continuity of stresses across the interface.

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Finally, the variables v, w satisfy the initial conditions v(·, 0) = v0 in Ωf

(2.12)

w(·, 0) = w0 in Ωe

(2.13)

wt (·, 0) = w1 in Ωe .

(2.14)

We also observe that η by definition satisfies the initial condition η(x, 0) = x in Ωf ,

(2.15)

∇η(x, 0) = I in Ωf .

(2.16)

and thus

Since the system is recast in the Lagrangian variables, it is advantageous to reformulate it so it does not depend on the bijectivity of η. For this purpose, observe that (2.8) implies ∂t a = −a : ∇v : a, a(x, 0) = I,

(x, t) ∈ Ωf × [0, T ]

x ∈ Ωf

(2.17) (2.18)

where the symbol : denotes the matrix product. We define η by means of the equations ∂t η = v(x, t), η(x, 0) = x,

(x, t) ∈ Ωf × [0, T ] x ∈ Ωf .

(2.19) (2.20)

Now, instead of (2.8), we take (2.17)–(2.20) to be a part of our system. The relationship (2.8) may then be deduced by the uniqueness theorem for ordinary differential equations (c.f. proof of Lemma 3.1 below). Throughout the paper we denote V = {v ∈ H 1 (Ωf ) : div v = 0, v|Γf = 0} and H = {v ∈ L2 (Ωf ) : div v = 0, v · N |Γf = 0}. Note that all spaces H s , L2 pertaining to v and w are in fact (H s )n , (L2 )n , where n = 2, 3, but we omit the exponent n for the sake of simplicity. The following theorem is our main result. Theorem 2.1. Assume that v0 ∈ V ∩ H 3 (Ωf ) and w1 ∈ H 2 (Ωe ) while w0 = η|Ωe (·, 0) = I satisfy the compatibility conditions w1i = v0i on Γc Z ∂j v i N j φi dσ(x) = 0, Γc

(2.21) φ ∈ V.

(2.22)


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Then for any δ0 > 0 there exists a local-in-time solution v ∈ L2 ([0, T ]; H 3 (Ωf )) ∩ C([0, T ]; H 11/4−δ0 (Ωf )) vt ∈ L2 ([0, T ]; H 2 (Ωf )) ∩ C([0, T ]; H 1 (Ωf )) w ∈ C([0, T ]; H 11/4−δ0 (Ωe )) wt ∈ C([0, T ]; H 7/4−δ0 (Ωe )) q ∈ L2 ([0, T ]; H 2 (Ωf )) ∩ C([0, T ]; H 7/4−δ0 (Ωf )) qt ∈ L2 ([0, T ]; H 1 (Ωf )) η|Ωf ∈ C([0, T ]; H 3 (Ωf )), to the system (2.5)–(2.7) with boundary conditions (2.9)–(2.11) and with initial conditions (2.12)–(2.14) for a time T > 0 depending on the initial data and δ0 > 0. Remark 1. Regarding uniqueness of solutions, some further assumptions are necessary to obtain uniqueness in a particular class of functions. The question of uniqueness is addressed in [21]. Remark 2. A regularity result when v0 ∈ H 3 (Ωf ) that does not involve a loss of regularity in the solutions to the wave equation with respect to the initial data is also possible as in [19]. In particular, it is possible to obtain solutions v ∈ C([0, T ]; H 5/2+k (Ωf )) w ∈ C([0, T ]; H 5/2+k (Ωe )) wt ∈ C([0, T ]; H 3/2+k (Ωe )) q ∈ C([0, T ]; H 3/2+k (Ωf )) η|Ωf ∈ C([0, T ]; H 5/2+k (Ωf ))

√ given initial data v0 ∈ H 3 (Ωf ) ∩ V and w1 ∈ H 3/2+k (Ωe ) with k ∈ (0, ( 2 − 1)/2) and under the same compatibility conditions (c.f. [21]). 3. Preliminary lemmas. We shall repeatedly use the following lemma which provides bounds on various norms of the coefficients a and the variable η, as well as for the ellipticity of the matrix aij . Lemma 3.1. Assume that k∇vkL∞ ([0,T ];H 1 (Ωf )) + k∇vkL2 ([0,T ];H 2 (Ωf )) ≤ M , where T > 0 and M ≥ 1, and let p ∈ [1, ∞]. Assume also T ≤ 1/M 4/3 . Then the following statements hold. (i) k∇η(·, t)kL∞ (Ωf ) ≤ C for t ∈ [0, T ]. (ii) det(∇η(x, t)) = 1 for all t ∈ [0, T ] and x ∈ Ωf . (iii) ka(·, t)kL∞ (Ωf ) ≤ C for t ∈ [0, T ]. (iv) kat (·, t)kLp (Ωf ) ≤ Ck∇v(t)kLp (Ωf ) for p ∈ [1, ∞] and t ∈ [0, T ]. (v) k∂i a(·, t)kLp (Ωf ) ≤ Ck∇∂i η(·, t)kLp (Ωf ) for i ∈ {1, 2, 3} and t ∈ [0, T ]. (vi) k∂ij a(·, t)kLp (Ωf ) ≤ Ck∇∂i η(·, t)kL2p (Ωf ) k∇∂j η(·, t)kL2p (Ωf ) + Ck∇∂ij η(·, t)kLp (Ωf ) for i, j ∈ {1, 2, 3} and t ∈ [0, T ]; in particular, 1/2

3/2

k∂ij a(·, t)kL2 (Ωf ) ≤ CkηkH 2 (Ωf ) kηkH 3 (Ωf ) + CkηkH 3 (Ωf ) . (vii) k∂i at (·, t)kLp (Ωf ) ≤ Ck∇v(·, t)kLp2 (Ωf ) k∇∂i η(·, t)kLp1 (Ωf ) +Ck∇∂i v(·, t)kLp (Ωf ) for t ∈ [0, T ] and i ∈ {1, 2, 3} with p1 , p2 ∈ [1, ∞] such that 1/p1 + 1/p2 = 1/p.

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(viii) katt (·, t)kLp (Ωf ) ≤ Ck∇v(·, t)k2L2p (Ωf ) + Ck∇vt (·, t)kLp (Ωf ) , for p ∈ [1, ∞] and t ∈ [0, T ]. (ix) For every ∈ (0, 1/2] there exists a constant C depending on such that for all t ≤ T ∗ = min{1/CM 2 , T } the form ∂j (ajl akl ∂k ) satisfies the estimates kδkj − ajl akl k2H s (Ωf ) ≤ ,

(3.23)

kδjl − ajl k2H s (Ωf ) ≤ ,

(3.24)

for j, k ∈ {1, 2, 3} and

for s ∈ [0, 2] and j, l = 1, 2, 3. In particular, the form ∂j (ajl akl ∂k ) satisfies the inequality 2 1 ξ ∈ Rn (3.25) ajl akl ξji ξki ≥ |ξ|2 , C for all t ∈ [0, T ∗ ] and x ∈ Ωf . Proof of Lemma 3.1. Note that (2.17) implies ∂t a = B : a where B = −a : ∇v. Now, by (2.6), we have Tr B = 0, and thus det a(·, t) = 1 for t ∈ [0, T ). In particular a is invertible for all t ∈ [0, T ). Now, applying the product rule and using (2.19) and (2.17), we get ∂t (a : ∇η) = −a : ∇v : a : ∇η + a : ∇v whence ∂t (I − a : ∇η) = (−a : ∇v)(I − a : ∇η). By the uniqueness theorem for the linear ordinary differential systems, a is the inverse of ∇η for all t ∈ [0, T ]. Also, η is a C 1 diffeomorphism; indeed, it is a local C 1 diffeomorphism by the implicit function theorem, while it is bijective by the ODE existence theorem. (i) By (2.19), we have

Z t

∇ k∇η(·, t)kL∞ (Ωf ) = v(·, s) ds + ∇η(·, 0)

∞ 0 L (Ωf ) Z t Z t 1/2 1/2 ≤ k∇vkL∞ (Ωf ) ds + 1 ≤ C k∇vkH 1 (Ωf ) k∇vkH 2 (Ωf ) ds + 1 0

0

where we used Agmon’s inequality in the last step. Therefore, Z t 1/2 k∇η(·, t)kL∞ (Ωf ) ≤ CM 1/2 k∇vkH 2 (Ωf ) ds + 1 ≤ CM

1/2 3/4

t

0 1/2 k∇vkL2 ([0,T ];H 2 (Ωf ))

+ 1 ≤ CM t3/4 + 1

and (i) is established since T ≤ 1/M 4/3 . (ii) This assertion follows from a : ∇η = I and det(a) = 1. (iii) Denoting the cofactor matrix of ∇η by cof (∇η), we have for t ≤ T

1

k cof (∇η)(·, t)kL∞ (Ωf ) ka(·, t)kL∞ (Ωf ) ≤ C det(∇η(·, t)) ∞ L

(Ωf )

≤ Ck cof (∇η)(·, t)kL∞ (Ωf ) ≤ Ck∇η(·, t)k2L∞ (Ωf ) ≤ C by the parts (i) and (ii) of this lemma.


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(iv) Using (2.17), we obtain kat (·, t)kLp (Ωf ) ≤ Cka(·, t)k2L∞ (Ωf ) k∇v(·, t)kLp (Ωf ) ≤ Ck∇v(·, t)kLp (Ωf )

(3.26)

for t ∈ [0, T ] which follows from part (iii) of this lemma. (v) Applying ∂i to the identity a : ∇η = I, we obtain ∂i a : ∇η + a : ∇∂i η = 0, from where ∂i a = −a : ∇∂i η : a.

(3.27)

In order to obtain the desired inequality, we estimate ∂i a in norm using the above expression. We get k∂i a(·, t)kLp (Ωf ) ≤ Cka(·, t)k2L∞ (Ωf ) k∇∂i η(·, t)kLp (Ωf ) ≤ Ck∇∂i η(·, t)kLp (Ωf ) , for t ∈ [0, T ] as claimed. (vi) Differentiating (3.27), we get ∂ij a = −∂i a : ∇∂j η : a − a : ∇∂i ∂j η : a − a : ∇∂j η : ∂i a = a : ∇∂i η : a : ∇∂j η : a − a : ∇∂i ∂j η : a + a : ∇∂j η : a : ∇∂i η : a where we used (3.27) in the last step. We thus obtain k∂ij akLp (Ωf ) ≤ Ckak3L∞ (Ωf ) k∇∂i ηkL2p (Ωf ) k∇∂j ηkL2p (Ωf ) + Ckak2L∞ (Ωf ) k∇∂i ∂j ηkLp (Ωf ) ≤ Ck∇∂i ηkL2p (Ωf ) k∇∂j ηkL2p (Ωf ) + Ck∇∂i ∂j ηkLp (Ωf ) and the part (vi) is established. (vii) Differentiating (3.27) in time we have ∂i at = −at : ∇∂i η : a − a : ∇∂i v : a − a : ∇∂i η : at = a : ∇v : a : ∇∂i η : a − a : ∇∂i v : a + a : ∇∂i η : a : ∇v : a. Estimating the terms in norm, we get k∂i at kLp (Ωf ) ≤ Ckak3L∞ (Ωf ) k∇vkLp1 (Ωf ) k∇∂i ηkLp2 (Ωf ) + Ckak2L∞ (Ωf ) k∇∂i vkLp (Ωf ) ≤ Ck∇vkLp1 (Ωf ) k∇∂i ηkLp2 (Ωf ) + Ck∇∂i vkLp (Ωf )

(3.28)

where we used the H¨ older inequality with 1/p1 + 1/p2 = 1/p and part (iii) of this lemma. (viii) Differentiating (2.17) in time, we obtain att = 2a : ∇v : a : ∇v : a − a : ∇vt : a

(3.29)

whence, by (i), katt (·, t)kLp (Ωf ) ≤ Cka(·, t)k3L∞ (Ωf ) k∇v(·, t)k2L2p (Ωf ) + Cka(·, t)k2L∞ (Ωf ) k∇vt (·, t)kLp (Ωf ) ≤ Ck∇v(·, t)k2L2p (Ωf ) + Ck∇vt (·, t)kLp (Ωf ) for t ∈ [0, T ]. (ix) Since ∇v ∈ L2 ([0, T ]; H 2 (Ωf )), we have that ∇η ∈ C([0, T ]; H 2 (Ωf )), which implies a ∈ C([0, T ]; H 2 (Ωf )) for all t < T . Observe that at t = 0 we have ajl = δjl for j, l = 1, 2, 3.

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Our starting point is the identity (2.17) which implies ∂t (a − I) = −a : ∇v : a = −(a − I) : ∇v : a − ∇v : (a − I) − ∇v = −(a − I) : ∇v : (a − I) − ∇v : (a − I) − (a − I) : ∇v − ∇v. Therefore, letting g(x, t) = a(x, t) − I we have

d

g(·, t) ≤ Cka − Ik2L∞ (Ωf ) k∇vkH 2 (Ωf )

dt

2 H (Ωf )

+ Ck∇vkL∞ (Ωf ) ka − IkL∞ (Ωf ) ka − IkH 2 (Ωf ) + Ck∇vkH 2 (Ωf ) ka − IkH 2 (Ωf ) + Ck∇vkH 2 (Ωf ) ≤ Cka − IkH 2 (Ωf ) ka − IkL∞ (Ωf ) k∇vkH 2 (Ωf ) + Cka − IkH 2 (Ωf ) k∇vkH 2 (Ωf ) + Ck∇vkH 2 (Ωf ) . Hence, kg(t)kH 2 (Ωf )

Z t

d

≤ ds

dt g(s) 2 0 H (Ωf ) Z T ≤C ka(s) − IkH 2 (Ωf ) k∇v(s)kH 2 (Ωf ) ds 0

Z

T

k∇v(s)kH 2 (Ωf ) ds.

+C 0

Applying Gronwall’s inequality, we have that kδjl − ajl (t)kH 2 (Ωf ) ≤ C for t ≤ T ∗ = 2 /CM 2 . Next, we prove the inequality (3.23) for s = 2. We have kδkj − ajl akl kH 2 (Ωf ) ≤ kδkj − ajl δkl kH 2 (Ωf ) + kajl δkl − ajl akl kH 2 (Ωf ) ≤ C for t ∈ (0, T ∗ ). We thus obtain (3.23) by replacing with /C in the definition of T ?. From here on, the constant T ∗ is relabeled as T . Also, we simplify the notation for all the norms by omitting the indication for the domain as it is always clear from the context. For instance, we write kvkL2 = kvkL2 (Ωf ) and kwkH 1 = kwkH 1 (Ωe ) . Next we state a crucial result we need is a pressure estimate for the Stokes system with variable coefficients. Lemma 3.2. Assume that v and q satisfy the system − ∂j (ajl akl ∂k v i ) + ∂k (aki q) = f i in Ωf

(3.30)

aki ∂k v i = i i

(3.31)

h in Ωf

v = g on Γc = ∂Ωe

(3.32)

for given coefficients aji ∈ L∞ (Ωf ) with i, j = 1, 2, 3 satisfying parts (iii) and (ix) of Lemma 3.1 with a sufficiently small constant > 0. Then we have the following statements. (i) The functions v and q satisfy kvkH s+2 + kqkH s+1 ≤ C kf kH s + kgkH s+3/2 (Γc ) + khkH s+1 (3.33)


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for s = −1, 0, 1. (ii) If in addition, v, q, f , g, and aji are time dependent and a satisfies the condition in part (iv) of Lemma 3.1, we have kvt kH 2 + kqt kH 1 ≤ Ckft kL2 + Ckgt kH 3/2 (Γc ) + Ckht kH 1 1/2

1/2

1/2

1/2

+ CkvkH 2 kvkH 3 (kηkH 2 kηkH 3 + 1) (kvkH 2 + kqkH 1 ) . (3.34) Before the proof, we apply this lemma (with s = 0, 1) to the system (2.5)–(2.6) with the term vt considered as force. First, using (i) with s = 0, we obtain kvkH 2 + kqkH 1 ≤ Ckvt kL2 + Ckv|Γc kH 3/2 (Γc ) ≤ Ckvt kL2 + C

2 X

k∂j v|Γc kH 1/2 (Γc )

j=1

≤ Ckvt kL2 + C

2 X

k∂j vkH 1 = Ckvt kL2 + CkD0 vkH 1

j=1

where D0 v = (∂1 v, ∂2 v), and thus kvkH 2 + kqkH 1 ≤ Ckvt kL2 + CkD0 vkH 1 .

(3.35)

Also, using (i) with s = 1, we get kvkH 3 + kqkH 2 ≤ Ckvt kH 1 + Ck(D0 )2 vkH 1

(3.36)

where (D0 )2 v denotes the matrix (∂ij v)i,j=1,2 . Similarly, using (ii), kvt kH 2 + kqt kH 1 ≤ Ckvtt kL2 + Ckvt |Γc kH 3/2 (Γc ) 1/2

1/2

1/2

1/2

+ CkvkH 2 kvkH 3 (kηkH 2 kηkH 3 + 1)(kvkH 2 + kqkH 1 )

(3.37)

and thus using (3.36) and ku|Γc kH 3/2 (Γc ) ≤

2 X

k∂j u|Γc kH 1/2 (Γc ) ≤ CkD0 ukH 1

(3.38)

j=1

it follows kvt kH 2 + kqt kH 1 ≤ Ckvtt kL2 + Ck(D0 )2 vkH 1 + CkD0 vt kH 1 + Ckvt kH 1 + CkvkH 2 (kηkH 2 kηkH 3 + 1)(kvk2H 2 + kqk2H 1 ).

(3.39)

Proof of Lemma 3.2. (i) Let φ be a solution of the elliptic problem ∆φ = −(δki − aki )∂k v i − h

in Ωf

(3.40)

∂φ = −g · N ∂ν

on Γc .

(3.41)

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Then u = v + ∇φ solves −∆ui + ∂i q = −∆∂i φ − ∂j ((δkj − ajl akl )∂k v i ) + ∂k ((δki − aki )q) + f i ∂i ui = 0

in Ωf (3.42) in Ωf (3.43)

u = g + ∇φ

on Γ. (3.44)

We then invoke the Stokes estimate from [30, Proposition 2.3] and obtain kukH s+2 + kqkH s+1 ≤ Ck∆∇φkH s + k∂j ((δkj − ajl akl )∂k v i )kH s + k∂k ((δki − aki )q)kH s + Ckf kH s + kgkH s+3/2 (Γc ) + k∇φ|Γc kH s+3/2 (Γc ) .

(3.45)

Since k∆∇φkH s ≤ CkφkH s+3 ≤ Ck(δki − aki )∂k v i kH s+1 + CkhkH s+1 + CkgkH s+3/2 (Γc ) we get kvkH s+2 + kqkH s+1 ≤ Ck(δki − aki )∂k v i kH s+1 + Ck∂j ((δkj − ajl akl )∂k v i )kH s + Ck(δki − aki )qkH s+1 + CkhkH s+1 + Ckf kH s + CkgkH s+3/2 (Γc ) .

(3.46)

For s = −1 this becomes kvkH 1 + kqkL2 ≤ Ck(δki − aki )∂k v i kL2 + Ck(δkj − ajl akl )∂k v i kL2 + Ck(δki − aki )qkL2 + CkhkL2 + Ckf kH −1 + CkgkH 1/2 (Γc ) . (3.47) Now we use the properties (3.23) and (3.24) of a to obtain kvkH 1 + kqkL2 ≤ Ckf kH −1 + CkgkH 1/2 (Γc ) + CkvkH 1 + CkhkL2 + CkqkL2 , and if is a sufficiently small positive constant, we obtain (3.33) with s = −1. For s = 0, the inequality (3.46) becomes kvkH 2 + kqkH 1 ≤ Ck(δki − aki )∂k v i kH 1 + Ck(δkj − ajl akl )∂k v i kH 1 + Ck(δki − aki )qkH 1 + CkhkH 1 + Ckf kL2 + CkgkH 3/2 (Γc ) . (3.48)


1365

Now, we estimate each term involving a. For the second term on the right side of (3.48), we have k(δkj − ajl akl )∂k v i kH 1 ≤ Ckδkj − ajl akl kL∞ kvkH 2 + Ck∇(δkj − ajl akl )kL3 k∇vkL6 ≤ Ckδkj − ajl akl kL∞ kvkH 2 1/2

1/2

+ Ck∇(δkj − ajl akl )kL2 k∇(δkj − ajl akl )kH 1 k∇vkL6 ≤ CkvkH 2 where we used the Sobolev inequality and the assumptions on a. Similarly, we estimate the first and the third terms on the right side of (3.48) in order to obtain k(δki − aki )∂k v i kH 1 ≤ CkvkH 2 , and k(δki − aki )qkH 1 ≤ CkqkH 1 . Absorbing these terms into the left side of (3.48), and choosing sufficiently small, we obtain (3.33) for s = 0. We next treat the case s = 1 by again appealing to the inequality (3.46) obtaining kvkH 3 + kqkH 2 ≤ Ck(δki − aki )∂k v i kH 2 + Ck∂j ((δkj − ajl akl )∂k v i )kH 1 + Ck(δki − aki )qkH 2 + CkhkH 2 + Ckf kH 1 + CkgkH 5/2 (Γc ) . (3.49) We again estimate separately the terms involving the coefficients ajl . We thus have k∂j ((δkj − ajl akl )∂k v i )kH 1 ≤ kδkj − ajl akl kH 2 k∇vkL∞ + kδkj − ajl akl kL∞ k∇vkH 2 ≤ CkvkH 3

(3.50)

where we used the assumptions on a in (3.23) and (3.24) in addition to the Sobolev inequality. Similarly, we have k∂j ((δki − aki )∂k v i )kH 1 ≤ CkvkH 3 , and k∂k ((δki − aki )q)kH 1 ≤ CkqkH 2 . The inequality then follows for s = 1. (ii) We differentiate the system (3.30)–(3.32) in time to get −∂j (ajl akl ∂k vti ) + ∂k (aki qt ) = fti − ∂j (∂t (ajl akl )∂k v i ) − ∂k (∂t aki q) aki ∂k vti vti

= ht − =

gti

∂t aki ∂k v i

in Ωf

(3.51)

in Ωf

(3.52)

on Γc

(3.53)

and thus, using (i) with s = 0, kvt kH 2 + kqt kH 1 ≤ Ckft kL2 + Ck∂j (∂t (ajl akl )∂k v i )kL2 + Ck∂k (∂t aki q)kL2 + Ckgt kH 3/2 (Γc ) + Ckht kH 1 + k∂t aki ∂k v i kH 1 .

(3.54)

1366


Proceeding to bound the terms containing factors of a we have k∂j (∂t (ajl akl )∂k v i )kL2 ≤ C (k∂t akL∞ k∇akL3 + k∂t ∇akL3 ) k∇vkL6 + Ck∂t akL∞ kvkH 2 ≤ Ck∇vkL∞ kD2 ηkL3 kvkH 2 + C(k∇vkL∞ kD2 ηkL3 + kD2 vkL3 )kvkH 2 + Ck∇vkL∞ kvkH 2 3/2

1/2

1/2

1/2

3/2

1/2

≤ CkvkH 2 kvkH 3 kηkH 2 kηkH 3 + CkvkH 2 kvkH 3 . Similarly, k∂k (∂t aki q)kL2 ≤ Ck∂t ∇akL3 kqkL6 + k∂t akL∞ k∇qkL2 ≤ C(k∇vkL∞ kD2 ηkL3 + kD2 vkL3 )kqkH 1 + Ck∇vkL∞ kqkH 1 1/2

1/2

1/2

1/2

≤ CkvkH 2 kvkH 3 kηkH 2 kηkH 3 kqkH 1 . Also, k∂t aki ∂k v i kH 1 ≤ Ck∂t ∇akL3 k∇vkL6 + Ck∂t akL∞ kvkH 2 . Hence, (3.34) is established. Lemma 3.3. Let δ0 ∈ (0, 1/2], and let v, q, w, wt satisfy the system (2.5)–(2.7), then we have the inequalities Z t kwkH 11/4−δ0 ≤ Ckwtt kH 3/4−δ0 + C kv(s)kH 11/4−δ0 ds + Ckw(0)kH 11/4−δ0 (3.55) 0

and kwt kH 7/4−δ0 ≤ Ckwttt kH −1/4−δ0 + CkvkH 7/4−δ0 for every t ∈ (0, T ).

(3.56)

Proof of Lemma 3.3. Observe that w solves the elliptic problem ∆w = wtt ,

on Ωe

with the Dirichlet boundary condition Z t Z t w= wt ds + w0 = v ds + w0 on Γc . 0

0

Hence, w satisfies the estimate

Z t

kwkH 11/4−δ0 ≤ Ckwtt kH 3/4−δ0 + C v ds + w 0

0

H 9/4−δ0 (Γc )

and (3.55) follows by applying the trace inequality on the second term. Similarly, the function wt satisfies the elliptic problem ∆wt = wttt with the Dirichlet boundary condition wt = v on Γc . Therefore, kwt kH 7/4−δ0 ≤ Ckwttt kH −1/4−δ0 + CkvkH 5/4−δ0 (Γc ) and (3.56) follows by applying the trace inequality. The following estimates are similar to those in Lemma 3.2 except for using the Neumann-stress boundary conditions.


1367

Lemma 3.4. Assume that v, q, w, wt are as in Theorem 2.1. Then we have kv(t)kH 11/4−δ0 + kq(t)kH 7/4−δ0 Z t k∇vt (s)kL2 ds ≤ Ck∇vt (t)kL2 + Ckwtt (t)kH 3/4−δ0 + C 0 Z t +C kwtt (s)kH 3/4−δ0 ds + Ckw(0)kH 11/4−δ0

(3.57)

0

with an additional assumption T ≤ 1/C for a sufficiently large constant C > 0. Proof of Lemma 3.4. First, we consider the Stokes problem (3.30)–(3.31) with the alternative boundary condition of Neumann type akl ajl ∂j v i N k − aki qN k = g i on Γc = ∂Ωe

(3.58)

in place of the Dirichlet condition (3.32), and with the Dirichlet condition v = 0 on Γf ; proceeding as in proof of Lemma 3.2 but using results on for the constant coefficients Stokes problem [25], we have the analogous estimate to (3.33) given by kvkH s+2 + kqkH s+1 ≤ C kf kH s + kgkH s+1/2 + khkH s+1 (3.59) for all s ∈ [−1, 1]. We then apply the above estimate to the case of f i = vti and g i = ∂j wi N j with s = 3/4 − δ0 and obtain kvkH 11/4−δ0 + kqkH 7/4−δ0 ≤ C kvt kH 1 + k∇w · N kH 5/4−δ0 (Γc ) (3.60) for every t ∈ [0, T ). We next apply the trace inequality and get kv(t)kH 11/4−δ0 + kq(t)kH 7/4−δ0 ≤ C kvt (t)kH 1 + kw(t)kH 11/4−δ0

(3.61)

whence by the elliptic estimate (3.55) kv(t)kH 11/4−δ0 + kq(t)kH 7/4−δ0 ≤ C kvt (t)kH 1 + kwtt (t)kH 3/4−δ0 Z t + kv(s)kH 11/4−δ0 ds + kw(0)kH 11/4−δ0 .

(3.62)

0

Integrating the last inequality, we get Z t kv(t)kH 11/4−δ0 ds 0 Z t Z t ≤C kvt (t)kH 1 ds + C kwtt (t)kH 3/4−δ0 ds 0 0 Z t + Ct kv(s)kH 11/4−δ0 ds + Ctkw(0)kH 11/4−δ0 . 0

If T is smaller than a sufficiently small constant, the third term on the right side can be absorbed in the left side. Replacing the resulting inequality in (3.62) we then obtain (3.57).

1368


4. Proof of the main theorem. 4.1. A priori estimates. Lemma 4.1. The time derivatives vt , wt , and wtt satisfy the estimate Z t k∇vt k2L2 ds kvt (t)k2L2 + kwtt (t)k2L2 + k∇wt (t)k2L2 + 0 Z t 1/2 1/2 ≤C kvkH 1 kvkH 2 kvkH 2 + kqkH 1 k∇vt kL2 ds 0 Z t +C k∂t qkL2 k∇vt kL2 ds + CE(0)

(4.63)

0

for all t ∈ [0, T ) where E(0) = kv0 k2H 3 + kw0 k2H 3 + kw1 k2H 2 . Proof of Lemma 4.1. We differentiate the system (2.5)–(2.7) in time and obtain i vtt − ∂t ∂j (ajl akl ∂k v i ) + ∂t ∂k (aki q) = 0 in Ωf × (0, T )

∂t aki ∂k v i

+ aki ∂k vti i wttt − ∆wti

(4.64)

= 0 in Ωf × (0, T )

(4.65)

= 0 in Ωe × (0, T ),

(4.66)

while the boundary conditions (2.9)–(2.11) under time differentiation read i vti = wtt

on Γc × (0, T )

(4.67)

on Γf × (0, T )

(4.68)

on Γc × (0, T ),

(4.69)

vti = 0 ∂j wti N j

=

∂t (ajl akl ∂k v i )N j

−

∂t (qaki )N k

for i = 1, 2, 3. We now multiply (4.64) by vti , integrate over the domain Ωf , and sum in i to obtain Z 1 d kvt k2L2 + ∂t (ajl akl )∂k v i ∂j vti dx 2 dt Ωf Z Z + ajl akl ∂k vti ∂j vti dx + ∂t (ajl akl ∂k v i )vti N j dσ(x) Ωf

Z

Γc

∂t q aki ∂k vti dx −

− Ωf

Z

q∂t aki ∂k vti dx −

Ωf

Z

∂t (qaki )vti N k dσ(x) = 0.

Γc

(4.70) i Similarly multiplying (4.66) by wtt and integrating by parts over Ωe then summing over i = 1, 2, 3 we have Z 1 d 1 d 2 2 i kwtt kL2 + k∇wt kL2 − ∂j wti wtt N j dσ(x) = 0. (4.71) 2 dt 2 dt Γc

We now add (4.70) and (4.71) and apply the boundary conditions (4.67)–(4.69) to cancel all the boundary terms from the equation obtaining Z 1 d 1 d 1 d kvt k2L2 + kwtt k2L2 + k∇wt k2L2 + ∂t (ajl akl )∂k v i ∂j vti dx 2 dt 2 dt 2 dt Ωf Z Z Z + ajl akl ∂k vti ∂j vti dx − ∂t q aki ∂k vti dx − q∂t aki ∂k vti dx = 0. Ωf

Ωf

Ωf

(4.72)


1369

Rearranging terms so that only positive terms remain on the right side and appealing to the ellipticity (3.25) we get 1 1 d kvt k2L2 + kwtt k2L2 + k∇wt k2L2 + k∇vt k2L2 2 dt Z C Z Z j k i i k i k i ≤ ∂t (al al )∂k v ∂j vt dx + ∂t qai ∂k vt dx + q∂t ai ∂k vt dx Ωf Ωf Ωf = A1 + A2 + A3 .

(4.73)

We next estimate each of these terms which appear on the right side of (4.73). Starting with A1 , we have by Hölder’s inequality and Lemma 3.1 A1 ≤ Ck∂t akL6 kakL∞ k∇vkL3 k∇vt kL2 ≤ Ck∇vkL6 k∇vkL3 k∇vt kL2 1/2

3/2

≤ CkvkH 1 kvkH 2 k∇vt kL2

(4.74)

for t ≤ T . Regarding A2 , we first apply Hölder’s inequality and obtain A2 ≤ k∂t qkL2 kakL∞ k∇vt kL2 ≤ Ck∂t qkL2 k∇vt kL2 ,

t < T.

(4.75)

We finally estimate A3 as A3 ≤ CkqkL6 k∂t akL3 k∇vt kL2 ≤ CkqkH 1 k∇vkL3 k∇vt kL2 1/2

1/2

≤ CkqkH 1 kvkH 1 kvkH 2 k∇vt kL2

(4.76)

for t ∈ [0, T ) Using (4.74), (4.75), and (4.76) in (4.73), integrating the resulting inequality in time, we obtain (4.63).

Let S be a product of time and tangential derivatives. Then the same derivation as in the above proof gives an identity 1 d kSvk2L2 + kSwt k2L2 + k∇Swk2L2 2 dt Z Z + Ωf

S(ajl akl ∂k v i )S∂j v i −

S(aki q)S∂k v i = 0

(4.77)

Ωf

for t ∈ (0, T ). We shall apply this equality when S is a product of the time and a tangential derivative (Lemma 4.2) or two tangential derivatives (Lemma 4.3). In the next lemma, we obtain estimates on the mixed time tangential derivatives of the solution.

1370


Lemma 4.2. With m = 1, 2 and for any ∈ (0, 1/2] we have kD

0

0

wtt (t)k2L2

0

1 + C

Z

t

+ kD + k∇D k∇D0 vt k2L2 ds 0 Z t Z t 2 2 ≤C k∇vkL∞ kvkH 2 ds + C kηk2H 3 k∇vt kL2 kvt kH 2 ds 0 0 Z t +C (k∇vkL2 k∇vkH 1 kηk2H 3 + kD0 vk2H 1 )k∇vk2L∞ ds 0 Z t kqk2L∞ (k∇vk2L∞ kηk2H 2 + k∇D0 vkL2 ) ds +C 0 Z t Z t (kqt k2H 1 kqk2H 1 k∇vk2L∞ ds + +C 0 0 Z t + kvt k2H 2 ) ds + C kηk2H 2 kηk6H 3 + kηk4H 3 kvt k2H 1 ds 0 Z t + C (kvkH 1 kvkH 2 kηk2H 3 + kD0 vk4H 1 )k∇vkL∞ ds 0 Z t + C (kηk2H 3 kvt kH 1 kvt kH 2 + k∇vk2L∞ k∇∂m vk2L2 ) ds

vt (t)k2L2

wt (t)k2L2

0

+ CE(0)

(4.78)

for all t ∈ [0, T ). Proof of Lemma 4.2. Fix m ∈ {1, 2}; the summation convention in this proof does not apply to m. Using the identity (4.77) with S = ∂t ∂m , we get 1 d k∂t ∂m vk2L2 + k∂t ∂m wt k2L2 + k∂t ∂m ∇wk2L2 2 dt Z Z + Ωf

∂t ∂m (ajl akl ∂k v i )∂t ∂m ∂j v i dx −

∂t ∂m (aki q)∂t ∂m ∂k v i dx = 0.

Ωf

By (3.25), we have the estimate Z 1 1 d 2 2 2 k∂m vt kL2 + k∂m wtt kL2 + k∇∂m wt kL2 + k∇∂m vti k2L2 dx 2 dt C Ωf Z Z j k j k i i i i ≤ ∂t (al al )∂k ∂m v ∂j ∂m vt dx + ∂m (al al )∂k vt ∂j ∂m vt dx Ωf Ωf Z Z j k i i k i + ∂m ∂t (al al )∂k v ∂j ∂m vt dx + q∂m ∂t (ai )∂k ∂m vt dx Ωf Ωf Z Z + ∂m q∂t (aki )∂k ∂m vti dx + qt ∂m aki ∂k ∂m vti dx Ωf Ωf Z + ∂m qt aki ∂k ∂m vti dx Ωf = A1 + A2 + A3 + A4 + A5 + A6 + A7 .

(4.79)


1371

We estimate each of these terms using Lemma 3.1 and Hölder’s inequality. For the first term A1 , we have A1 ≤ Ck∂t (ajl akl )kL∞ k∇∂m vkL2 k∇∂m vt kL2 ≤ 0 k∇∂m vt k2L2 + C0 k∂t ak2L∞ kak2L∞ k∇∂m vk2L2 ≤ 0 k∇∂m vt k2L2 + C0 k∇vk2L∞ kvk2H 2 with 0 ∈ (0, 1/2] to be determined. Similarly, we bound A2 by Hölder’s inequality and Lemma 3.1 to get A2 ≤ k∂m (ajl akl )kL6 k∇vt kL3 k∇∂m vt kL2 ≤ 0 k∇∂m vt k2L2 + C0 k∂m ak2L6 kak2L∞ k∇vt k2L3 ≤ 0 k∇∂m vt k2L2 + C0 k∇∂m ηk2L6 k∇vt kL2 k∇vt kH 1 ≤ 0 k∇∂m vt k2L2 + C0 kηk2H 3 k∇vt kL2 kvt kH 2 .

(4.80)

Regarding A3 , we have A3 ≤ Ck∂m ∂t (ajl akl )kL2 k∇vkL∞ k∇∂m vt kL2 ≤ 0 k∇∂m vt k2L2 + C0 (k∂t ∂m ak2L2 kak2L∞ + k∂t ak2L3 k∂m ak2L6 )k∇vk2L∞ ≤ 0 k∇∂m vt k2L2 + C0 (k∇vkL2 k∇vkH 1 kηk2H 3 + kD0 vk2H 1 )k∇vk2L∞ where we used in the third inequality k∂t a3 kL3 k∂m akL6 ≤ Ck∇vkL3 kD2 ηkL6 and k∂t ∂m akL2 ≤ Ck∇vkL3 k∇∂m ηkL6 + Ck∇∂m vkL2 1/2

1/2

≤ Ck∇vkL2 k∇vkH 1 kηkH 3 + Ck∂m vkH 1 which both result from Lemma 3.1. We again estimate the pressure by (3.35) to get A4 ≤ kqkL∞ k∂m ∂t akL2 k∇∂m vt kL2 ≤ 0 k∇∂m vt k2L2 + C0 kqk2L∞ (k∇vk2L∞ kηk2H 2 + k∇D0 vk2L2 ). We next estimate A5 by H¨ older’s inequality to obtain A5 ≤ k∂m qkL2 k∂t akL∞ k∇∂m vt kL2 ≤ 0 k∇∂m vt k2L2 + C0 kqk2H 1 k∇vk2L∞ . Applying Lemma 3.2 (ii), we have Z Z Z 2 k A6 = qt ∂m aki ∂k ∂m vti dx = ∂m qt ∂m aki ∂k vti dx + qt ∂m ai ∂k vti dx Ωf Ωf Ωf 2 k ≤ Ck∂m qt kL2 k∂m aki kL6 k∂k vti kL3 + Ckqt kL6 k∂m ai kL2 k∂k vti kL3 1/2

3/2

1/2

1/2

≤ Ckqt kH 1 (kηkH 2 kηkH 3 + kηkH 3 )kvt kH 1 kvt kH 2 and thus

A6 ≤ (kqt k2H 1 + kvt k2H 2 ) + C kηk2H 2 kηk6H 3 + kηk4H 3 kvt k2H 1 . We finally treat the last term A7 by invoking the divergence condition ∂m ∂t aki ∂k v i + ∂m aki ∂k vti + ∂t aki ∂k ∂m v i + aki ∂k ∂m vti = 0 in Ωf × (0, T )

(4.81)

1372


which is the result of time and space differentiation of the condition (2.6). Hence, we have Z A7 = ∂m qt aki ∂k ∂m vt dx Ωf Z k k k = ∂m qt (∂m ∂t ai ∂k v + ∂m ai ∂k vt + ∂t ai ∂k ∂m v) dx . Ωf Estimating the right side using Hölder’s inequality, we obtain A7 ≤ Ck∂m qt kL2 k∂m ∂t akL2 k∇vkL∞ + Ck∂m qt kL2 k∂m akL6 k∇vt kL3 + Ck∂m qt kL2 k∂t akL∞ k∂m ∇vkL2 ≤ Ck∂m qt kL2 (k∇vkL3 kD2 ηkL6 + kD0 vk2H 1 )k∇vkL∞ 1/2

1/2

+ Ck∂m qt kL2 kηkH 3 kvt kH 1 kvt kH 2 + Ck∂m qt kL2 k∇vkL∞ k∇∂m vkL2 ≤ k∂m qt k2L2 + C (kvkH 1 kvkH 2 kηk2H 3 + kD0 vk4H 1 )k∇vk2L∞ + C kηk2H 3 kvt kH 1 kvt kH 2 + C k∇vk2L∞ k∇∂m vk2L2 . We now substitute all the estimates on A1 through A7 in (4.79), choose 0 sufficiently small, and then integrate in time the resulting inequality in order to obtain (4.78). Lemma 4.3. For all , δ0 ∈ (0, 1/2], we have 0 2

k(D )

0 2

wt (t)k2L2

0 2

1 + C

Z

t

+ k(D ) + k∇(D ) k∇(D0 )2 v(t)k2L2 ds 0 Z t Z t 2 ≤ kqkH 2 ds + C (kηkH 2 kηk3H 3 + kηk2H 3 )kqk2L∞ ds 0 0 Z t Z t +C kηk4H 3 kD0 vk2H 1 ds + kqk2H 3/2 kηk2H 3 ds 0 0 Z t + C (kηkH 2 kηk3H 3 + kηk2H 3 )kvk2H 5/2+δ0 ds + CE(0) (4.82)

v(t)k2L2

w(t)k2L2

0

for all t ∈ [0, T ). 2 Proof of Lemma 4.3. It is sufficient to treat the derivative ∂m for m = 1, 2. Fix m ∈ {1, 2} (again, there is no summation convention for the index m). Using the 2 identity (4.77) with S = ∂m , we obtain

1 d 2 2 2 k∂m vk2L2 + k∂m wt k2L2 + k∇∂m wk2L2 2 dt Z Z + Ωf

2 2 ∂m (ajl akl ∂k v i )∂m ∂j v i dx −

Ωf

2 2 ∂m (aki q)∂m ∂k v i dx = 0


1373

whence 1 1 d 2 2 2 2 k∂m vk2L2 + k∂m v(t)k2L2 wt k2L2 + k∇∂m wkL2 + k∇∂m 2 dt C Z Z j k j k i 2 i 2 i 2 i ∂m (al al )∂k ∂m v ∂j ∂m v dx ∂m (al al )∂k v ∂m ∂j v dx + 2 ≤ Ωf Ωf Z Z 2 k 2 i 2 i q∂m ∂m q∂m aki ∂k ∂m + ai ∂k ∂m v dx + 2 v dx Ωf Ωf Z 2 2 i + ∂m q aki ∂k ∂m v dx Ωf = A1 + A2 + A3 + A4 + A5 .

(4.83)

Estimating the terms on the right side we have 2 2 A1 ≤ C(k∂m akL2 + k∂m ak2L4 )k∇vkL∞ k∇∂m vkL2 2 2 ≤ 0 k∇∂m vk2L2 + C0 (k∂m ak2L2 + k∂m ak4L4 )k∇vk2L∞

≤ 0 k(D0 )2 vk2H 1 + C0 (kηkH 2 kηk3H 3 + kηk2H 3 )k∇vk2L∞ with 0 ∈ (0, 1/2] to be determined. Next, 2 A2 ≤ Ck∂m akL6 k∇∂m vkL3 k∇∂m vkL2 1/2

1/2

2 ≤ Ck∇∂m ηkL6 k∂m vkH 1 k∂m vkH 2 k∇∂m vkL2 2 ≤ 0 k∇∂m vk2L2 + 0 k∂m vk2H 2 + C0 kηk4H 3 kD0 vk2H 1 .

Next, we estimate A3 . We have 2 2 A3 ≤ CkqkL∞ k∂m akL2 k∇∂m vkL2 2 ≤ 0 k∇∂m vk2L2 + C0 kqk2L∞ (kηkH 2 kηk3H 3 + kηk2H 3 ).

For A4 we have 2 2 A4 ≤ Ck∂m qkL3 k∂m akL6 k∇∂m vkL2 ≤ 0 k∇∂m vk2L2 + C0 k∂m qk2L3 k∂m ak2L6 2 ≤ 0 k∇∂m vk2L2 + C0 kqk2H 3/2 kηk2H 3 .

In order to bound A5 , we use the divergence-free condition (2.6) which gives 2 k 2 i ∂m ai ∂k v i + 2∂m aki ∂k ∂m v i + aki ∂k ∂m v = 0.

Therefore, we may rewrite Z Z Z 2 k 2 2 2 k i 2 k i A5 = ∂m qai ∂k ∂m v dx = ∂m q∂m ai ∂k v dx + 2 ∂m q∂m ai ∂k ∂m v dx . Ωf Ωf Ωf Applying H¨ older’s inequality to A5 , we get 2 2 A5 ≤ Ck∂m qkL2 (k∂m akL2 k∇vkL∞ + k∂m akL6 k∇∂m vkL3 ) 1/2

3/2

≤ CkqkH 2 (kηkH 2 kηkH 3 + kηkH 3 )kvkH 5/2+δ0 ≤ kqk2H 2 + C (kηkH 2 kηk3H 3 + kηk2H 3 )kvk2H 5/2+δ0 . The lemma is then obtained by collecting the estimates on A1 through A5 and 2 choosing 0 so small that the terms 0 k∇∂m vk2L2 may be absorbed by the half of the second term on the left side of (4.83).

1374


Lemma 4.4. For all ∈ (0, 1/2] we have t

Z

kvtt k2L2 ds +

0

1 k∇vt (t)k2L2 + kwttt (t)k2H −1/4−δ + kwtt (t)k2H 3/4−δ C 1/2

≤ CkvkL2 (kvt kL2 + kD0 vkH 1 )7/2 Z t +C k∇vkL∞ k∇vkL2 + k∇vt kL2 k∇vkL∞ k∇vt kL2 ds 0 Z t k∂t qkL2 k∇vkL∞ + kqkL∞ k∇vt kL2 +C 0 × k∇vt kL2 + k∇vkL2 k∇vkL∞ ds Z t Z t kvt k2H 2 ds + C + C k∇vt k2L2 ds + CkD0 v(t)k2H 1 0

0

+ C kvt (t)k2L2 + f2 (E(0)) (4.84) for all t ∈ [0, T ). i , and sum Proof of Lemma 4.4. We differentiate (2.5) in time, then multiply by vtt in i and obtain

Z

i i vtt vtt dx −

Z

Ωf

i ∂t ∂j (ajl akl ∂k v i )vtt dx +

Ωf

Z

i ∂t ∂k (qaki )vtt dx = 0.

Ωf

Integrating by parts over Ωf , we get

kvtt k2L2 +

Z Ωf

i akl ∂k vti ajl ∂j vtt dx +

Z Ωf

i ∂t (ajl akl )∂k v i ∂j vtt dx

Z

i ∂t (ajl akl ∂k v i )vtt N j dσ(x) ZΓc Z i i − ∂t (qaki )∂k vtt dx − ∂t (qaki )N k vtt dσ(x) = 0.

+

Ωf

Γc

We now appeal to the boundary condition (4.69) so that

kvtt k2L2 +

Z Z 1 1 d ajl akl ∂k vti ∂j vti dx − ∂t (ajl akl )∂k vti ∂j vti dx 2 dt Ωf 2 Ωf Z Z j k i i i + ∂t (al al )∂k v ∂j vtt dx − ∂t (qaki )∂k vtt dx Ωf

Z + Γc

Ωf

i ∂j wti N j vtt dσ(x) = 0.

(4.85)


1375

Integrating in time, we then have

Z

t

Z 1 ds + aj (t)∂j vti (t)akl (t)∂k vti (t) dx 2 Ωf l Z Z Z 1 t 1 ∂t (ajl akl )∂k vti ∂j vti dx ds ajl (0)∂j vti (0)akl (0)∂k vti (0) dx + = 2 Ωf 2 0 Ωf Z tZ Z tZ j k i i i − ∂t (al al )∂k v ∂j vtt dx ds + ∂t (qaki )∂k vtt dx ds

kvtt k2L2

0

0

Ωf

Z tZ − 0

0

Ωf

i ∂j wti N j vtt dσ(x) ds

Γc

from where, using Lemma 3.1 (ix) and ∇a(0, ·) = I

Z

t

1 k∇vt (t)k2L2 C

kvtt k2L2 ds +

0

Z Z 1 1 t j k 2 i i ≤ k∇vt (0)kL2 + ∂t (al al )∂k vt ∂j vt dx ds 2 2 0 Ωf Z Z t i + ∂t (ajl akl )∂k v i ∂j vtt dx ds 0 Ωf Z Z Z Z t t i j i k i ∂j wt N vtt dσ(x) ds + ∂t (qai )∂k vtt dx ds + 0 Ωf 0 Γc =

1 k∇vt (0)k2L2 + A1 + A2 + A3 + A4 . 2

We bound A1 by H¨ older’s inequality and Lemma 3.1 to obtain t

Z

k∂t akL∞ kakL∞ k∇vt k2L2 ds ≤ C

A1 ≤ C

Z

0

t

k∇vkL∞ k∇vt k2L2 ds.

0

To treat A2 , we first integrate by parts in time so that Z Z t j k i i A2 = ∂t (al al )∂k v ∂j vtt dx ds 0 Ωf Z Z = ∂t (ajl (t)akl (t))∂k v i (t)∂j vti (t) dx − Ωf

Ωf

Z tZ − 0

Ωf

∂tt (ajl akl )∂k v i ∂j vti

∂t (ajl (0)akl (0))∂k v i (0)∂j vti (0) dx Z tZ

dx ds − 0

Ωf

∂t (ajl akl )∂k vti ∂j vti

dx ds .

1376


We now estimate the terms in A2 using Hölder’s inequality and the bounds in Lemma 3.1 to get A2 ≤ Ck∂t a(t)kL3 ka(t)kL∞ k∇v(t)kL6 k∇vt (t)kL2 + Ck∂t a(0)kL∞ ka(0)kL∞ k∇v(0)kL2 k∇vt (0)kL2 Z t k∂tt akL2 kakL∞ k∇vkL∞ k∇vt kL2 ds +C 0 Z t k∂t ak2L∞ k∇vkL2 k∇vt kL2 ds +C 0 Z t +C k∂t akL∞ kakL∞ k∇vt k2L2 ds 0

≤ 0 k∇vt (t)k2L2 + C0 k∇vkL2 k∇vk3H 1 + Ck∇v0 kL∞ k∇v0 kL2 k∇vt (0)kL2 Z t +C (k∇vkL∞ k∇vkL2 + k∇vt kL2 )k∇vkL∞ k∇vt kL2 ds 0

with 0 ∈ (0, 1/2] to be determined below. Now, for the second term on the far right, we use 1/2

7/2

1/2

k∇vkL2 k∇vk3H 1 ≤ CkvkL2 kvkH 2 ≤ CkvkL2 (kvt kL2 + kD0 vkH 1 )

7/2

where we used (3.35). Therefore, 7/2

1/2

A2 ≤ 0 k∇vt (t)k2L2 + C0 kvkL2 (kvt kL2 + kD0 vkH 1 )

+ Ck∇v0 kL∞ k∇v0 kL2 k∇vt (0)kL2 Z t +C (k∇vkL∞ k∇vkL2 + k∇vt kL2 )k∇vkL∞ k∇vt kL2 ds. 0

In order to estimate the pressure term A3 , we use i aki ∂k vtt + 2∂t aki ∂k vti + ∂tt aki ∂k v i = 0

which follows from (2.6) by time differentiation. We also integrate by parts in time obtaining Z Z Z tZ t k i k i A3 = ∂t qai ∂k vtt dx ds + q∂t ai ∂k vtt dx ds 0 Ωf 0 Ωf Z tZ Z tZ = − ∂t q(2∂t aki ∂k vti + ∂tt aki ∂k v i ) dx ds − ∂t q∂t aki ∂k vti dx ds 0

Ωf

0

Z tZ − 0

Z − Ωf

Z ≤C 0

q∂tt aki ∂k vti dx ds +

Ωf

Z

Ωf

q(t)∂t aki (t)∂k vti (t) dx

Ωf

q(0)∂t aki (0)∂k vti (0) dx

t

k∂t qkL2 (k∂t akL∞ k∇vt kL2 + k∂tt akL2 k∇vkL∞ ) ds Z t +C k∇vt kL2 k∂tt akL2 kqkL∞ ds 0

+ kq0 kL2 k∇v0 kL∞ k∇vt (0)kL2 + kq(t)kL6 k∇vkL3 k∇vt (t)kL2


1377

whence Z

t

k∂t qkL2

A3 ≤ C

k∇vkL∞ k∇vt kL2 + k∇vkL2 k∇vk2L∞

0

+ kqkL∞ k∇vt kL2 (k∇vt kL2 + k∇vkL∞ k∇vkL2 ) ds + kq(t)kL6 k∇vkL3 k∇vt (t)kL2 + kq0 kL2 k∇v0 kL∞ k∇vt (0)kL2 . We then have Z t k∂t qkL2 k∇vkL∞ + kqkL∞ k∇vt kL2 k∇vkL2 k∇vkL∞ + k∇vt kL2 ds A3 ≤ C 0 1/2

+ C(kvt kL2 + kD0 vkH 1 )3/2 kvkH 1 k∇vt (t)kL2 + kq0 kL2 k∇v0 kL∞ k∇vt (0)kL2 where we used (3.35). The second term on the right is dominated by 0 k∇vt k2L2 + C0 (kvt kL2 + kD0 vkH 1 )3 kvkH 1 7/2

1/2

7/2

1/2

≤ 0 k∇vt k2L2 + C0 kvt kL2 kvkL2 + C0 kD0 vkH 1 kvkL2 where we used the inequality 1/2

1/2

1/2

kvkH 1 ≤ CkvkL2 kvkH 2 ≤ CkvkL2 (kvt kL2 + kD0 vkH 1 )1/2 resulting from (3.35) in the last step. Therefore, Z t A3 ≤ C k∂t qkL2 k∇vkL∞ + kqkL∞ k∇vt kL2 k∇vt kL2 + k∇vkL2 k∇vkL∞ ds 0 7/2

1/2

7/2

1/2

+ 0 k∇vt k2L2 + C0 kvt kL2 kvkL2 + C0 kD0 vkH 1 kvkL2 + kq0 kL2 k∇v0 kL∞ k∇vt (0)kL2 .

We finally treat the boundary term A4 following the technique developed in [18]. We integrate by parts in time in order to get Z t Z i j i ∂j wt N vtt dσ(x) ds A4 = 0

Γ

Z t Zc = − 0

i ∂j wtt N j vti dσ(x) ds +

Γc

Z

∂j wti (t)N j vti (t) dσ(x) ∂j w1i N j vti (0) dσ(x) . Γc

Z − Γc

(4.86)

With δ ∈ (0, 1/8], we obtain

∂wtt

∂wt (t)

A4 ≤ kvt kH 1/4+δ (Γc ×[0,t]) + kvt (t)kL2 (Γc ) ∂ν H −1/4−δ0 (Γc ×[0,t]) ∂ν L2 (Γc )

∂w1

+ kvt (0)kL2 (Γc ) . ∂ν L2 (Γc ) Applying the standard trace theory estimates to the initial conditions as well as the inequality 1/2

1/2

1/2

1/2

kukL2 (Γc ) ≤ CkukL2 kukH 1 ≤ CkukL2 k∇ukL2 ,

1378


we obtain

∂wtt 2

+ C kvt k2H 1/2 (Γc ×[0,t]) A4 ≤ ∂ν H −1/4−δ (Γc ×[0,t]) 1/2

1/2

+ kwt (t)kH 3/2+δ kvt (t)kL2 k∇vt (t)kL2 + C(kw1 k2H 2 + k∇vt (0)k2L2 ). At this point, we appeal to the hidden regularity theorem [23] which gives

∂wtt 2

+ kwttt (t)k2H −1/4−δ0 + kwtt (t)k2H 3/4−δ0

∂ν −1/4−δ H (Γc ×[0,t]) ≤ C(kvt k2H 3/4−δ0 (Γc ×[0,t]) + kwtt (·, 0)k2H 3/4−δ0 + kwttt (·, 0)k2H −1/4−δ0 ) (4.87) for t ∈ [0, T ), which follows from the hidden regularity theorem [23] applied to wtt . Consequently, A4 ≤ Ckvt k2H 3/4−δ0 (Γc ×[0,t]) + C kvt k2H 1/2 (Γc ×[0,t]) + 0 k∇vt (t)k2L2 + 1 kwt (t)k2H 3/2+δ0 + C0 ,1 kvt (t)k2L2 − kwttt (t)k2H −1/4−δ0 − kwtt (t)k2H 3/4−δ0 + CE(0)

(4.88)

E(0) = kw0 k2H 3 + kw1 k2H 2 + k∇v0 k2H 2 .

(4.89)

where

To treat the first two terms on the right of (4.88), we appeal to the trace interpolation inequalities kvt k2H 3/4−δ0 (Γc ×[0,T ]) ≤ C(kvtt k2L2 ([0,T ],L2 ) + kvt k2L2 ([0,T ];H 2 ) ),

(4.90)

and kvt k2H 1/2 (Γc ×[0,T ]) ≤ 2 kvtt k2L2 ([0,T ];L2 ) +

C kvt k2L2 ([0,T ];H 1 ) , 2

(4.91)

for 2 ∈ (0, 1/2] (c.f. [18] for the proof). Also, to treat the fourth term 1 kwt (t)kH 3/2+δ0 on the right side of (4.88), we use (3.35) and (3.56) which give kwt (t)k2H 3/2+δ0 ≤ Ckwttt k2H −1/2+δ0 + CkvkH 3/2+δ0 ≤ Ckwttt k2H −1/4−δ0 + Ckvt k2L2 + CkD0 vk2H 1 . We thus get A4 ≤ Ckvtt k2L2 ([0,t];L2 ) + Ckvt k2L2 ([0,t];H 2 ) C k∇vt k2L2 ([0,t];L2 ) + 0 k∇vt k2L2 2 + C1 kvt k2L2 + C1 kD0 vk2H 1 + C0 ,1 kvt (t)k2L2

+ C 2 kvtt k2L2 ([0,t];L2 ) + + C1 kwttt k2H −1/4−δ0

− kwttt (t)k2H −1/4−δ0 − kwtt (t)k2H 3/4−δ0 + Cf1 (E(0))

(4.92)

where fj (E(0)) denote certain explicitly computable function of E(0). Combining the inequalities for A1 through A4 , we get with 0 > 0 fixed sufficiently small so


1379

that the terms 0 k∇vt k2L2 can be absorbed Z

t

1 k∇vt (t)k2L2 + kwttt (t)k2H −1/4−δ0 + kwtt (t)k2H 3/4−δ0 C

kvtt k2L2 ds +

0

1/2

≤ CkvkL2 (kvt kL2 + kD0 vkH 1 )7/2 Z t k∇vkL∞ k∇vkL2 + k∇vt kL2 k∇vkL∞ k∇vt kL2 ds +C 0 Z t k∂t qkL2 k∇vkL∞ + kqkL∞ k∇vt kL2 +C 0 × k∇vt kL2 + k∇vkL2 k∇vkL∞ ds 7/2

1/2

7/2

1/2

+ Ckvt kL2 kvkL2 + CkD0 vkH 1 kvkL2 + Ckvtt k2L2 (0,t;L2 ) C k∇vt k2L2 (0,t;L2 ) 2 + C1 kD0 vk2H 1

+ Ckvt k2L2 (0,t;H 2 ) + C 2 kvtt k2L2 (0,t;L2 ) + + C1 kwttt k2H −1/4−δ0 + C1 kvt k2L2 + C1 kvt (t)k2L2 + f2 (E(0)).

Now, we choose 1 > 0 so small that the tenth term C1 kwttt k2H −1/4−δ0 can be absorbed by the half of the third term on the left. Also, we choose 2 > 0 so that C 2 ≤ , and the inequality (4.84) then follows Lemma 4.5. We have kD0 v(t)k2H 1 +

t

Z

kD0 vt k2L2 ds

0

Z ≤C 0

t

1/2

1/2

k∇vkH 1 k∇vkH 2 k∇D0 vk2L2 ds Z t 1/2 1/2 +C kηkH 3 kvkH 1 kvkH 2 k∇∂m vt kL2 ds 0 Z t 1/2 1/2 +C kqkH 1 (kηkH 2 kηkH 3 + 1)k∇D0 vt kL2 ds 0 Z t 1/2 1/2 3/4 3/4 +C (k∇vkL∞ + kqkL∞ )(kηkH 2 kηkH 3 + kηkH 2 kηkH 3 ) 0

1/2

1/2

× kD0 vt kL2 kD0 vt kH 1 ds Z +C 0

t

1/2

1/2

(kvkH 5/2 + kqkH 3/2 )kD0 vt kL2 kD0 vt kH 1 ds

(4.93)

for all t ∈ [0, T ). Proof of Lemma 4.5. Fix m ∈ {1, 2} (with no summation convention on this index). We differentiate (2.5) in the direction xm and obtain ∂m vti − ∂m ∂j (ajl akl ∂k v i ) + ∂m ∂k (aki q) = 0.

1380


Multiplying by ∂m vti , integrating over Ωf and summing over i = 1, 2, 3, we get after integrating by parts Z Z k∂m vti k2L2 + ∂m (ajl akl )∂k v i ∂j ∂m vti dx + akl ∂k ∂m v i ajl ∂j ∂m vti dx Ωf

Z + Γc

Z

Ωf

∂m (ajl akl ∂k v i )N j ∂m vti dσ(x) −

Z

∂m (qaki )∂k ∂m vti dx

Ωf

∂m (qaki )N k ∂m vti dσ(x) = 0.

− Γc

Rewriting the expression, we have Z Z 1 1 ∂t (ajl akl ∂k ∂m v i ∂j ∂m v i ) dx − ∂t (ajl akl )∂k ∂m v i ∂j ∂m v i dx k∂m vt k2L2 + 2 Ωf 2 Ωf Z + ∂m (ajl akl )∂k v i ∂j ∂m vti dx Ωf

Z

∂m (qaki )∂k ∂m vti dx

= Ωf

Z + Γc

∂m (ajl akl ∂k v i )N j ∂m vti

Z dσ(x) −

∂m (qaki )N k ∂m vti dσ(x)

Γc

whence Z 1 d aj ak ∂k ∂m v i ∂j ∂m v i dx 2 dt Ωf l l Z Z 1 j k j k i i i i ∂m (al al )∂k v ∂j ∂m vt dx ∂t (al al )∂k ∂m v ∂j ∂m v dx + ≤ Ωf 2 Ωf Z Z j k i j i k i ∂m (al al ∂k v )N ∂m vt dσ(x) + ∂m (qai )∂k ∂m vt dx + Ωf Γc Z ∂m (qaki )N k ∂m vti dσ(x) +

k∂m vt k2L2 +

Γc

= A1 + A2 + A3 + A4 + A5 .

(4.94)

We next treat the terms on the right starting with A1 . Bounding A1 by Hölder’s inequality, we have A1 ≤ Ck∂t akL∞ kakL∞ k∇∂m vk2L2 ≤ Ck∇vkL∞ k∇∂m vk2L2 1/2

1/2

≤ Ck∇vkH 1 k∇vkH 2 k∇D0 vk2L2

(4.95)

where we used Agmon’s inequality in the last step. For A2 , we have 1/2

1/2

A2 ≤ Ck∂m akL6 kakL∞ k∇vkL3 k∇∂m vt kL2 ≤ CkηkH 3 kvkH 1 kvkH 2 k∇∂m vt kL2 (4.96) where we used Lemma 3.1 (v). Proceeding to estimate A3 , we have by Hölder’s inequality A3 ≤ k∂m qkL2 kakL∞ k∇∂m vt kL2 + CkqkL6 k∂m akL3 k∇∂m vt kL2 1/2

1/2

≤ CkqkH 1 (kηkH 2 kηkH 3 + 1)k∇D0 vt kL2 .

(4.97)


1381

We finally treat the boundary terms starting with A4 . Namely, A4 ≤ Ck∂m (ajl akl ∂k v i )N j kL2 (Γ) k∂m vt kL2 (Γc ) ≤ C k∂m akL2 (Γc ) kakL∞ (Γc ) k∇vkL∞ (Γc ) + kak2L∞ (Γc ) k∇∂m vkL2 (Γc ) k∂m vt kL2 (Γc ) 1/2 1/2 3/4 3/4 ≤ C kηkH 2 kηkH 3 k∇vkL∞ + kηkH 2 kηkH 3 k∇vkL∞ + kvkH 5/2 1/2

1/2

× k∂m vt kL2 k∂m vt kH 1 where we used 1/2

1/2

1/2

1/2

3/2

k∂m akL2 (Γc ) ≤ Ck∂m akL2 k∂m akH 1 ≤ CkηkH 2 (kηkH 3 + kηkH 2 kηkH 3 )1/2 in the last step. Similarly, we estimate A5 and get A5 ≤ C(k∂m qkL2 (Γc ) kakL∞ (Γc ) k∂m vt kL2 (Γc ) + kqkL∞ (Γc ) k∂m akL2 (Γc ) k∂m vt kL2 (Γc ) ) 1/2 1/2 3/4 3/4 1/2 1/2 ≤ C kqkH 3/2 + kηkH 2 kηkH 3 kqkL∞ + kηkH 2 kηkH 3 kqkL∞ k∂m vt kL2 k∂m vt kH 1 . Integrating the resulting inequality, we obtain the lemma. The next lemma is a simple corollary of Lemmas 4.1 and 4.5 Lemma 4.6. The time derivatives vt , wt , and wtt satisfy the estimate kvt (t)k2L2 + kv(t)k2H 2 + kwtt (t)k2L2 + k∇wt (t)k2L2 Z t Z t + k∇vt k2L2 ds + kD0 vt k2L2 ds 0 0 Z t 1/2 1/2 ≤C kvkH 1 kvkH 2 kvkH 2 + kqkH 1 k∇vt kL2 ds 0 Z t +C k∂t qkL2 k∇vt kL2 ds 0 Z t 1/2 1/2 +C k∇vkH 1 k∇vkH 2 k∇D0 vk2L2 ds 0 Z t 1/2 1/2 +C kηkH 3 kvkH 1 kvkH 2 k∇∂m vt kL2 ds 0 Z t 1/2 1/2 +C kqkH 1 (kηkH 2 kηkH 3 + 1)k∇D0 vt kL2 ds 0 Z t +C (k∇vkL∞ + kqkL∞ ) 0 1/2

1/2

3/4

3/4

1/2

1/2

× (kηkH 2 kηkH 3 + kηkH 2 kηkH 3 )kD0 vt kL2 kD0 vt kH 1 ds Z t 1/2 1/2 +C (kvkH 5/2 + kqkH 3/2 )kD0 vt kL2 kD0 vt kH 1 ds + CE(0) 0

(4.98) for all t ∈ [0, T ) where E(0) = kv0 k2H 3 + kw0 k2H 3 + kw1 k2H 2 . Proof of Lemma 4.5. The inequality follows by adding (4.63) and (4.84) and then using (3.35).

1382


Proof of Theorem 2.1. In order to prove the necessary a priori estimate giving the local existence, we need to combine Lemmas 3.4, 4.1, 4.2, 4.3, 4.4, and 4.6. Letting X(t) = k∇vt (t)k2L2 + kv(t)k2H 11/4−δ0 + kq(t)k2H 7/4−δ0 + kwttt (t)k2H −1/4−δ0 + kwtt (t)k2H 3/4−δ0 , and Z Y (t) = 0

t

kvk2H 3 ds +

Z

t

kqk2H 2 ds +

Z

0

t

kvt k2H 2 ds +

0

Z

t

kqt k2H 1 ds

0

we combine estimates (4.78), (4.82), (4.98), and (4.84) with the Stokes estimates (3.36), (3.39), and (3.57) and get after a careful inspection of all the terms Z t X(t) + Y (t) ≤ C(X(t) + Y (t)) + P (X(t), kηkH 3 ) ds 0

where P denotes a polynomial function in the arguments indicated; the most critical term to control is the first term in (4.84), and in order to bound it, we raise the inequality (4.98) to 7/4. Rt Now, note that η = 0 v ds + η(0), which implies Z t kηkH 3 ≤ kvkH 3 ds + kη(0)kH 3 0

and therefore kη(t)k2H 3 ≤ Ct

Z

t

kvk2H 3 ds + C.

0

For sufficiently small t, the first term on the right is dominated by Y . Therefore, we get an a priori bound Z t ˜ ˜ ds X(t) + Y (t) ≤ C(X(t) + Y (t)) + P (X) (4.99) 0

˜ = X + kηkH 3 . Fix a sufficiently small constant > 0 so that the first where X term on the right of (4.99) is absorbed by the first two terms on the left. Then we multiply the resulting inequality with a constant in order to get Z t ˜ + Y (t) ≤ C ˜ ds. X(t) P (X) 0

This estimate is adequate for the application of the Gronwall lemma, which implies the a priori bound. The regularity of w and wt follows from (3.55) and (3.56), while Rt the full regularity of vt and q follows from vt (t) = 0 vtt (s) ds + vt (0) and (3.57). 4.2. Justification of the a priori estimates. We first show the existence of solutions to the system (2.5)–(2.7) when the matrix a is given and invertible, with C ∞ (Ωf ×[0, T ]) entries ajl for j, l = 1, 2, 3. Then we apply the retarded mollification (as in [8]) and compactness to obtain solutions to our problem. We first note that when the coefficients of the matrix a are time independent, all the regularity results for the linear problem with constant coefficients apply globally since the form ∂j (ajl akl ∂k ·) is uniformly elliptic on Ωf × [0, T ] [3, 4, 1, 2, 18, 19, 20]. We next state and prove a theorem on existence when the coefficients are given and time dependent.


1383

Theorem 4.7. Assume that (v0 , w0 , w1 ) ∈ V ∩ H 3 (Ωf ) × H 3 (Ωe ) × H 2 (Ωe ) satisfy appropriate compatibility conditions, and let a(x, t) be a given invertible matrix with ¯ f × [0, T ]) for j, l = 1, 2, 3. Then for every δ0 > 0 there exists a entries ajl in C ∞ (Ω solution v ∈ L2 ([0, T ]; H 3 (Ωf )) ∩ C([0, T ]; H 11/4−δ0 (Ωf )) w ∈ C([0, T ]; H 11/4−δ0 (Ωe )) wt ∈ C([0, T ]; H 7/4−δ0 (Ωe )) q ∈ L2 ([0, T ]; H 2 (Ωf )) ∩ C([0, T ]; H 7/4−δ0 (Ωf )) to the system (2.5)–(2.7) with boundary conditions (2.9)–(2.11) and with initial conditions (2.12)–(2.14). Note that the conditions on a are sufficient to conclude that the matrix a : aT is positive definite for all (x, t) ∈ Ωf × [0, T ], and hence the form ∂j (ajl akl ∂k v i ) is uniformly elliptic. Before we prove the above theorem, we establish the following existence theorem pertaining to the non-homogeneous linear Stokes-wave problem with time independent coefficients and non-zero divergence condition. Theorem 4.8. Consider the coupled Stokes-elasticity system vti − ∂j (bjl bkl ∂k v i ) + ∂k (bki q) = f i bki ∂k v i i wtt − ∆wi

in Ωf × (0, T )

(4.100)

=g

in Ωf × (0, T )

(4.101)

=0

in Ωe × (0, T )

for i = 1, 2, 3 with b(x) a given matrix such that det b 6= 0 and with C independent entries bjl satisfying bjl bkl ξji ξki ≥

1 2 |ξ| , Ce

ξ ∈ Rn

(4.102) ∞

time-

2

where 1/Ce > 0 is the ellipticity constant and kbjl bkl kH 3 (Ωf ) ≤ M. The corresponding boundary conditions read v = wt v=0 ∂j wi N j =

bjl bkl ∂k v i N j 3

− bki qN k + hi 3

on Γc × (0, T )

(4.103)

on Γf × (0, T )

(4.104)

on Γc × (0, T ).

(4.105)

2

If (v0 , w0 , w1 ) ∈ V ∩ H (Ωf ) × H (Ωe ) × H (Ωe ) satisfies appropriate compatibility conditions, and f ∈ L2 ([0, T ]; H 1 (Ωf )), ft ∈ L2 ([0, T ]; L2 (Ωf )), g ∈ L2 ([0, T ]; H 2 (Ωf )), gtt ∈ L2 ([0, T ]; H −1 (Ωf )), h ∈ L2 ([0, T ]; H 3/2 (Γc )), and ht ∈ L2 ([0, T ]; H 1/2 (Γc )) for some time T > 0, then there exists a unique solution v ∈ L2 ([0, T ]; H 3 (Ωf )) ∩ C([0, T ]; H 11/4−δ0 (Ωf )) w ∈ C([0, T ]; H 11/4−δ0 (Ωe )) wt ∈ C([0, T ]; H 7/4−δ0 (Ωe )) q ∈ L2 ([0, T ]; H 2 (Ωf )) ∩ C([0, T ]; H 7/4−δ0 (Ωf ))

1384


Proof of Theorem 4.8. In order to prove this theorem, we shall appeal to the existence results pertaining to the homogeneous linear problem [3, 2, 1, 12]. We first change variables, so that we have the zero divergence condition. We let u = v − z where z and q¯ satisfy the Stokes problem ∂j (bjl bkl ∂k z i ) + ∂k (bki q¯) = 0

in Ωf × (0, T )

(4.106)

=g

in Ωf × (0, T )

(4.107)

z=0

on Γc × (0, T )

(4.108)

z=0

on Γf × (0, T )

(4.109)

bki ∂k z i

(c.f. Lemma 3.2 for the relevant estimates). Then u and w satisfy the system uit − ∂j (bjl bkl ∂k ui ) + ∂k (bki q) = −zt − ∂k (bki q¯) + f

in Ωf × (0, T )

(4.110)

bki ∂k ui

=0

in Ωf × (0, T )

(4.111)

wtt − ∆w = 0

in Ωe × (0, T ),

(4.112)

on Γc × (0, T )

(4.113)

on Γf × (0, T )

(4.114)

in Γc × (0, T ).

(4.115)

subject to the boundary conditions u = wt u=0 bjl bkl ∂k ui N j

−

bki qN k

i

j

= ∂j w N −

bjl bkl ∂k z i N j

i

+h

By [3, 12], this system has a weak solution given the initial conditions u0 = v0 − z ∈ Hb = {u ∈ L2 (Ωf ) : bki ∂k ui = 0} and (w0 , w1 ) ∈ H 1 (Ωe ) × L2 (Ωe ), and the non-homogeneous terms satisfy f + zt + ∇¯ q ∈ L2 ([0, T ]; V 0 ) and −∇z · 2 −1/2 N + h ∈ L ([0, T ]; H (Γc )) for which it suffices that g ∈ L2 ([0, T ]; H 1 (Ωf )) and 2 0 gt ∈ L ([0, T ]; V ) while f ∈ L2 ([0, T ]; V 0 ) and h ∈ L2 ([0, T ]; H −1/2 (Γc )). Note that although the cited results pertain to the case b = I, the results are applicable to the general space dependent coefficients. We now proceed to show that given more regular initial data (u0 , w0 , w1 ) ∈ H 3 (Ωf ) ∩ V × H 3 (Ωe ) × H 2 (Ωe ) and regular functions f , g, and h, the solution becomes smooth with the desired regularity. In particular, by mollifying the solution, we can perform the same a priori estimates above (for the linear constant coefficients case) and then pass through the limit to obtain the desired regularity and in particular the estimate kvk2L2 ([0,T ];H 3 (Ωf )) + kvt k2H 1 ([0,T ];H 1 (Ωf )) + kqk2L2 ([0,T ];H 2 (Ωf )) + kqk2H 1 ([0,T ];H 1 (Ωf )) + kvtt k2L2 ([0,T ];L2 (Ωf )) + kwk2L∞ ([0,T ];H 11/4−δ0 (Ωe )) + kwt k2L∞ ([0,T ];H 7/4− (Ωe )) (4.116) ≤ Ckf k2L2 ([0,T ];H 1 (Ωf )) + Ckf k2H 1 ([0,T ];L2 (Ωf )) + Ckgk2L2 ([0,T ];H 2 (Ωf )) + Ckgk2H 2 ([0,T ];H −1 (Ωf )) + Ckhk2L2 ([0,T ];H 3/2 (Γc )) + Ckhk2H 1 ([0,T ];H 1/2 (Γc )) + Ckv0 k2H 3 (Ωf ) + Ckw0 k2H 3 (Ωe ) + Ckw1 k2H 2 (Ωe )

(4.117)


1385

with constants only depending on Ce and the uniform bound M on the coefficients. This proves Theorem 4.8. We now use this result to prove Theorem 4.7. Proof of Theorem 4.7. Let b(x) = I f i = −∂j ((δkj − ajl akl )∂k v˜i ) + ∂k ((δki − aki )˜ q) g = (δkj − akj )∂k v˜j hi = −(δkj − ajl akl )∂k v˜i N j + (δki − aki )˜ qN k with v˜ ∈ L2 ([0, T ]; H 3 (Ωf )) ∩ H 2 ([0, T ]; L2 (Ωf )) ∩ H 1 ([0, T ]; H 2 (Ωf )) and q˜ ∈ L2 ([0, T ]; H 2 (Ωf )) ∩ H 1 ([0, T ]; H 1 (Ωf )). We then have that the solution map for given initial data Λ : (˜ v , q˜) → (v, q) is continuous on XT = L2 ([0, T ]; H 3 (Ωf )) ∩ 2 1 H ([0, T ]; H (Ωf )) × L2 ([0, T ]; H 2 (Ωf )) ∩ H 1 ([0, T ]; H 1 (Ωf )). We now show the existence of a fixed point (v, q) by showing that the map Λ is a contraction for some time T > 0 depending on the coefficient matrix a. In particular, we have the inequality kajl (t)akl (t) − ajl (s)akl (s)kH 3 (Ωf ) ≤ C|t − s|

(4.118)

for all t, s ∈ [0, T ] by uniform continuity and smoothness of the coefficients. This allows us to show that the map Λ is a contraction on the space XT for a sufficiently small T1 > 0. The argument can then be repeatedly applied to the system when b(x) = a(x, mT1 ) using the fact that the constants in estimate (4.116) only depend on the uniform ellipticity constant for a and its global bound M . Using this theorem, we now obtain existence for the main problem by using the retarded mollification of the coefficients ajl . We introduce Z y τ , ψδ (ajl )(x, t) = δ −4 ψ a ¯j (x − y, t − τ ) dy dτ δ δ l R4 where ψ is a smooth compactly supported function with support inside {|x|2 < R t} × [1, 2] and R4 ψ dx dt = 1. The function a ¯ is the continuous extension of a by the identity for t < 0 and by any continuous extension into Ωe and the rest of R3 . Note that the value of ψδ (ajl ) at time t depends on the values of a before t − δ. For a fixed δ > 0, we solve the system on the interval [0, δ] with coefficients ajl = δjl which just produces the linear Stokes-elasticity problem with the divergence (δ) free condition. We then use the solution to R t (δ)v guaranteed by the above theorem (δ) compute the corresponding η = 0 v ds + x and the coefficient matrix a(δ) = (∇η (δ) )−1 on the interval which is possible since det(aδ ) = det(a(δ) (·, 0))e−

Rt 0

Tr(aδ :∇v)

ds > 0

by Liouville’s formula. This determines the coefficient matrix ψδ (a(δ) ) on the next interval [δ, 2δ]. Note that the mollified form ∂j (ajl akl ∂k ) corresponding to the entries of ψδ (a(δ) ) is still uniformly elliptic with the same ellipticity constant as aδ . We then apply the above theorem which guarantees the existence of solution and allows to extend v (δ) to the next interval [δ, 2δ], using the predetermined mollified coefficients of the matrix ψδ (a(δ) ).

1386


By the above theorem, we obtain a sequence of solutions v (δ) ∈ C([0, T ]; H 2 (Ωf ))∩ (δ) L2 ([0, T ]; H 3 (Ωf )), q (δ) ∈ L2 ([0, T ]; H 2 (Ωf )), and ηf ∈ C([0, T ]; H 3 (Ωf )), while (δ)

w(δ) ∈ C([0, T ]; H 11/4−δ0 (Ωe )) and wt ∈ C([0, T ]; H 7/4−δ0 (Ωe )) for time T inde(δ) (δ) pendent of δ. Moreover, vt ∈ L2 ([0, T ]; H 2 (Ωf )) and vtt ∈ L2 ([0, T ]; H 1 (Ωf )) (δ) while wtt ∈ L∞ ([0, T ]; H 1 (Ωe )). We then apply the above a priori time and tangential estimates to this sequence of solutions thus obtaining uniform estimates on the sequence in terms of norm of the initial data in H 3 (Ωf ) × H 3 (Ωe ) × H 2 (Ωe ) for a time T 0 also depending on the initial data. In addition, the Stokes estimate in (δ) Lemma 3.2(ii) provides uniform estimates on qt in L2 ([0, T 0 ]; H 1 (Ωf )). Observe (δ) that the mollified coefficient matrix ψδ (a ) satisfies the estimates of Lemma 3.1. Therefore, by passing to an appropriate subsequence the a priori estimates imply

v (δ) → v weakly in L2 ([0, T 0 ]; H 3 (Ωf )) w

∞

0

→ w weakly in L ([0, T ]; H

(Ωe ))

(4.120)

(δ)

→ wt weakly∗ in L∞ ([0, T 0 ]; H 7/4−δ0 (Ωe ))

(4.121)

wt

(δ) ηf

∗

∞

0

11/4−δ0

(4.119)

(δ)

3

→ ηf weakly∗ in L ([0, T ]; H (Ωf ))

(δ)

(4.122)

wtt → wtt weakly∗ in L∞ ([0, T 0 ]; H 3/4−δ0 (Ωe ))

(4.123)

(δ) vt

(4.124)

(δ) vtt

→ vt weakly in L2 ([0, T 0 ]; H 2 (Ωf )) 0

2

2

→ vtt weakly in L ([0, T ]; L (Ωf ))

(4.125)

q (δ) → q weakly in L2 ([0, T 0 ]; H 2 (Ωf ))

(4.126)

(δ) qt

(4.127)

2

0

1

→ qt weakly in L ([0, T ]; H (Ωf )).

Using Aubin’s Theorem, we conclude that v (δ) → v strongly in L2 ([0, T 0 ]; H s (Ωf )) (δ) for s < 3 while vt → vt strongly in L2 ([0, T 0 ]; H s (Ωf )) for s < 2. Moreover, since ηt = v, we also conclude that η (δ) → η strongly in C([0, T 0 ]; H s (Ωf )) for s < 3 and det ∇η (δ) → det ∇η strongly in L∞ ([0, T 0 ]; L∞ (Ωf )). We then note that since a(δ) = (∇η (δ) )−1 = (det ∇η (δ) )−1 cof (∇η (δ) ), we deduce the weak∗ convergence of a(δ) to a in L∞ ([0, T 0 ]; H 2 (Ωf )), and accordingly of ψδ (a(δ) ) to a. Moreover, ψδ (a(δ) ) → a strongly in L2 ([0, T 0 ]; H s (Ωf )) for s < 2. As for the pressure, q (δ) converges to q strongly in L2 ([0, T 0 ]; H s ) for s < 2. Based on these convergence results, we may pass through the limit in the system (2.4)–(2.6) satisfied by the sequence to show that the limit still satisfies the system. The convergence is straightforward in linear terms. As for the viscosity term, we have

j,(δ)

k∂j (ψδ (al

k,(δ)

)ψδ (al

)∂k v i,(δ) ) − ∂j (ajl akl ∂k v i )kL2 (Ωf ) j,(δ)

≤ k∂j ((ψδ (al

k,(δ)

) − ajl )ψδ (al k,(δ)

+ k∂j (ajl (ψδ (al

)∂k v i,(δ) )kL2 (Ωf )

) − akl )∂k v i,(δ) )kL2 (Ωf )

+ k∂j (ajl akl ∂k (v i,(δ) − v i ))kL2 (Ωf ) .


1387

Applying H¨ older’s inequality, we get j,(δ)

k∂j (ψδ (al

k,(δ)

)ψδ (al

)∂k v i,(δ) ) − ∂j (ajl akl ∂k v i )kL2 (Ωf )

≤ Ck∂j (ψδ (a(δ) ) − a)kL3 (Ωf ) kψδ (a(δ) )kL∞ (Ωf ) k∇v (δ) kL6 (Ωf ) + Ckψδ (a(δ) ) − akL∞ (Ωf ) k∇ψδ (a(δ) )kL3 (Ωf ) k∇v (δ) kL6 (Ωf ) 2 + Ck(ψδ (a(δ) ) − a)kL∞ (Ωf ) kψδ (a(δ) )kL∞ (Ωf ) k∂j,k v (δ) kL2 (Ωf )

+ Ck∇(ψδ (a(δ) ) − a)kL3 (Ωf ) kakL∞ (Ωf ) k∇v (δ) kL6 (Ωf ) + Ck(ψδ (a(δ) ) − a)kL∞ (Ωf ) k∇akL3 (Ωf ) k∇v (δ) kL6 (Ωf ) 2 + Ck(ψδ (a(δ) ) − a)kL∞ (Ωf ) kakL∞ (Ωf ) k∂j,k v (δ) kL2 (Ωf )

+ Ck∇akL3 (Ωf ) kakL∞ (Ωf ) k∂k (v i,(δ) − v i ))kL6 (Ωf ) . Applying the Sobolev inequality and appealing to the fact that v (δ) is bounded uniformly in L∞ ([0, T 0 ]; H 2 (Ωf )) and ψδ (a(δ) ) is bounded uniformly in L∞ ([0, T 0 ]; H 2 (Ωf )) while v (δ) converges strongly to v in L2 ([0, T 0 ]; H 3−δ0 (Ωf )) and ψδ (a(δ) ) converges strongly to a = (∇ηf )−1 in C([0, T 0 ]; H 2−δ0 (Ωf )), we obtain the dej,(δ) k,(δ) sired result. Namely, ∂j (ψδ (al )ψδ (al )∂k v i,(δ) ) converges to ∂j (ajl akl ∂k v i ) in 2 0 2 L ([0, T ]; L (Ωf )) strongly. Similarly, for the pressure term we have k,(δ)

k∂k (ψδ (ai

)q (δ) − aki q)kL2 (Ωf ) k,(δ)

≤ k∂k (ψδ (ai

) − aki )q (δ) kL2 (Ωf ) + k∂k (aki (q (δ) − q))kL2 (Ωf )

≤ Ck∇(ψδ (a(δ) ) − a)kL3 (Ωf ) kq (δ) kL6 (Ωf ) + Ck(ψδ (a(δ) ) − a)kL∞ (Ωf ) k∇q (δ) kL2 (Ωf ) + Ck∇akL3 (Ωf ) k(q (δ) − q)kL6 (Ωf ) + CkakL∞ (Ωf ) k∇(q (δ) − q)kL6 (Ωf ) . Again, since q (δ) converges to q strongly in L2 ([0, T 0 ]; H 1 (Ωf )) and since ψδ (a(δ) ) converges strongly to a in L2 ([0, T 0 ]; H 2−δ0 ) while q (δ) is bounded uniformly in L∞ ([0, T 0 ]; H 2 (Ωf )) and ψδ (a(δ) ) is bounded uniformly in L∞ ([0, T 0 ]; H 2 (Ωf )), we k,(δ) conclude from the above estimate that ∂k (ψδ (ai )q (δ) ) converges to ∂k (ψδ (aki )q) strongly in L2 ([0, T 0 ], L2 (Ωf )). Showing convergence of the boundary terms and the divergence condition follows similar arguments. Acknowledgments. I.K. was supported in part by the NSF grant DMS-1009769 and A.T. was supported in part by the Petroleum Institute Research Grant Ref. Number 11014. REFERENCES [1] G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J., 15 (2008), 403–437. [2] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, in “Fluids and Waves,” 15–54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [3] V. Barbu, Z. Gruji´ c, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in “Fluids and Waves,” 55–82, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007.

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Received September 2010; revised June 2011. E-mail address: [email protected] E-mail address: [email protected]