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On the local existence of the free-surface Euler equation with surface tension

Abstract

We address the existence of solutions for the free-surface Euler equation with surface tension in a bounded domain. Considering the problem in Lagrangian variables we provide a priori estimates leading to existence of local solutions with the initial velocity in H3.5 for which the trace on the free boundary belongs to H3.5.

1.Introduction

In this paper, we address the local existence of solutions to the 3D free-surface incompressible Euler equations

(1.1)tu+u·u+p=0in Ω(t)(1.2)·u=0in Ω(t)
where the free boundary Ω(t) evolves according to the fluid velocity field u(x,t), and the pressure obeys
(1.3)p(x,t)=σHon Ω(t).
Here σ>0 is the surface tension, while H represents twice the mean curvature of the boundary Ω(t).

Problems related to local or global existence of solutions of free surface evolution under the Euler flow, with or without surface tension, have attracted considerable attention in the last decades. For both cases different approaches have been developed; however, the search is still in progress for the lowest regularity spaces where the existence or uniqueness of solutions hold. For the history of both problems, cf. [2,17,32] and references therein.

While in the zero surface tension case the problem is known to be unstable, and thus the Rayleigh-Taylor stability condition has to be imposed, this is not necessary when the surface tension is nonzero since the surface tension provides a stabilizing effect close to the boundary.

We may divide the existing results of the rotational case, i.e., when the vorticity is nontrivial, into the Eulerian approach and the Lagrangian one. In the Eulerian approach, Schweitzer has obtained in [42] a local existence result with the initial velocity in H4.5 and with a smallness assumption on the height of the interface. The primary tools in [42] are tangential and time differentiation, up to order three. In [17], Coutand and Shkoller used the Lagrangian formulation to obtain the local existence with initial data in H4.5. The method used by Coutand and Shkoller is, as in [42], differentiation in space and time up to three times; however, the Lagrangian approach allowed to bypass the smallness assumption on the initial surface. We would like to stress that simple integrations by parts are not by themselves sufficient to close the estimates; additional care, including a careful treatment of the vorticity and the pressure equations, is necessary to close the estimates. In addition, in [42], a harmonic change of variables was used to overcome a lack of 1/2 derivative in the estimates resulting from tangential and time differentiation.

In [43] the authors employ ideas inspired by the geometrical description of Euler flows as geodesics on the infinite dimensional group of volume preserving diffeomorphisms to obtain conditional a priori energy estimates for the solutions when the initial velocity belongs to H3. They also provide estimates which are uniform in surface tension, if additionally a Raleigh-Taylor condition is satisfied. Recently, in [20,21], using a different method, the authors established the local existence when the initial velocity belongs to H3.5+ϵ for every ϵ>0.

Our goal is to revisit the Lagrangian approach to the free-surface rotational Euler equations and provide a priori estimates leading to local existence for the velocity in H3.5 such that the trace on the free boundary belongs to H3.5, lowering the regularity requirements from [17] and [42]. While the basic framework still involves time and tangential differentiation used in [17,42], we introduce two improvements which allow us to lower the required regularity. The first improvement is the use of Cauchy invariance [11,14,15,23,35,44] recently used in the zero surface tension case in [32,33]. The second improvement is a simple and direct treatment of the pressure, employing the Laplace problem with Neumann boundary conditions.

Before discussing the organization of the paper, we briefly recall the history of free surface Euler equation problems. Early works on the free surface Euler equations involve results on small analytic data [19,38,51]. The important work [7] considered the viscous case, employing a Lagrangian set-up, subsequently used in many works on the inviscid problem. In [47,48] Wu obtained existence of solutions of the free surface Euler equations in 2D and 3D cases respectively, both addressing irrotational, no surface-tension cases. Positive surface tension was considered by Ambrose and Masmoudi in [4,5], who also studied the zero surface tension limit. The works [17,42,43] then constructed local solutions for the nonzero-surface tension Euler equations; cf. also works [29,40,41,46,50] for the positive surface-tension Navier-Stokes system. For other works on the zero-surface tension case, see [2,3,810,13,18,22,27,30,31,34,3638,45,52,53], for other works on non-zero surface tension, cf. [1,39] while for global existence of solutions, see [2426,28,49].

The paper is organized as follows. In Section 2, we introduce the Lagrangian setting of the problem and state the main result, Theorem 2.1. Section 3 contains a preliminary lemma containing a priori estimates on the Lagrangian map η and the cofactor matrix a. Section 4 contains the proof of the main statement. It is subdivided into four subsections containing the vttt, vtt, vt estimates, and the div-curl estimate. In the final section, we collect all the available inequalities and apply the Gronwall lemma.

2.The main result

We consider the 3D Euler equation in the Lagrangian framework over a fixed domain Ω. Let η(·,t):ΩΩ(t) be the flow map under which the initial domain configuration Ω evolves with time, such that Ω(t)=η(Ω,t). For simplicity, we assume that the initial domain Ω is flat, i.e.,

(2.1)Ω={x=(x1,x2,x3):(x1,x2)R2,0<x3<1}
with periodic boundary conditions with period 1 in the lateral directions. We denote the top of Ω (corresponding to the free-surface) by
(2.2)Γ1=T2×{x3=1}
and the stationary bottom by
(2.3)Γ0=T2×{x3=0}.
Then the incompressible Euler equation has the form
(2.4)vti+k(aikq)=0in Ω×(0,T),i=1,2,3(2.5)aikkvi=0in Ω×(0,T),
where v(x,t)=ηt(x,t)=u(η(x,t),t) and q(x,t)=p(η(x,t),t) denote the Lagrangian velocity and the pressure of the fluid over the initial domain Ω. The dynamics of the Lagrangian matrix a(x,t)=[η(x,t)]1 and the flow map η(x,t) are described by the ODEs
(2.6)at=a:v:ain Ω×(0,T)(2.7)ηt=vin Ω×(0,T)
where the symbol : denotes the matrix multiplication, with the initial conditions
(2.8)a(x,0)=I(2.9)η(x,0)=x
in Ω. The condition (2.6) can be written in coordinates as
(2.10)taki=alkjvlaij,i,k=1,2,3.
We assume v·N=0 on Γ0×(0,T) and
(2.11)aikqNk=Δ2ηion Γ1×(0,T)
for i=1,2,3, where N=(N1,N2,N3) is the unit outward normal with respect to Ω and Δ2=12+22. Note that we have set the surface tension to be 1, for simplicity.

We now state the main result of this paper.

Theorem 2.1.

Assume that v(·,t)=v0H3.5(Ω) is divergence-free and is such that v0|Γ1H3.5(Γ1). Then there exists a local-in-time solution (v,q,a,η) to (2.4)–(2.11) which satisfies

vL([0,T];H3.5(Ω))vtL([0,T];H2.5(Ω))vttL([0,T];H1.5(Ω))vtttL([0,T];L2(Ω))
with qL([0,T];H3.5(Ω)), qtL([0,T];H2.5(Ω)), qttL([0,T];H1(Ω)), aL([0,T];H2.5(Ω)), and ηC([0,T];H3.5(Ω)).

3.Preliminary results

In this section, we give formal a priori estimates on time derivatives of the unknown functions needed in the proof of Theorem 2.1. We begin with an auxiliary result providing bounds on the flow map η and the matrix a.

Lemma 3.1.

Assume that vL([0,T];H3.5)M. Let p[1,] and i,j=1,2,3. With T[0,1/CM], where C is a sufficiently large constant, the following statements hold:

  • (i) ηH3.5C for t[0,T];

  • (ii) aH2.5C for t[0,T];

  • (iii) atLpCvLp for t[0,T];

  • (iv) iatLpCvLp1iaLp2+CivLp for i=1,2,3 and t[0,T] where 1p,p1,p2 are such that 1/p=1/p1+1/p2;

  • (v) atHrvHr, for r[0,2.5] and t[0,T];

  • (vi) attHσCvHσvL+CvtHσ for σ[0,1.5] and t[0,T] and attH1(Ω)CvH5/42+CvtH1;

  • (vii) atttLpCvLpvL2+CvtLpvL+CvttLp for t[0,T];

  • (viii) for every ϵ(0,1/2] and all tT=min{ϵ/CM2,T}, we have

    (3.1)δjkaljalkH2.52ϵ,j,k=1,2,3
    and
    (3.2)δjkakjH2.52ϵ,j,k=1,2,3.

In particular, the form aljalkξjiξki satisfies the ellipticity estimate
(3.3)aljalkξjiξki1C|ξ|2,ξRn2
for all t[0,T] and xΩ, provided ϵ1/C with C sufficiently large.

Above and in the sequel, if the domain of the norm is not specified, it is understood to be Ω.

Proof of Lemma 3.1.

The assertion (i) follows immediately from (2.7), while for (ii), we have by (2.6)

(3.4)a(t)H2.5C+0ta(s)H2.52v(s)H2.5ds
and (ii) is obtained by using the Gronwall lemma, provided T1/CM. Next, (2.6) implies
(3.5)atLpCaLvLpaLCvLp
using (ii) in the last inequality. The inequality (v) is proved analogously, using the Sobolev multiplicative inequality instead of (3.5). The estimates (iv) and (vi) are proven similarly. For (viii), we write
(3.6)δjkaljalk=0tt(aljalk)ds
where j,k{1,2,3}. Therefore,
δjkaljalkH2.50tt(aljalk)H2.5ds(3.7)C0taljH2.5talkH2.5dsC0tatH2.5dsCMt.
The estimate (3.1) then follows if CM2T2ϵ. The other assertions in (viii) are obtained analogously. □

In order to estimate the second derivative of the pressure, we need the following regularity lemma for an elliptic equation with Neumann boundary condition in a smooth (bounded) domain Ω. Assume that bij satisfies bLM and that bij(x)ξiξjM1|ξ|2 for all xΩ and ξRn, where n{2,3,}.

Lemma 3.2

Lemma 3.2([16]).

Let q be an H1 solution of the

(3.8)i(bijjq)=divπin Ω(3.9)bmkkqNm=gon Ω
where π,divπL2(Ω) and gH1/2(Ω) with the compatibility condition
(3.10)Ω(π·Ng)=0.
If
(3.11)bILϵ0
where ϵ0>0 is a sufficiently small constant depending on M, then we have
(3.12)qq¯H1CπL2+Cgπ·NH1/2(Ω)
where q¯=(1/|Ω|)qdx.

The existence of solutions of this problem under the given conditions has been established in [6]. However, we believe that the inequality (3.12), which does not contain the L2-norm of divπ, is new.

Proof of Lemma 3.2.

First, using (3.8)–(3.9), we have

(3.13)Ωbmkkqmϕ=Ωϕdivπ+Ωgϕ,ϕH1(Ω)
and thus
(3.14)Ωbmkkqmϕ=Ωπ·ϕ+Ω(gπ·N)ϕ,ϕH1(Ω).
Using the Cauchy–Schwarz inequality, we obtain
(3.15)|Ωbmkkqmϕ|C(πL2+gπ·NH1/2(Ω))ϕH1,ϕH1(Ω).
Since also Ω|ϕ||q|ϕL2qL2 for all ϕL2(Ω), we get
(3.16)qH1C(πL2+gπ·NH1/2(Ω)+qL2).
Next, we aim to improve this inequality by estimating the L2-norm of q. For this purpose, for every fL2(Ω) such that Ωf=0, solve
Δϕ˜f=fin Ωϕ˜fN=0on Ω(3.17)Ωϕ˜f=0.
Note that, by the energy inequality,
(3.18)Ω|ϕ˜f|2CfL22.
Since ϕ˜f/N=0 on Ω, we have ΩqΔϕ˜f+Ωq·ϕ˜f=0 and thus
(3.19)|Ωqf|=|Ωq·ϕ˜f|.
In order to estimate Ωq·ϕ˜f, we write
(3.20)Ωiqiϕ˜f=Ωbmkkqmϕ˜f+Ω(δmkbmk)kqmϕ˜f.
Using (3.15) on the first term, we get
(3.21)|Ωqf|C(πL2+gπ·NH1/2(Ω))ϕ˜fL2+Cϵ0qL2ϕ˜fL2
whence
(3.22)|Ωqf|C(πL2+gπ·NH1/2(Ω)+ϵ0qL2)fL2.
Since this inequality holds for all fL2(Ω) such that Ωf=0, we obtain
(3.23)qq¯L2C(πL2+gπ·NH1/2(Ω)+ϵ0qL2).
On the other hand, by (3.16), we also have
(3.24)(qq¯)L2C(πL2+gπ·NH1/2(Ω)+qq¯L2+q¯L2).
Combining (3.23) and (3.24) and then choosing ϵ0 sufficiently small then leads to (3.12). □

Now, let Ω be as in (2.1), and let q be as in Lemma 3.2. In order to bound |q¯|, let H be a solution of the Dirichlet/Neumann problem

(3.25)ΔH=1in Ω(3.26)H=0on Γ1(3.27)HN=0on Γ0.
Using
(3.28)ΩqΔH+Ωq·H=ΩHNq
we obtain
(3.29)|Ωqdx|C(qq¯)L2+CqL2(Γ1)
which combined with (3.12) leads to
(3.30)qH1CπL2+Cgπ·NH1/2(Ω)+CqL2(Γ1)
for solutions of the problem (3.8)–(3.9) under given boundedness and ellipticity conditions.

The bounds on the pressure and its derivatives are obtained by solving a linear elliptic equation with Neumann boundary conditions.

Lemma 3.3.

Assume that (v,q,a,η) solves the system (2.4)–(2.11) for a given coefficient matrix aH2.5(Ω) satisfying (i)–(viii) from Lemma 3.1, with a sufficiently small constant ϵ=1/C. Then the estimate

(3.31)qH3.5CvH2.5vH2.5+CvtH2(Γ1)+C
holds for all t(0,T). Moreover, the time derivatives qt and qtt satisfy
qtH2.5CvH1.5+ϵ(qH2.5+vtH1.5)+C(vH1.5vL+vtH1.5)vH1.5+ϵ(3.32)+CvttH1(Γ1)+CvH2.5+CvtH2.5
and
qttH1C(vH1.5vL+vtH1.5)(qH2+vtH1)+CvL(qtH1+vttL2)+C(vH1.5vL2+vtH1.5vL+vttH1.5)vH1+CvtttL2(3.33)+CvH23/2vH31/2+CvLvH2.5+CvtH2.5
for all t(0,T), where T1/CM for a sufficiently large constant C.

Proof.

Applying the Lagrangian divergence to the evolution equation (2.4) leads to

(3.34)Δq=m((δkmaimaik)kq)+taimmvi.
In order to obtain the boundary condition for q, we multiply the equation (2.4) with aimNm and sum. We get
(3.35)qN=(δkmaimaik)kqNmaimtviNm
which holds on Γ0Γ1. As in [17, Lemma 12.1, p. 866], we have a regularity estimate for
Δq=fin Ω(3.36)q·N=gon Ω
which reads
(3.37)qHsCfHs2+CgHs1.5(Ω)+CqL2
and is valid for s2, with the constant C depending on s. Using (3.29), we then get
(3.38)qHsCfHs2+CgHs1.5(Ω)+CqL2(Γ1)
for any s2. We use this estimate with
(3.39)f=m((δkmaimaik)kq)+m(taimvi)
and
(3.40)g=(δkmaimaik)kqNmaimtviNmon Γ0Γ1.
In order to obtain (3.31), we apply the estimate (3.38) with s=3.5. We thus have
fH1.5(IaTa)qH2.5+atvH2.5IaTaH2.5qH3.5+atH2.5vH2.5(3.41)ϵqH3.5+CvH2.5vH2.5
and, similarly,
gH2(Γ1)IaTaH2.5qH3.5+avtH2(Γ1)(3.42)ϵqH3.5+CvtH2(Γ1)
by using (3.1), (3.2) and part (v) from Lemma 3.1. Also,
(3.43)qL2(Γ1)CηH2(Γ1)CηH2.5C.
Next, for qt we apply (3.38) for the time differentiated problem (3.36) with s=2.5. We get
ftH0.5(IaTa)tqH1.5+(IaTa)qtH1.5+attvH1.5+atvtH1.5CatH1.5+ϵqH2.5+IaTaH1.5+ϵqtH2.5+attH1.5vH1.5+ϵ+atH1.5+ϵvtH1.5CvH1.5+ϵqH2.5+ϵqtH2.5(3.44)+C(vH1.5vL+vtH1.5)vH1.5+ϵ+vH1.5+ϵvtH1.5
where we utilized the multiplicative Sobolev inequality and the parts (ii), (vi), and (viii) from Lemma 3.1. As in (3.43), we have
(3.45)qtL2(Γ1)CatL4(Ω)ηH2.5(Ω)+CaLηtH2(Ω)CvH2.5.
Lastly, we consider the twice differentiated in time system (3.34). First, we rewrite it as
(3.46)m((aimaik)kq)=taimmvi
while the boundary condition (3.35) is
(3.47)aimaikkqNm=aimtviNm.
The twice differentiated system then reads
(3.48)m((aimaik)kqtt)=m(tt(aimaik)kq)2m(t(aimaik)kqt)+m(tt(taimvi))
with the boundary condition
(3.49)aimaikkqttNm=tt(aimaik)kqNm2t(aimaik)kqtNmtt(aimtviNm).
Applying the inequality (3.30), we obtain
qttH1Cmtt(aimaik)kqL2+Cmt(aimaik)kqtL2+Cmtt(taimvi)L2+Ctt(aimtviNm+taimviNm)H1/2(Ω)+CqttL2(Γ1)(3.50)=I1+I2+I3+I4+I5.
In order to estimate the last term I5 in (3.50), we use (2.11), which, when rewritten as
(3.51)Niq=(δikaik)qNkΔ2ηi
on Γ1, leads to
(3.52)q=(1a33)qΔ2η3on Γ1.
Therefore,
(3.53)qL4(Γ1)CηH3C
by Lemma 3.1. Using (3.45) and (3.52)
qttL2(Γ1)Ctta33qL2(Γ1)+Cta33tqL2(Γ1)+CvtH2(Γ1)(3.54)CattL4(Γ1)qL4(Γ1)+Cta33tqL2(Γ1)+CvtH2(Γ1).
In order to estimate the first term on the far right side, we use
(3.55)attL4(Γ1)CattH1(Ω)CvH5/42+CvtH1.
Replacing this inequality in (3.54), we get
qttL2(Γ1)CvH5/42+CvtH1+CatLvH2.5+CvtH2.5CvH5/42+CvtH1+CatLvH2.5+CvtH2.5(3.56)CvH23/2vH31/2+CvtH1+CatLvH2.5+CvtH2.5.
In order to bound I4, we write
(3.57)I4=Cttt(aimvi)NmH1/2(Ω)Cmttt(aimvi)L2(Ω)
the last inequality following from m(aimvi)=0. Therefore,
I1+I2+I3+I4C(aTa)ttqL2+C(aTa)tqtL2+CatttvL2+CattvtL2+CatvttL2+CavtttL2C(vH0.5vL+vtH0.5)qH1+CvLqtL2+atttL3vL6+C(vH0.5vL+vtH0.5)vtH1(3.58)+CvLvttL2+CvtttL2.
The third term on the far right side is then estimated as
(3.59)atttL3vL6C(vL3vL2+vtL3vL+vttL3)vH1
and (3.33) follows. □

4.Local in time solutions

4.1.L2 estimate on vttt

Applying t3 to (2.4), multiplying the resulting equation by vttt, and integrating in space and time gives

(4.1)12vttt(t)L22=12vttt(0)L220t(aikkq)tttvttti=12vttt(0)L220tk(aikq)tttvttti
where we utilized the Piola identity
(4.2)kaik=0,i=1,2,3.
In order to bound the integral on the right side, we integrate by parts,
(4.3)0tk(aikq)tttvttti=0tΩ(aikq)tttvtttiNk+0t(aikq)tttkvttti=I1+I2.
Since v3=0 on Γ0, we have ¯v3=0, where
(4.4)¯=(1,2)
and vttt3=0 on Γ0. Also, ¯η3=0 on Γ0, which implies that a13=a23=0 on Γ0. As a consequence,
(4.5)0tΓ0(aikq)tttvtttiNk=0tΓ0(ai3q)tttvttti=0.
Thus, for the boundary term in (4.3), we obtain
(4.6)I1=0tΓ1(aikqNk)tttvttti=0tΓ1Δ2ηtttivttti=12¯vtt(t)L2(Γ1)2+12¯vtt(0)L2(Γ1)2
by using (2.7), (2.11), and integrating by parts in the tangential direction.

Now, we bound the second integral

I2=0taikqtttkvttti+30t(aik)tqttkvttti+30t(aik)ttqtkvttti+0t(aik)tttqkvttti(4.7)=I21+I22+I23+I24.
Using the incompressibility condition to write
aikkvttti=(aikkvi)ttt3(aik)tkvtti3(aik)ttkvti(aik)tttkvi(4.8)=3(aik)tkvtti3(aik)ttkvti(aik)tttkvi,
we get
I21=0t3(aik)ttkvtiqttt0t3(aik)tkvttiqttt0t(aik)tttkviqttt(4.9)=I211+I212+I213.
For I211 we integrate by parts in time:
I211=3(aik)ttkvtiqtt|0t+30tt((aik)ttkvti)qttCatt(0)L2vt(0)L3qtt(0)L6+Catt(t)L2vt(t)L3qtt(t)L6+C0tatttL3vtL6qttL2+C0tattL6vttL3qttL2P(v0H3.5,v0H3.5(Γ1))+Catt(t)L2vt(t)L3qtt(t)L6(4.10)+0tP(qttL2,vttH1.5,vtH2,vH3.5).
Integrating by parts I212=3(aik)tkvttiqttt in space, we have
(4.11)I212=30tΩ(aik)tvttiqtttNk+30t(aik)tvttikqttt=I2121+I2122
where we used (4.2). Observe that
I2121=30tΓ1aijjvlalkvttiqtttNk=30tΓ1aijjvlΔ2ηtttlvtti(4.12)0tΓ1(3(alk)tqttNk+3(alk)ttqtNk+(alk)tttqNk)aijjvlvtti
by using (2.6) in the first and (2.11) in the second equality; also note that the integral over Γ0 vanishes. Integrating by parts in the tangential directions, we obtain
30tΓ1aijjvlΔ2ηtttlvtti=3k=120tkηtttlk(aijjvlvtti)(4.13)C0tP(vttH1.5,vH3),
while the lower order terms are bounded as
0tΓ1(3(alk)tqttNk+3(alk)ttqtNk+(alk)tttqNk)aijjvlvtti(4.14)0tP(qttH1,vttH1.5,vH3.5,qtH1.5,vtH2,qH2.5).
Next, integrating by parts in time gives
I2122=3(aik)tvttikqtt|0t30t(aik)ttvttikqtt30t(aik)tvtttikqttP(v0H3.5)+Cqtt(t)H1vtt(t)L3v(t)L6(4.15)+0tP(qttH1,vtttL2,vttH1,vtH1.5,vH3).
By (2.6) and integrating by parts in time we have
(4.16)I213=0taijjvttlalkkviqttt+R.
Until the end of this paper, we denote by R the remainder terms. In (4.16), the lower order terms are of the form
(4.17)0t(attva+atvat+atvta)vqttt,
(written in a symbolic way, omitting all the indices) which can be bounded by
RP(v0H3.5)+P(v(t)eH3,qtt(t)H1)vt(t)H1.5(4.18)+0tP(qttH1,vH3,vtH1.5,vttH1.5)dt
after integrating by parts in time. Integrating by parts in space, the leading term of (4.16) becomes
0taijjvttlalkkviqttt=0taijvttlalkjkviqttt0taijvttlalkkvijqttt0taijvttljalkkviqttt+0tΓ1aijvttlalkkviqtttNj(4.19)=I2131+I2132+I2133+I2134,
where we have omitted the term when the j-th derivatives fall on aij which equals to zero by (4.2). First, observe that the boundary term I2134 can be treated exactly as I2121 above. Now, using aijjkvi=kaijjvi for k=1,2,3, we write
I2131=0tkaijvttlalkjviqttt=kaijvttlalkjviqtt|0t+0t(kaijvttlalkjvi)tqttP(v0H3.5,v0H3.5(Γ1))+Cqtt(t)L6vtt(t)L2v(t)L3,(4.20)+0tP(qttH1,vtttL2,vttH1,vtH1.5,vH3),
while
I2132=aijvttlalkkvijqtt|0t+0t(aijvttlalkkvi)tjqttP(v0H3.5,v0H3.5(Γ1))+Cqtt(t)H1vtt(t)L3v(t)L6(4.21)+0tP(qttH1,vtttL2,vttH1,vtH1.5,vH3).
Note that the lower order term I2133 is also bounded by the right side of (4.21). For I22 we integrate by parts in space
I22=30tΓ1taikqttvtttiNk30t(aik)tkqttvttti=30tΓ1(aijjvlalk)qttvtttiNk30t(aik)tkqttvttti=30tΓ1(aijjvlΔ2ηttlvttti+aijjvl((alk)ttq+2(alk)tqt)Nkvttti)30t(aik)tkqttvttti(4.22)=I221+I222.
We denote the first boundary term in I221 by I2211. The other two terms in I221 are easy to bound. Integrating by parts in the tangential directions, we get
(4.23)I2211=30tΓ1aijjvl¯vtl¯vttti30tΓ1¯(aijjvl)¯vtlvttti
which after an additional integration by parts in time leads to
I2211=3Γ1(aijjvl¯vtl¯vtti+¯(aijjvl)¯vtlvtti)|0t(4.24)+30tΓ1(aijjvl¯vtl)t¯vtti+(¯(aijjvl)¯vtl)tvtti.
Thus,
I2211P(v0H3.5,v0H3.5(Γ1))+Cvtt(t)H1.5vt(t)H2v(t)H2.5(4.25)+0tP(vttH1.5,vtH2.5,vH3).

Next, for I23 we proceed as in I22 by first integrating by parts in space

I23=30tΓ1(aijjvlalk)tqtvtttiNk30t(aik)ttkqtvttti(4.26)=30tΓ1aijjvtlΔ2ηtlvttti+R30t(aik)ttkqtvttti,
where the remainder term
R=30tΓ1(aij)tjvlalkqtvtttiNk30tΓ1aijjvl(alk)tqtvtttiNk(4.27)+30tΓ1aijjvtl(alk)tqvtttiNk
is bounded by
RP(v0H3.5,v0H3.5(Γ1))+Cqt(t)H2.5vtt(t)H1.5v(t)H2.52+Cq(t)H1.5vtt(t)H1.5vt(t)H1.5v(t)H2.5(4.28)+0tP(qttH1,qtH1.5,qH2.5,vttH1.5,vtH2.5,vH3).
The first boundary term on the far right sides in (4.26) can be bounded similarly as I2211 above, by integrating by parts in time. We omit further details.

Lastly, we consider I24. We use that (aik)ttt=(aijjvlalk)tt=aijjvttlalk+l.o.t., where the lower order terms are of the form attva, atvta, atvat (and the resulting integrals are clearly easy to bound). Thus, we estimate only the leading term in I24. We have

(4.29)I24=0taijjvttlalkkvtttiq+R
and observe that
t(aijjvttlalkkvtti)=aijjvttlalkkvttti+aijjvtttlalkkvtti+(aij)tjvttlalkkvtti+aijjvttl(alk)tkvtti(4.30)=2aijjvttlalkkvttti+2(aij)tjvttlalkkvtti.
Hence,
I24=12aijjvttlalkkvttiq|0t120taijjvttlalkkvttiqt0t(aij)tjvttlalkkvttiq+l.o.t.P(v0H3.5,v0H3.5(Γ1))+Cvtt(t)H1.5vtt(t)H1q(t)H1(4.31)+0tP(vttH1,vtH2,vH3,qtH2,qH2).
Therefore, we conclude
vttt(t)L22+¯vtt(t)L2(Γ1)2P(v0H3.5,v0H3.5(Γ1))+ϵ(vtt(t)H1.52+qtt(t)H12)(4.32)+0tP(vtttL2,vttH1.5,vtH2.5,vH3,qttH1,qtH2,qH2).

4.2.Tangential H1 estimate on vtt

Applying mt2 to the equation (2.4), multiplying by mvtt, summing for m=1,2, and integrating in space and time, we get

(4.33)12¯vtt(t)L22=12¯vtt(0)L220tm(aikkq)ttmvtti=12¯vtt(0)L220taikkmqttmvtti+R.
Here and in the next section, for simplicity of notation, we modify the summation convention for repeated indices in m with m=1,2 (while other indices are still summed for 1,2,3). Note that the remainder term R on the right of (4.33) is bounded by
R=0t(m(aik)ttkq+(aik)ttkmq+2m(aik)tkqt+2(aik)tkmqt+maikkqtt)mvtti(4.34)0tP(vttH1,vtH2.5,vH3,qttH1,qtH2,qH2.5).
Now, we integrate by parts in the higher order term
(4.35)0taikkmqttmvtti=0tΩaikmqttmvttiNk+0taikmqttkmvtti=I1+I2.
For I1, the integral over Γ0 vanishes, while on Γ1 we use
(4.36)aikmqttNk=mtt(aikqNk)m((aik)ttqNk)2m((aik)tqtNk)maikqttNk
(to check this, write m(aikqttNk)=aikmqttNk+maikqttNk and rewrite the second term) and get
I1=0tΓ1Δ2mηttmvtti+0tΓ1(m((aik)ttq)+2m((aik)tqt)+maikqtt)mvttiNk=12¯2vt(t)L2(Γ1)2+12¯2vt(0)L2(Γ1)2(4.37)+0tΓ1(m((aik)ttq)+2m((aik)tqt)+maikqtt)mvttiNk,
and the last term on the right side can be bounded by
(4.38)0tP(vttH1.5,qttH1,qtH1.5,vtH2.5,qH2.5,vH3.5).
For I2, we use the divergence free condition to write
(4.39)aikkmvtti=maikkvttim(2(aik)tkvti+(aik)ttkvi).
Thus, we obtain
I2=0t(maikkvtti+m(2(aik)tkvti+(aik)ttkvi))mqtt(4.40)0tP(qttH1,vttH1.5,vtH2,vH3).
We conclude
¯vtt(t)L22+¯2vt(t)L2(Γ1)2(4.41)¯2vt(0)L2(Γ1)2+0tP(vttH1.5,qttH1,qtH2,vtH2.5,qH2.5,vH3.5).

4.3.Tangential H2 estimate on vt

Applying lmt to (2.4), multiplying by lmvt, summing for l,m=1,2, and integrating in space and time, we get

12¯2vt(t)L22=12¯2vt(0)L220tlm(aikkq)tlmvti(4.42)=12¯2vt(0)L220taikklmqtlmvtil.o.t.,
where the lower order terms on the right are bounded by 0tP(vtH2,qH3,vH3.5,qtH2.5). Next, integrating by parts, we get similarly as in the previous section
(4.43)0taikklmqtlmvti=0tΩaiklmqtlmvtiNk+0taiklmqtklmvti=I1+I2,
where
I1=0tΩΔ2lmηtlmvtil.o.t.(4.44)=12¯3v(t)L2(Γ1)2+12¯3v(0)L2(Γ1)2l.o.t.
and, by using the divergence free condition,
(4.45)I20tP(qtH2,vtH2,vH3).
Therefore, we conclude
(4.46)¯2vt(t)L22+¯3v(t)L2(Γ1)2¯2vt(0)L22+¯3v(0)L2(Γ1)2+0tP(vtH2,qH3,vH3.5,qtH2.5).

4.4.Div-curl estimates

We use the elliptic estimate (cf. [12,17])

(4.47)fHsCfL2+CcurlfHs1+CdivfHs1+Cf·NHs0.5(Ω)
for s1.

We recall that ¯vttL2(Γ1), so in particular vtt3H1(Γ1). By (4.47) with s=1.5, we have

(4.48)vttH1.5CvttL2+CcurlvttH0.5+CdivvttH0.5+Cvtt3H1(Γ1),
where we also used vtt3=0 on Γ0. Similarly, applying (4.47) with s=2.5 and s=3.5 respectively, we have
(4.49)vtH2.5CvtL2+CcurlvtH1.5+CdivvtH1.5+Cvt3H2(Γ1)
and
(4.50)vH3.5CvL2+CcurlvH2.5+CdivvH2.5+Cv3H3(Γ1).
The first term on the right side of (4.48) (same for (4.49) and (4.50)) is of lower order and can be written as
(4.51)vtt(t)L2=vtt(0)L2+t1/20tvtttL2.
By the multiplicative Sobolev inequality, for divv we have
(4.52)divvH2.5=(δikaik)kviH2.5ϵvH3.5,
as well as
(4.53)divvtH1.5=(δikaik)kvti(aik)tkviH1.5ϵvtH2.5+CatH1.5+δvH2.5
and
divvttH0.5=(δikaik)kvtti2(aik)tkvti(aik)ttkviH0.5(4.54)ϵvttH1.5+CatH1.5+δvtH1.5+CattH0.5vH2.5+δ.

Recall the Cauchy invariance (cf. [32] for instance)

(4.55)ϵijkjvlkηl=curlv0i,i=1,2,3,
for t0, where ϵijk is the antisymmetric tensor defined by ϵ123=1 with ϵijk=ϵjik and ϵijk=ϵjki. Thus, we have
(4.56)(curlv)i=ϵijkjvk=ϵijkjvl(δlkkηl)+curlv0i,
where
(4.57)δlkkηl=0tkηtl=0tkvl,k,l=1,2,3,
which implies
(4.58)curlvH2.5CvH3.50tvH3.5+curlv0H2.5.
Differentiating (4.55) in time, we have
(4.59)0=(ϵijkjvlkηl)t=ϵijkjvtlkηl+ϵijkjvlkηtl,
where the second term on the right vanishes because it is equal to ϵijkjvlkvl and ϵijk=ϵikj. Thus, we also get
(4.60)ϵijkjvtlkηl=0
from where
(4.61)(curlvt)i=ϵijkjvtk=ϵijkjvtl(δlkkηl).
Therefore,
(4.62)curlvtH1.5CvtH1.5IηH1.5+δCvtH2.50tvH2.5+δ.
Differentiating (4.60), using (2.7), and rearranging the terms in the equality, we obtain
(4.63)ϵijkjvttlkηl=ϵijkjvtlkvl.
Then, we may write
(4.64)(curlvtt)i=ϵijkjvttk=ϵijkjvttl(δlkkηl)ϵijkjvtlkvl,
from where
(4.65)curlvttH0.5CvttH1.50tvH2.5+δ+CvtH1.5vH2.5+δ.
Now, we gather the div-curl inequalities to obtain Sobolev estimates on v, vt, and vtt. Namely, we have
vH3.5C(v(0)L2+curlv0H2.5)+t1/20tvtL2(4.66)+CvH3.50tvH3.5+Cv3H3(Γ1)
and
vtH2.5Cvt(0)L2+Ct1/20tvttL2(4.67)+CvtH2.50tvH2.5+δ+CatH1.5+δvH2.5+Cvt3H2(Γ1).
Finally,
vttH1.5Cvtt(0)L2+Ct1/2vtttL2+CvttH1.50tvH2.5+δ+CvtH1.5vH2.5+δ(4.68)+CatH1.5+δvtH1.5+CattH0.5vH2.5+δ+Cvtt3H1(Γ1).

5.Closing the estimates

Squaring the estimate (4.66) and using (4.46) for the bound of v3H3(Γ1), we have

vH3.52P(v0H3.5,v0H3.5(Γ1))+CvH3.520tvH3.52(5.1)+0tP(vtH2,qH3,vH3.5,qtH2.5).
By the pressure estimate (3.31),
(5.2)qH3.5C(vH3.5+1)(v0H2.5+C0tvtH2.5)+CvtH2(Γ1).
This, combined with (4.41) for the bound of vtH2(Γ1), gives
qH3.52P(v0H3.5,v0H3.5(Γ1))+CvH3.52(v0H2.52+C0tvtH2.52)(5.3)+0tP(vttH1.5,qttH1,qtH2,vtH2.5,qH2.5,vH3.5).
Similarly, squaring (4.67) and using (4.41),
vtH2.52P(v0H3.5,v0H3.5(Γ1))+CvtH2.520tvH2.5+δ2+CvH2.5+δ2(v0H2.52+0tvtH2.52)(5.4)+0tP(vttH1.5,qttH1,qtH2,vtH2.5,qH2.5,vH3.5),
while combining the square of the estimate (3.32),
qtH2.52CvH2.5+δ2(P(v0H3.5,v0H3.5(Γ1))+C0t(vttH1.52+qtH2.52))+CvH2.52+CvtH2.52(5.5)+C(vH2.5+δ4+vtH2.52)(v0H1.5+δ2+0tvtH1.5+δ2)+CvttH1(Γ1)2
(4.32) and (5.4), we obtain
qtH2.52P(v0H3.5,v0H3.5(Γ1))+Cϵ(vttH1.52+qttH12+vH3.52)+P(vtH2.5,vH2.5+δ)0tP(vH2.5+δ,vtH2.5)(5.6)+0tP(qttH1,vtttL2,vH3.5,vttH1.5,vtH2.5,qtH2,qH2).
Lastly, squaring (4.68) and using (4.32),
vttH1.52P(v0H3.5,v0H3.5(Γ1))+Cϵ(vttH1.52+qttH12+vH3.52)+CvttH1.520tvH2.5+δ2+CvH2.5+δ20tP(vttH1.5,vtH2.5)(5.7)+0tP(qttH1,vtttL2,vH3.5,vttH1.5,vtH2.5,qtH2,qH2),
while squaring (3.33) and (4.32) give
qttH12P(v0H3.5,v0H3.5(Γ1))+Cϵ(vttH1.52+qttH12+vH3.52)+CvttH1.52(v0H12+0tvtH12)+P(vH3.5,vtH2.5)0tP(qttH1,vtttL2,qtH2,vttH1,vtH2.5,vH3.5)(5.8)+0tP(qttH1,vtttL2,vH3.5,vttH1.5,vtH2.5,qtH2,qH2).
Combining all the estimates, we obtain a Gronwall type inequality yielding the a priori estimates for the local in time existence.

Acknowledgements

I.K. was supported in part by the NSF grant DMS-1311943 and DMS-1615239. We would like to thank Peter Constantin for useful discussions, and especially for Lemma 3.2 and the inequality (3.30). We also thank Marcelo Disconzi and the referee for useful suggestions.

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