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

Article type: Research Article

Authors: Ignatova, Mihaelaa | Kukavica, Igorb; *

Affiliations: [a] Department of Mathematics, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] | [b] Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Keywords: Euler equations, surface tension, free boundary

DOI: 10.3233/ASY-161386

Journal: Asymptotic Analysis, vol. 100, no. 1-2, pp. 63-86, 2016

Published: 7 November 2016
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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 [2,3,8–10,13,18,22,27,30,31,34,36–38,45,52,53], for other works on non-zero surface tension, cf. [1,39] while for global existence of solutions, see [24–26,28,49].

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):Ω→Ω(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)aik∂kvi=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=−alk∂jvlaij,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)=v0∈H3.5(Ω) is divergence-free and is such that v0|Γ1∈H3.5(Γ1). Then there exists a local-in-time solution (v,q,a,η) to (2.4)–(2.11) which satisfies

v∈L∞([0,T];H3.5(Ω))vt∈L∞([0,T];H2.5(Ω))vtt∈L∞([0,T];H1.5(Ω))vttt∈L∞([0,T];L2(Ω))
with q∈L∞([0,T];H3.5(Ω)), qt∈L∞([0,T];H2.5(Ω)), qtt∈L∞([0,T];H1(Ω)), a∈L∞([0,T];H2.5(Ω)), and η∈C([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 ‖v‖L∞([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) ‖η‖H3.5⩽C for t∈[0,T];

  • (ii) ‖a‖H2.5⩽C for t∈[0,T];

  • (iii) ‖at‖Lp⩽C‖∇v‖Lp for t∈[0,T];

  • (iv) ‖∂iat‖Lp⩽C‖∇v‖Lp1‖∂ia‖Lp2+C‖∇∂iv‖Lp for i=1,2,3 and t∈[0,T] where 1⩽p,p1,p2⩽∞ are such that 1/p=1/p1+1/p2;

  • (v) ‖at‖Hr⩽‖∇v‖Hr, for r∈[0,2.5] and t∈[0,T];

  • (vi) ‖att‖Hσ⩽C‖∇v‖Hσ‖∇v‖L∞+C‖∇vt‖Hσ for σ∈[0,1.5] and t∈[0,T] and ‖att‖H1(Ω)⩽C‖∇v‖H5/42+C‖∇vt‖H1;

  • (vii) ‖attt‖Lp⩽C‖∇v‖Lp‖∇v‖L∞2+C‖∇vt‖Lp‖∇v‖L∞+C‖∇vtt‖Lp for t∈[0,T];

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

    (3.1)‖δjk−aljalk‖H2.52⩽ϵ,j,k=1,2,3
    and
    (3.2)‖δjk−akj‖H2.52⩽ϵ,j,k=1,2,3.

In particular, the form aljalkξjiξki satisfies the ellipticity estimate
(3.3)aljalkξjiξki⩾1C|ξ|2,ξ∈Rn2
for all t∈[0,T∗] and x∈Ω, provided ϵ⩽1/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)‖H2.5⩽C+∫0t‖a(s)‖H2.52‖∇v(s)‖H2.5ds
and (ii) is obtained by using the Gronwall lemma, provided T⩽1/CM. Next, (2.6) implies
(3.5)‖at‖Lp⩽C‖a‖L∞‖∇v‖Lp‖a‖L∞⩽C‖∇v‖Lp
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)δjk−aljalk=−∫0t∂t(aljalk)ds
where j,k∈{1,2,3}. Therefore,
‖δjk−aljalk‖H2.5⩽∫0t‖∂t(aljalk)‖H2.5ds(3.7)⩽C∫0t‖alj‖H2.5‖∂talk‖H2.5ds⩽C∫0t‖at‖H2.5ds⩽CMt.
The estimate (3.1) then follows if CM2T2⩽ϵ. 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 ‖b‖L∞⩽M and that bij(x)ξiξj⩾M−1|ξ|2 for all x∈Ω and ξ∈Rn, where n∈{2,3,…}.

Lemma 3.2

Lemma 3.2([16]).

Let q be an H1 solution of the

(3.8)∂i(bij∂jq)=divπin Ω(3.9)bmk∂kqNm=gon ∂Ω
where π,divπ∈L2(Ω) and g∈H−1/2(∂Ω) with the compatibility condition
(3.10)∫∂Ω(π·N−g)=0.
If
(3.11)‖b−I‖L∞⩽ϵ0
where ϵ0>0 is a sufficiently small constant depending on M, then we have
(3.12)‖q−q¯‖H1⩽C‖π‖L2+C‖g−π·N‖H−1/2(∂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)∫Ωbmk∂kq∂mϕ=−∫Ωϕdivπ+∫∂Ωgϕ,ϕ∈H1(Ω)
and thus
(3.14)∫Ωbmk∂kq∂mϕ=∫Ωπ·∇ϕ+∫∂Ω(g−π·N)ϕ,ϕ∈H1(Ω).
Using the Cauchy–Schwarz inequality, we obtain
(3.15)|∫Ωbmk∂kq∂mϕ|⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω))‖ϕ‖H1,ϕ∈H1(Ω).
Since also ∫Ω|ϕ||q|⩽‖ϕ‖L2‖q‖L2 for all ϕ∈L2(Ω), we get
(3.16)‖q‖H1⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω)+‖q‖L2).
Next, we aim to improve this inequality by estimating the L2-norm of q. For this purpose, for every f∈L2(Ω) such that ∫Ωf=0, solve
Δϕ˜f=fin Ω∂ϕ˜f∂N=0on ∂Ω(3.17)∫Ωϕ˜f=0.
Note that, by the energy inequality,
(3.18)∫Ω|∇ϕ˜f|2⩽C‖f‖L22.
Since ∂ϕ˜f/∂N=0 on ∂Ω, we have ∫ΩqΔϕ˜f+∫Ω∇q·∇ϕ˜f=0 and thus
(3.19)|∫Ωqf|=|∫Ω∇q·∇ϕ˜f|.
In order to estimate ∫Ω∇q·∇ϕ˜f, we write
(3.20)∫Ω∂iq∂iϕ˜f=∫Ωbmk∂kq∂mϕ˜f+∫Ω(δmk−bmk)∂kq∂mϕ˜f.
Using (3.15) on the first term, we get
(3.21)|∫Ωqf|⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω))‖∇ϕ˜f‖L2+Cϵ0‖∇q‖L2‖∇ϕ˜f‖L2
whence
(3.22)|∫Ωqf|⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω)+ϵ0‖∇q‖L2)‖f‖L2.
Since this inequality holds for all f∈L2(Ω) such that ∫Ωf=0, we obtain
(3.23)‖q−q¯‖L2⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω)+ϵ0‖∇q‖L2).
On the other hand, by (3.16), we also have
(3.24)‖∇(q−q¯)‖L2⩽C(‖π‖L2+‖g−π·N‖H−1/2(∂Ω)+‖q−q¯‖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)∂H∂N=0on Γ0.
Using
(3.28)∫ΩqΔH+∫Ω∇q·∇H=∫∂Ω∂H∂Nq
we obtain
(3.29)|∫Ωqdx|⩽C‖∇(q−q¯)‖L2+C‖q‖L2(Γ1)
which combined with (3.12) leads to
(3.30)‖q‖H1⩽C‖π‖L2+C‖g−π·N‖H−1/2(∂Ω)+C‖q‖L2(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 a∈H2.5(Ω) satisfying (i)–(viii) from Lemma 3.1, with a sufficiently small constant ϵ=1/C. Then the estimate

(3.31)‖q‖H3.5⩽C‖∇v‖H2.5‖v‖H2.5+C‖vt‖H2(Γ1)+C
holds for all t∈(0,T). Moreover, the time derivatives qt and qtt satisfy
‖qt‖H2.5⩽C‖∇v‖H1.5+ϵ(‖q‖H2.5+‖vt‖H1.5)+C(‖∇v‖H1.5‖∇v‖L∞+‖∇vt‖H1.5)‖v‖H1.5+ϵ(3.32)+C‖vtt‖H1(Γ1)+C‖v‖H2.5+C‖vt‖H2.5
and
‖qtt‖H1⩽C(‖v‖H1.5‖∇v‖L∞+‖vt‖H1.5)(‖q‖H2+‖vt‖H1)+C‖∇v‖L∞(‖qt‖H1+‖vtt‖L2)+C(‖v‖H1.5‖∇v‖L∞2+‖vt‖H1.5‖∇v‖L∞+‖vtt‖H1.5)‖v‖H1+C‖vttt‖L2(3.33)+C‖v‖H23/2‖v‖H31/2+C‖∇v‖L∞‖v‖H2.5+C‖vt‖H2.5
for all t∈(0,T), where T⩽1/CM for a sufficiently large constant C.

Proof.

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

(3.34)Δq=∂m((δkm−aimaik)∂kq)+∂taim∂mvi.
In order to obtain the boundary condition for q, we multiply the equation (2.4) with aimNm and sum. We get
(3.35)∂q∂N=(δkm−aimaik)∂kqNm−aim∂tviNm
which holds on Γ0∪Γ1. 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)‖q‖Hs⩽C‖f‖Hs−2+C‖g‖Hs−1.5(∂Ω)+C‖q‖L2
and is valid for s⩾2, with the constant C depending on s. Using (3.29), we then get
(3.38)‖q‖Hs⩽C‖f‖Hs−2+C‖g‖Hs−1.5(∂Ω)+C‖q‖L2(Γ1)
for any s⩾2. We use this estimate with
(3.39)f=∂m((δkm−aimaik)∂kq)+∂m(∂taimvi)
and
(3.40)g=(δkm−aimaik)∂kqNm−aim∂tviNmon Γ0∪Γ1.
In order to obtain (3.31), we apply the estimate (3.38) with s=3.5. We thus have
‖f‖H1.5⩽‖(I−aTa)∇q‖H2.5+‖atv‖H2.5⩽‖I−aTa‖H2.5‖q‖H3.5+‖at‖H2.5‖v‖H2.5(3.41)⩽ϵ‖q‖H3.5+C‖∇v‖H2.5‖v‖H2.5
and, similarly,
‖g‖H2(Γ1)⩽‖I−aTa‖H2.5‖q‖H3.5+‖avt‖H2(Γ1)(3.42)⩽ϵ‖q‖H3.5+C‖vt‖H2(Γ1)
by using (3.1), (3.2) and part (v) from Lemma 3.1. Also,
(3.43)‖q‖L2(Γ1)⩽C‖η‖H2(Γ1)⩽C‖η‖H2.5⩽C.
Next, for qt we apply (3.38) for the time differentiated problem (3.36) with s=2.5. We get
‖ft‖H0.5⩽‖(I−aTa)t∇q‖H1.5+‖(I−aTa)∇qt‖H1.5+‖attv‖H1.5+‖atvt‖H1.5⩽C‖at‖H1.5+ϵ‖q‖H2.5+‖I−aTa‖H1.5+ϵ‖qt‖H2.5+‖att‖H1.5‖v‖H1.5+ϵ+‖at‖H1.5+ϵ‖vt‖H1.5⩽C‖∇v‖H1.5+ϵ‖q‖H2.5+ϵ‖qt‖H2.5(3.44)+C(‖∇v‖H1.5‖∇v‖L∞+‖∇vt‖H1.5)‖v‖H1.5+ϵ+‖∇v‖H1.5+ϵ‖vt‖H1.5
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)‖qt‖L2(Γ1)⩽C‖at‖L4(∂Ω)‖η‖H2.5(∂Ω)+C‖a‖L∞‖ηt‖H2(∂Ω)⩽C‖v‖H2.5.
Lastly, we consider the twice differentiated in time system (3.34). First, we rewrite it as
(3.46)∂m((aimaik)∂kq)=∂taim∂mvi
while the boundary condition (3.35) is
(3.47)aimaik∂kqNm=−aim∂tviNm.
The twice differentiated system then reads
(3.48)∂m((aimaik)∂kqtt)=−∂m(∂tt(aimaik)∂kq)−2∂m(∂t(aimaik)∂kqt)+∂m(∂tt(∂taimvi))
with the boundary condition
(3.49)aimaik∂kqttNm=−∂tt(aimaik)∂kqNm−2∂t(aimaik)∂kqtNm−∂tt(aim∂tviNm).
Applying the inequality (3.30), we obtain
‖qtt‖H1⩽C∑m‖∂tt(aimaik)∂kq‖L2+C∑m‖∂t(aimaik)∂kqt‖L2+C∑m‖∂tt(∂taimvi)‖L2+C‖∂tt(aim∂tviNm+∂taimviNm)‖H−1/2(∂Ω)+C‖qtt‖L2(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=(δik−aik)qNk−Δ2ηi
on Γ1, leads to
(3.52)q=(1−a33)q−Δ2η3on Γ1.
Therefore,
(3.53)‖q‖L4(Γ1)⩽C‖η‖H3⩽C
by Lemma 3.1. Using (3.45) and (3.52)
‖qtt‖L2(Γ1)⩽C‖∂tta33q‖L2(Γ1)+C‖∂ta33∂tq‖L2(Γ1)+C‖vt‖H2(Γ1)(3.54)⩽C‖att‖L4(Γ1)‖q‖L4(Γ1)+C‖∂ta33∂tq‖L2(Γ1)+C‖vt‖H2(Γ1).
In order to estimate the first term on the far right side, we use
(3.55)‖att‖L4(Γ1)⩽C‖att‖H1(Ω)⩽C‖∇v‖H5/42+C‖∇vt‖H1.
Replacing this inequality in (3.54), we get
‖qtt‖L2(Γ1)⩽C‖∇v‖H5/42+C‖∇vt‖H1+C‖at‖L∞‖v‖H2.5+C‖vt‖H2.5⩽C‖∇v‖H5/42+C‖∇vt‖H1+C‖at‖L∞‖v‖H2.5+C‖vt‖H2.5(3.56)⩽C‖v‖H23/2‖v‖H31/2+C‖∇vt‖H1+C‖at‖L∞‖v‖H2.5+C‖vt‖H2.5.
In order to bound I4, we write
(3.57)I4=C‖∂ttt(aimvi)Nm‖H−1/2(∂Ω)⩽C∑m‖∂ttt(aimvi)‖L2(Ω)
the last inequality following from ∂m(aimvi)=0. Therefore,
I1+I2+I3+I4⩽C‖(aTa)tt∇q‖L2+C‖(aTa)t∇qt‖L2+C‖atttv‖L2+C‖attvt‖L2+C‖atvtt‖L2+C‖avttt‖L2⩽C(‖∇v‖H0.5‖∇v‖L∞+‖∇vt‖H0.5)‖∇q‖H1+C‖∇v‖L∞‖∇qt‖L2+‖attt‖L3‖v‖L6+C(‖∇v‖H0.5‖∇v‖L∞+‖∇vt‖H0.5)‖vt‖H1(3.58)+C‖∇v‖L∞‖vtt‖L2+C‖vttt‖L2.
The third term on the far right side is then estimated as
(3.59)‖attt‖L3‖v‖L6⩽C(‖∇v‖L3‖∇v‖L∞2+‖∇vt‖L3‖∇v‖L∞+‖∇vtt‖L3)‖v‖H1
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)12‖vttt(t)‖L22=12‖vttt(0)‖L22−∫0t∫(aik∂kq)tttvttti=12‖vttt(0)‖L22−∫0t∫∂k(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)−∫0t∫∂k(aikq)tttvttti=−∫0t∫∂Ω(aikq)tttvtttiNk+∫0t∫(aikq)ttt∂kvttti=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)∫0t∫Γ0(aikq)tttvtttiNk=∫0t∫Γ0(ai3q)tttvttti=0.
Thus, for the boundary term in (4.3), we obtain
(4.6)I1=−∫0t∫Γ1(aikqNk)tttvttti=∫0t∫Γ1Δ2ηtttivttti=−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=∫0t∫aikqttt∂kvttti+3∫0t∫(aik)tqtt∂kvttti+3∫0t∫(aik)ttqt∂kvttti+∫0t∫(aik)tttq∂kvttti(4.7)=I21+I22+I23+I24.
Using the incompressibility condition to write
aik∂kvttti=(aik∂kvi)ttt−3(aik)t∂kvtti−3(aik)tt∂kvti−(aik)ttt∂kvi(4.8)=−3(aik)t∂kvtti−3(aik)tt∂kvti−(aik)ttt∂kvi,
we get
I21=−∫0t∫3(aik)tt∂kvtiqttt−∫0t∫3(aik)t∂kvttiqttt−∫0t∫(aik)ttt∂kviqttt(4.9)=I211+I212+I213.
For I211 we integrate by parts in time:
I211=−3∫(aik)tt∂kvtiqtt|0t+3∫0t∫∂t((aik)tt∂kvti)qtt⩽C‖att(0)‖L2‖∇vt(0)‖L3‖qtt(0)‖L6+C‖att(t)‖L2‖∇vt(t)‖L3‖qtt(t)‖L6+C∫0t‖attt‖L3‖∇vt‖L6‖qtt‖L2+C∫0t‖att‖L6‖∇vtt‖L3‖qtt‖L2⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖att(t)‖L2‖∇vt(t)‖L3‖qtt(t)‖L6(4.10)+∫0tP(‖qtt‖L2,‖vtt‖H1.5,‖vt‖H2,‖v‖H3.5).
Integrating by parts I212=−3∫(aik)t∂kvttiqttt in space, we have
(4.11)I212=−3∫0t∫∂Ω(aik)tvttiqtttNk+3∫0t∫(aik)tvtti∂kqttt=I2121+I2122
where we used (4.2). Observe that
I2121=3∫0t∫Γ1aij∂jvlalkvttiqtttNk=−3∫0t∫Γ1aij∂jvlΔ2ηtttlvtti(4.12)−∫0t∫Γ1(3(alk)tqttNk+3(alk)ttqtNk+(alk)tttqNk)aij∂jvlvtti
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
−3∫0t∫Γ1aij∂jvlΔ2ηtttlvtti=3∑k=12∫0t∫∂kηtttl∂k(aij∂jvlvtti)(4.13)⩽C∫0tP(‖vtt‖H1.5,‖v‖H3),
while the lower order terms are bounded as
−∫0t∫Γ1(3(alk)tqttNk+3(alk)ttqtNk+(alk)tttqNk)aij∂jvlvtti(4.14)⩽∫0tP(‖qtt‖H1,‖vtt‖H1.5,‖v‖H3.5,‖qt‖H1.5,‖vt‖H2,‖q‖H2.5).
Next, integrating by parts in time gives
I2122=3∫(aik)tvtti∂kqtt|0t−3∫0t∫(aik)ttvtti∂kqtt−3∫0t∫(aik)tvttti∂kqtt⩽P(‖v0‖H3.5)+C‖qtt(t)‖H1‖vtt(t)‖L3‖∇v(t)‖L6(4.15)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖vtt‖H1,‖vt‖H1.5,‖v‖H3).
By (2.6) and integrating by parts in time we have
(4.16)I213=∫0t∫aij∂jvttlalk∂kviqttt+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∫(att∇va+at∇vat+at∇vta)∇vqttt,
(written in a symbolic way, omitting all the indices) which can be bounded by
R⩽P(‖v0‖H3.5)+P(‖v(t)‖eH3,‖qtt(t)‖H1)‖vt(t)‖H1.5(4.18)+∫0tP(‖qtt‖H1,‖v‖H3,‖vt‖H1.5,‖vtt‖H1.5)dt
after integrating by parts in time. Integrating by parts in space, the leading term of (4.16) becomes
∫0t∫aij∂jvttlalk∂kviqttt=−∫0t∫aijvttlalk∂jkviqttt−∫0t∫aijvttlalk∂kvi∂jqttt−∫0t∫aijvttl∂jalk∂kviqttt+∫0t∫Γ1aijvttlalk∂kviqtttNj(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 aij∂jkvi=−∂kaij∂jvi for k=1,2,3, we write
I2131=−∫0t∫∂kaijvttlalk∂jviqttt=−∫∂kaijvttlalk∂jviqtt|0t+∫0t∫(∂kaijvttlalk∂jvi)tqtt⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖qtt(t)‖L6‖vtt(t)‖L2‖∇v(t)‖L3,(4.20)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖vtt‖H1,‖vt‖H1.5,‖v‖H3),
while
I2132=−∫aijvttlalk∂kvi∂jqtt|0t+∫0t∫(aijvttlalk∂kvi)t∂jqtt⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖qtt(t)‖H1‖vtt(t)‖L3‖∇v(t)‖L6(4.21)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖vtt‖H1,‖vt‖H1.5,‖v‖H3).
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=3∫0t∫Γ1∂taikqttvtttiNk−3∫0t∫(aik)t∂kqttvttti=−3∫0t∫Γ1(aij∂jvlalk)qttvtttiNk−3∫0t∫(aik)t∂kqttvttti=3∫0t∫Γ1(aij∂jvlΔ2ηttlvttti+aij∂jvl((alk)ttq+2(alk)tqt)Nkvttti)−3∫0t∫(aik)t∂kqttvttti(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=−3∫0t∫Γ1aij∂jvl∂¯vtl∂¯vttti−3∫0t∫Γ1∂¯(aij∂jvl)∂¯vtlvttti
which after an additional integration by parts in time leads to
I2211=−3∫Γ1(aij∂jvl∂¯vtl∂¯vtti+∂¯(aij∂jvl)∂¯vtlvtti)|0t(4.24)+3∫0t∫Γ1(aij∂jvl∂¯vtl)t∂¯vtti+(∂¯(aij∂jvl)∂¯vtl)tvtti.
Thus,
I2211⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖vtt(t)‖H1.5‖vt(t)‖H2‖v(t)‖H2.5(4.25)+∫0tP(‖vtt‖H1.5,‖vt‖H2.5,‖v‖H3).

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

I23=−3∫0t∫Γ1(aij∂jvlalk)tqtvtttiNk−3∫0t∫(aik)tt∂kqtvttti(4.26)=3∫0t∫Γ1aij∂jvtlΔ2ηtlvttti+R−3∫0t∫(aik)tt∂kqtvttti,
where the remainder term
R=−3∫0t∫Γ1(aij)t∂jvlalkqtvtttiNk−3∫0t∫Γ1aij∂jvl(alk)tqtvtttiNk(4.27)+3∫0t∫Γ1aij∂jvtl(alk)tqvtttiNk
is bounded by
R⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖qt(t)‖H2.5‖vtt(t)‖H1.5‖v(t)‖H2.52+C‖q(t)‖H1.5‖vtt(t)‖H1.5‖vt(t)‖H1.5‖v(t)‖H2.5(4.28)+∫0tP(‖qtt‖H1,‖qt‖H1.5,‖q‖H2.5,‖vtt‖H1.5,‖vt‖H2.5,‖v‖H3).
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=(aij∂jvlalk)tt=aij∂jvttlalk+l.o.t., where the lower order terms are of the form att∇va, at∇vta, at∇vat (and the resulting integrals are clearly easy to bound). Thus, we estimate only the leading term in I24. We have

(4.29)I24=∫0t∫aij∂jvttlalk∂kvtttiq+R
and observe that
∂t(aij∂jvttlalk∂kvtti)=aij∂jvttlalk∂kvttti+aij∂jvtttlalk∂kvtti+(aij)t∂jvttlalk∂kvtti+aij∂jvttl(alk)t∂kvtti(4.30)=2aij∂jvttlalk∂kvttti+2(aij)t∂jvttlalk∂kvtti.
Hence,
I24=12∫aij∂jvttlalk∂kvttiq|0t−12∫0t∫aij∂jvttlalk∂kvttiqt−∫0t∫(aij)t∂jvttlalk∂kvttiq+l.o.t.⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖vtt(t)‖H1.5‖vtt(t)‖H1‖q(t)‖H1(4.31)+∫0tP(‖vtt‖H1,‖vt‖H2,‖v‖H3,‖qt‖H2,‖q‖H2).
Therefore, we conclude
‖vttt(t)‖L22+‖∂¯vtt(t)‖L2(Γ1)2⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+ϵ(‖vtt(t)‖H1.52+‖qtt(t)‖H12)(4.32)+∫0tP(‖vttt‖L2,‖vtt‖H1.5,‖vt‖H2.5,‖v‖H3,‖qtt‖H1,‖qt‖H2,‖q‖H2).

4.2.Tangential H1 estimate on vtt

Applying ∂m∂t2 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)‖L22−∫0t∫∂m(aik∂kq)tt∂mvtti=12‖∂¯vtt(0)‖L22−∫0t∫aik∂kmqtt∂mvtti+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)tt∂kq+(aik)tt∂kmq+2∂m(aik)t∂kqt+2(aik)t∂kmqt+∂maik∂kqtt)∂mvtti(4.34)⩽∫0tP(‖vtt‖H1,‖vt‖H2.5,‖v‖H3,‖qtt‖H1,‖qt‖H2,‖q‖H2.5).
Now, we integrate by parts in the higher order term
(4.35)−∫0t∫aik∂kmqtt∂mvtti=−∫0t∫∂Ωaik∂mqtt∂mvttiNk+∫0t∫aik∂mqtt∂kmvtti=I1+I2.
For I1, the integral over Γ0 vanishes, while on Γ1 we use
(4.36)aik∂mqttNk=∂m∂tt(aikqNk)−∂m((aik)ttqNk)−2∂m((aik)tqtNk)−∂maikqttNk
(to check this, write ∂m(aikqttNk)=aik∂mqttNk+∂maikqttNk and rewrite the second term) and get
I1=∫0t∫Γ1Δ2∂mηtt∂mvtti+∫0t∫Γ1(∂m((aik)ttq)+2∂m((aik)tqt)+∂maikqtt)∂mvttiNk=−12‖∂¯2vt(t)‖L2(Γ1)2+12‖∂¯2vt(0)‖L2(Γ1)2(4.37)+∫0t∫Γ1(∂m((aik)ttq)+2∂m((aik)tqt)+∂maikqtt)∂mvttiNk,
and the last term on the right side can be bounded by
(4.38)∫0tP(‖vtt‖H1.5,‖qtt‖H1,‖qt‖H1.5,‖vt‖H2.5,‖q‖H2.5,‖v‖H3.5).
For I2, we use the divergence free condition to write
(4.39)aik∂kmvtti=−∂maik∂kvtti−∂m(2(aik)t∂kvti+(aik)tt∂kvi).
Thus, we obtain
I2=−∫0t∫(∂maik∂kvtti+∂m(2(aik)t∂kvti+(aik)tt∂kvi))∂mqtt(4.40)⩽∫0tP(‖qtt‖H1,‖vtt‖H1.5,‖vt‖H2,‖v‖H3).
We conclude
‖∂¯vtt(t)‖L22+‖∂¯2vt(t)‖L2(Γ1)2(4.41)⩽‖∂¯2vt(0)‖L2(Γ1)2+∫0tP(‖vtt‖H1.5,‖qtt‖H1,‖qt‖H2,‖vt‖H2.5,‖q‖H2.5,‖v‖H3.5).

4.3.Tangential H2 estimate on vt

Applying ∂lm∂t 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)‖L22−∫0t∫∂lm(aik∂kq)t∂lmvti(4.42)=12‖∂¯2vt(0)‖L22−∫0t∫aik∂klmqt∂lmvti−l.o.t.,
where the lower order terms on the right are bounded by ∫0tP(‖vt‖H2,‖q‖H3,‖v‖H3.5,‖qt‖H2.5). Next, integrating by parts, we get similarly as in the previous section
(4.43)−∫0t∫aik∂klmqt∂lmvti=−∫0t∫∂Ωaik∂lmqt∂lmvtiNk+∫0t∫aik∂lmqt∂klmvti=I1+I2,
where
I1=∫0t∫∂ΩΔ2∂lmηt∂lmvti−l.o.t.(4.44)=−12‖∂¯3v(t)‖L2(Γ1)2+12‖∂¯3v(0)‖L2(Γ1)2−l.o.t.
and, by using the divergence free condition,
(4.45)I2⩽∫0tP(‖qt‖H2,‖vt‖H2,‖v‖H3).
Therefore, we conclude
(4.46)‖∂¯2vt(t)‖L22+‖∂¯3v(t)‖L2(Γ1)2⩽‖∂¯2vt(0)‖L22+‖∂¯3v(0)‖L2(Γ1)2+∫0tP(‖vt‖H2,‖q‖H3,‖v‖H3.5,‖qt‖H2.5).

4.4.Div-curl estimates

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

(4.47)‖f‖Hs⩽C‖f‖L2+C‖curlf‖Hs−1+C‖divf‖Hs−1+C‖f·N‖Hs−0.5(∂Ω)
for s⩾1.

We recall that ∂¯vtt∈L2(Γ1), so in particular vtt3∈H1(Γ1). By (4.47) with s=1.5, we have

(4.48)‖vtt‖H1.5⩽C‖vtt‖L2+C‖curlvtt‖H0.5+C‖divvtt‖H0.5+C‖vtt3‖H1(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)‖vt‖H2.5⩽C‖vt‖L2+C‖curlvt‖H1.5+C‖divvt‖H1.5+C‖vt3‖H2(Γ1)
and
(4.50)‖v‖H3.5⩽C‖v‖L2+C‖curlv‖H2.5+C‖divv‖H2.5+C‖v3‖H3(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+t1/2∫0t‖vttt‖L2.
By the multiplicative Sobolev inequality, for divv we have
(4.52)‖divv‖H2.5=‖(δik−aik)∂kvi‖H2.5⩽ϵ‖v‖H3.5,
as well as
(4.53)‖divvt‖H1.5=‖(δik−aik)∂kvti−(aik)t∂kvi‖H1.5⩽ϵ‖vt‖H2.5+C‖at‖H1.5+δ‖v‖H2.5
and
‖divvtt‖H0.5=‖(δik−aik)∂kvtti−2(aik)t∂kvti−(aik)tt∂kvi‖H0.5(4.54)⩽ϵ‖vtt‖H1.5+C‖at‖H1.5+δ‖vt‖H1.5+C‖att‖H0.5‖v‖H2.5+δ.

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

(4.55)ϵijk∂jvl∂kηl=curlv0i,i=1,2,3,
for t⩾0, where ϵijk is the antisymmetric tensor defined by ϵ123=1 with ϵijk=−ϵjik and ϵijk=ϵjki. Thus, we have
(4.56)(curlv)i=ϵijk∂jvk=ϵijk∂jvl(δlk−∂kηl)+curlv0i,
where
(4.57)δlk−∂kηl=−∫0t∂kηtl=−∫0t∂kvl,k,l=1,2,3,
which implies
(4.58)‖curlv‖H2.5⩽C‖v‖H3.5∫0t‖v‖H3.5+‖curlv0‖H2.5.
Differentiating (4.55) in time, we have
(4.59)0=(ϵijk∂jvl∂kηl)t=ϵijk∂jvtl∂kηl+ϵijk∂jvl∂kηtl,
where the second term on the right vanishes because it is equal to ϵijk∂jvl∂kvl and ϵijk=−ϵikj. Thus, we also get
(4.60)ϵijk∂jvtl∂kηl=0
from where
(4.61)(curlvt)i=ϵijk∂jvtk=ϵijk∂jvtl(δlk−∂kηl).
Therefore,
(4.62)‖curlvt‖H1.5⩽C‖∇vt‖H1.5‖I−∇η‖H1.5+δ⩽C‖vt‖H2.5∫0t‖v‖H2.5+δ.
Differentiating (4.60), using (2.7), and rearranging the terms in the equality, we obtain
(4.63)ϵijk∂jvttl∂kηl=−ϵijk∂jvtl∂kvl.
Then, we may write
(4.64)(curlvtt)i=ϵijk∂jvttk=ϵijk∂jvttl(δlk−∂kηl)−ϵijk∂jvtl∂kvl,
from where
(4.65)‖curlvtt‖H0.5⩽C‖vtt‖H1.5∫0t‖v‖H2.5+δ+C‖vt‖H1.5‖v‖H2.5+δ.
Now, we gather the div-curl inequalities to obtain Sobolev estimates on v, vt, and vtt. Namely, we have
‖v‖H3.5⩽C(‖v(0)‖L2+‖curlv0‖H2.5)+t1/2∫0t‖vt‖L2(4.66)+C‖v‖H3.5∫0t‖v‖H3.5+C‖v3‖H3(Γ1)
and
‖vt‖H2.5⩽C‖vt(0)‖L2+Ct1/2∫0t‖vtt‖L2(4.67)+C‖vt‖H2.5∫0t‖v‖H2.5+δ+C‖at‖H1.5+δ‖v‖H2.5+C‖vt3‖H2(Γ1).
Finally,
‖vtt‖H1.5⩽C‖vtt(0)‖L2+Ct1/2∫‖vttt‖L2+C‖vtt‖H1.5∫0t‖v‖H2.5+δ+C‖vt‖H1.5‖v‖H2.5+δ(4.68)+C‖at‖H1.5+δ‖vt‖H1.5+C‖att‖H0.5‖v‖H2.5+δ+C‖vtt3‖H1(Γ1).

5.Closing the estimates

Squaring the estimate (4.66) and using (4.46) for the bound of ‖v3‖H3(Γ1), we have

‖v‖H3.52⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖v‖H3.52∫0t‖v‖H3.52(5.1)+∫0tP(‖vt‖H2,‖q‖H3,‖v‖H3.5,‖qt‖H2.5).
By the pressure estimate (3.31),
(5.2)‖q‖H3.5⩽C(‖v‖H3.5+1)(‖v0‖H2.5+C∫0t‖vt‖H2.5)+C‖vt‖H2(Γ1).
This, combined with (4.41) for the bound of ‖vt‖H2(Γ1), gives
‖q‖H3.52⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖v‖H3.52(‖v0‖H2.52+C∫0t‖vt‖H2.52)(5.3)+∫0tP(‖vtt‖H1.5,‖qtt‖H1,‖qt‖H2,‖vt‖H2.5,‖q‖H2.5,‖v‖H3.5).
Similarly, squaring (4.67) and using (4.41),
‖vt‖H2.52⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C‖vt‖H2.52∫0t‖v‖H2.5+δ2+C‖v‖H2.5+δ2(‖v0‖H2.52+∫0t‖vt‖H2.52)(5.4)+∫0tP(‖vtt‖H1.5,‖qtt‖H1,‖qt‖H2,‖vt‖H2.5,‖q‖H2.5,‖v‖H3.5),
while combining the square of the estimate (3.32),
‖qt‖H2.52⩽C‖v‖H2.5+δ2(P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+C∫0t(‖vtt‖H1.52+‖qt‖H2.52))+C‖v‖H2.52+C‖vt‖H2.52(5.5)+C(‖v‖H2.5+δ4+‖vt‖H2.52)(‖v0‖H1.5+δ2+∫0t‖vt‖H1.5+δ2)+C‖vtt‖H1(Γ1)2
(4.32) and (5.4), we obtain
‖qt‖H2.52⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+Cϵ(‖vtt‖H1.52+‖qtt‖H12+‖v‖H3.52)+P(‖vt‖H2.5,‖v‖H2.5+δ)∫0tP(‖v‖H2.5+δ,‖vt‖H2.5)(5.6)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖v‖H3.5,‖vtt‖H1.5,‖vt‖H2.5,‖qt‖H2,‖q‖H2).
Lastly, squaring (4.68) and using (4.32),
‖vtt‖H1.52⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+Cϵ(‖vtt‖H1.52+‖qtt‖H12+‖v‖H3.52)+C‖vtt‖H1.52∫0t‖v‖H2.5+δ2+C‖v‖H2.5+δ2∫0tP(‖vtt‖H1.5,‖vt‖H2.5)(5.7)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖v‖H3.5,‖vtt‖H1.5,‖vt‖H2.5,‖qt‖H2,‖q‖H2),
while squaring (3.33) and (4.32) give
‖qtt‖H12⩽P(‖v0‖H3.5,‖v0‖H3.5(Γ1))+Cϵ(‖vtt‖H1.52+‖qtt‖H12+‖v‖H3.52)+C‖vtt‖H1.52(‖v0‖H12+∫0t‖vt‖H12)+P(‖v‖H3.5,‖vt‖H2.5)∫0tP(‖qtt‖H1,‖vttt‖L2,‖qt‖H2,‖vtt‖H1,‖vt‖H2.5,‖v‖H3.5)(5.8)+∫0tP(‖qtt‖H1,‖vttt‖L2,‖v‖H3.5,‖vtt‖H1.5,‖vt‖H2.5,‖qt‖H2,‖q‖H2).
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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