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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 $H^{3.5}$ for which the trace on the free boundary belongs to $H^{3.5}$ .

Keywords

Euler equations surface tension free boundary

1. Introduction

In this paper, we address the local existence of solutions to the 3D free-surface incompressible Euler equations $\begin{array}{l} (1.1) & \partial_{t} u + u \cdot \nabla u + \nabla p = 0 in Ω (t) \\ (1.2) & \nabla \cdot u = 0 in Ω (t) \end{array}$ where the free boundary $\partial Ω (t)$ evolves according to the fluid velocity field $u (x, t)$ , and the pressure obeys $\begin{array}{l} (1.3) & p (x, t) = σ H on \partial Ω (t) . \end{array}$ Here $σ > 0$ is the surface tension, while H represents twice the mean curvature of the boundary $\partial Ω (t)$ .

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 $H^{4.5}$ 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 $H^{4.5}$ . 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 $H^{3}$ . 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 $H^{3.5 + ϵ}$ 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 $H^{3.5}$ such that the trace on the free boundary belongs to $H^{3} .5$ , 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 $v_{t t t}$ , $v_{t t}$ , $v_{t}$ 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 $η (\cdot, t) : Ω \to Ω (t)$ 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., $\begin{matrix} (2.1) & Ω = {x = (x_{1}, x_{2}, x_{3}) : (x_{1}, x_{2}) \in R^{2}, 0 < x_{3} < 1} \end{matrix}$ with periodic boundary conditions with period 1 in the lateral directions. We denote the top of Ω (corresponding to the free-surface) by $\begin{array}{l} (2.2) & Γ_{1} = T^{2} \times {x_{3} = 1} \end{array}$ and the stationary bottom by $\begin{array}{l} (2.3) & Γ_{0} = T^{2} \times {x_{3} = 0} . \end{array}$ Then the incompressible Euler equation has the form $\begin{array}{l} (2.4) & v_{t}^{i} + \partial_{k} (a_{i}^{k} q) = 0 in Ω \times (0, T), i = 1, 2, 3 \\ (2.5) & a_{i}^{k} \partial_{k} v^{i} = 0 in Ω \times (0, T), \end{array}$ where $v (x, t) = η_{t} (x, t) = u (η (x, t), t)$ 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 $a (x, t) = {[\nabla η (x, t)]}^{- 1}$ and the flow map $η (x, t)$ are described by the ODEs $\begin{array}{l} (2.6) & a_{t} = - a : \nabla v : a in Ω \times (0, T) \\ (2.7) & η_{t} = v in Ω \times (0, T) \end{array}$ where the symbol : denotes the matrix multiplication, with the initial conditions $\begin{array}{l} (2.8) & a (x, 0) = I \\ (2.9) & η (x, 0) = x \end{array}$ in Ω. The condition (2.6) can be written in coordinates as $\begin{matrix} (2.10) & \partial_{t} a_{k}^{i} = - a_{l}^{k} \partial_{j} v^{l} a_{i}^{j}, i, k = 1, 2, 3 . \end{matrix}$ We assume $v \cdot N = 0$ on $Γ_{0} \times (0, T)$ and $\begin{array}{l} (2.11) & a_{i}^{k} q N_{k} = - Δ_{2} η^{i} on Γ_{1} \times (0, T) \end{array}$ for $i = 1, 2, 3$ , where $N = (N_{1}, N_{2}, N_{3})$ is the unit outward normal with respect to Ω and $Δ_{2} = \partial_{1}^{2} + \partial_{2}^{2}$ . 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 $v (\cdot, t) = v_{0} \in H^{3.5} (Ω)$ is divergence-free and is such that $v_{0} |_{Γ_{1}} \in H^{3.5} (Γ_{1})$ . Then there exists a local-in-time solution $(v, q, a, η)$ to ( 2.4 )–( 2.11 ) which satisfies $\begin{array}{l} v \in L^{\infty} ([0, T]; H^{3.5} (Ω)) \\ v_{t} \in L^{\infty} ([0, T]; H^{2.5} (Ω)) \\ v_{t t} \in L^{\infty} ([0, T]; H^{1.5} (Ω)) \\ v_{t t t} \in L^{\infty} ([0, T]; L^{2} (Ω)) \end{array}$ with $q \in L^{\infty} ([0, T]; H^{3.5} (Ω))$ , $q_{t} \in L^{\infty} ([0, T]; H^{2.5} (Ω))$ , $q_{t t} \in L^{\infty} ([0, T]; H^{1} (Ω))$ , $a \in L^{\infty} ([0, T]; H^{2.5} (Ω))$ , and $η \in C ([0, T]; H^{3.5} (Ω))$ .

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 $‖ v ‖_{L^{\infty} ([0, T]; H^{3.5})} ⩽ M$ . Let $p \in [1, \infty]$ and $i, j = 1, 2, 3$ . With $T \in [0, 1 / C M]$ , where C is a sufficiently large constant, the following statements hold:
$‖ η ‖_{H^{3.5}} ⩽ C$ for $t \in [0, T]$ ;

$‖ a ‖_{H^{2.5}} ⩽ C$ for $t \in [0, T]$ ;

$‖ a_{t} ‖_{L^{p}} ⩽ C ‖ \nabla v ‖_{L^{p}}$ for $t \in [0, T]$ ;

$‖ \partial_{i} a_{t} ‖_{L^{p}} ⩽ C ‖ \nabla v ‖_{L^{p_{1}}} ‖ \partial_{i} a ‖_{L^{p_{2}}} + C ‖ \nabla \partial_{i} v ‖_{L^{p}}$ for $i = 1, 2, 3$ and $t \in [0, T]$ where $1 ⩽ p, p_{1}, p_{2} ⩽ \infty$ are such that $1 / p = 1 / p_{1} + 1 / p_{2}$ ;

$‖ a_{t} ‖_{H^{r}} ⩽ ‖ \nabla v ‖_{H^{r}}$ , for $r \in [0, 2.5]$ and $t \in [0, T]$ ;

$‖ a_{t t} ‖_{H^{σ}} ⩽ C ‖ \nabla v ‖_{H^{σ}} ‖ \nabla v ‖_{L^{\infty}} + C ‖ \nabla v_{t} ‖_{H^{σ}}$ for $σ \in [0, 1.5]$ and $t \in [0, T]$ and $‖ a_{t t} ‖_{H^{1} (Ω)} ⩽ C ‖ \nabla v ‖_{H^{5 / 4}}^{2} + C ‖ \nabla v_{t} ‖_{H^{1}}$ ;

$‖ a_{t t t} ‖_{L^{p}} ⩽ C ‖ \nabla v ‖_{L^{p}} ‖ \nabla v ‖_{L^{\infty}}^{2} + C ‖ \nabla v_{t} ‖_{L^{p}} ‖ \nabla v ‖_{L^{\infty}} + C ‖ \nabla v_{t t} ‖_{L^{p}}$ for $t \in [0, T]$ ;

for every $ϵ \in (0, 1 / 2]$ and all $t ⩽ T^{} = min {ϵ / C M^{2}, T}$ , we have* $\begin{matrix} (3.1) & {‖ δ_{j k} - a_{l}^{j} a_{l}^{k} ‖}_{H^{2.5}}^{2} ⩽ ϵ, j, k = 1, 2, 3 \end{matrix}$ and $\begin{matrix} (3.2) & {‖ δ_{j k} - a_{k}^{j} ‖}_{H^{2.5}}^{2} ⩽ ϵ, j, k = 1, 2, 3 . \end{matrix}$
In particular, the form $a_{l}^{j} a_{l}^{k} ξ_{j}^{i} ξ_{k}^{i}$ satisfies the ellipticity estimate $\begin{matrix} (3.3) & a_{l}^{j} a_{l}^{k} ξ_{j}^{i} ξ_{k}^{i} ⩾ \frac{1}{C} | ξ |^{2}, ξ \in R^{n^{2}} \end{matrix}$ for all $t \in [0, T^{}]$ and* $x \in Ω$ , 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) $\begin{matrix} (3.4) & {‖ a (t) ‖}_{H^{2.5}} ⩽ C + \int_{0}^{t} {‖ a (s) ‖}_{H^{2.5}}^{2} {‖ \nabla v (s) ‖}_{H^{2.5}} d s \end{matrix}$ and (ii) is obtained by using the Gronwall lemma, provided $T ⩽ 1 / C M$ . Next, (2.6) implies $\begin{matrix} (3.5) & ‖ a_{t} ‖_{L^{p}} ⩽ C ‖ a ‖_{L^{\infty}} ‖ \nabla v ‖_{L^{p}} ‖ a ‖_{L^{\infty}} ⩽ C ‖ \nabla v ‖_{L^{p}} \end{matrix}$ 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 $\begin{matrix} (3.6) & δ_{j k} - a_{l}^{j} a_{l}^{k} = - \int_{0}^{t} \partial_{t} (a_{l}^{j} a_{l}^{k}) d s \end{matrix}$ where $j, k \in {1, 2, 3}$ . Therefore, $\begin{array}{l} {‖ δ_{j k} - a_{l}^{j} a_{l}^{k} ‖}_{H^{2.5}} & ⩽ \int_{0}^{t} {‖ \partial_{t} (a_{l}^{j} a_{l}^{k}) ‖}_{H^{2.5}} d s \\ (3.7) & ⩽ C \int_{0}^{t} {‖ a_{l}^{j} ‖}_{H^{2.5}} {‖ \partial_{t} a_{l}^{k} ‖}_{H^{2.5}} d s ⩽ C \int_{0}^{t} ‖ a_{t} ‖_{H^{2.5}} d s ⩽ C M t . \end{array}$ The estimate (3.1) then follows if $C M^{2} T^{2} ⩽ ϵ$ . 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 $b_{i j}$ satisfies $‖ b ‖_{L^{\infty}} ⩽ M$ and that $b_{i j} (x) ξ_{i} ξ_{j} ⩾ M^{- 1} | ξ |^{2}$ for all $x \in Ω$ and $ξ \in R^{n}$ , where $n \in {2, 3, \dots}$ .
Lemma 3.2 ([16]).

Let q be an $H^{1}$ solution of the $\begin{array}{l} (3.8) & \partial_{i} (b_{i j} \partial_{j} q) = div π in Ω \\ (3.9) & b_{m k} \partial_{k} q N^{m} = g on \partial Ω \end{array}$ where $π, div π \in L^{2} (Ω)$ and $g \in H^{- 1 / 2} (\partial Ω)$ with the compatibility condition $\begin{matrix} (3.10) & \int_{\partial Ω} (π \cdot N - g) = 0 . \end{matrix}$ If $\begin{matrix} (3.11) & ‖ b - I ‖_{L^{\infty}} ⩽ ϵ_{0} \end{matrix}$ where $ϵ_{0} > 0$ is a sufficiently small constant depending on M, then we have $\begin{matrix} (3.12) & ‖ q - \bar{q} ‖_{H^{1}} ⩽ C ‖ π ‖_{L^{2}} + C ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} \end{matrix}$ where $\bar{q} = (1 / | Ω |) \int q d x$ .

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 $L^{2}$ -norm of $div π$ , is new.

Proof of Lemma 3.2.
First, using (3.8)–(3.9), we have $\begin{matrix} (3.13) & \int_{Ω} b_{m k} \partial_{k} q \partial_{m} ϕ = - \int_{Ω} ϕ div π + \int_{\partial Ω} g ϕ, ϕ \in H^{1} (Ω) \end{matrix}$ and thus $\begin{matrix} (3.14) & \int_{Ω} b_{m k} \partial_{k} q \partial_{m} ϕ = \int_{Ω} π \cdot \nabla ϕ + \int_{\partial Ω} (g - π \cdot N) ϕ, ϕ \in H^{1} (Ω) . \end{matrix}$ Using the Cauchy–Schwarz inequality, we obtain $\begin{matrix} (3.15) & | \int_{Ω} b_{m k} \partial_{k} q \partial_{m} ϕ | ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)}) ‖ ϕ ‖_{H^{1}}, ϕ \in H^{1} (Ω) . \end{matrix}$ Since also $\int_{Ω} | ϕ | | q | ⩽ ‖ ϕ ‖_{L^{2}} ‖ q ‖_{L^{2}}$ for all $ϕ \in L^{2} (Ω)$ , we get $\begin{matrix} (3.16) & ‖ q ‖_{H^{1}} ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} + ‖ q ‖_{L^{2}}) . \end{matrix}$ Next, we aim to improve this inequality by estimating the $L^{2}$ -norm of q. For this purpose, for every $f \in L^{2} (Ω)$ such that $\int_{Ω} f = 0$ , solve $\begin{array}{l} Δ {\tilde{ϕ}}_{f} = f in Ω \\ \frac{\partial {\tilde{ϕ}}_{f}}{\partial N} = 0 on \partial Ω \\ (3.17) & \int_{Ω} {\tilde{ϕ}}_{f} = 0 . \end{array}$ Note that, by the energy inequality, $\begin{matrix} (3.18) & \int_{Ω} | \nabla {\tilde{ϕ}}_{f} |^{2} ⩽ C ‖ f ‖_{L^{2}}^{2} . \end{matrix}$ Since $\partial {\tilde{ϕ}}_{f} / \partial N = 0$ on $\partial Ω$ , we have $\int_{Ω} q Δ {\tilde{ϕ}}_{f} + \int_{Ω} \nabla q \cdot \nabla {\tilde{ϕ}}_{f} = 0$ and thus $\begin{matrix} (3.19) & | \int_{Ω} q f | = | \int_{Ω} \nabla q \cdot \nabla {\tilde{ϕ}}_{f} | . \end{matrix}$ In order to estimate $\int_{Ω} \nabla q \cdot \nabla {\tilde{ϕ}}_{f}$ , we write $\begin{array}{l} (3.20) & \int_{Ω} \partial_{i} q \partial_{i} {\tilde{ϕ}}_{f} = \int_{Ω} b_{m k} \partial_{k} q \partial_{m} {\tilde{ϕ}}_{f} + \int_{Ω} (δ_{m k} - b_{m k}) \partial_{k} q \partial_{m} {\tilde{ϕ}}_{f} . \end{array}$ Using (3.15) on the first term, we get $\begin{array}{l} (3.21) & | \int_{Ω} q f | & ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)}) ‖ \nabla {\tilde{ϕ}}_{f} ‖_{L^{2}} + C ϵ_{0} ‖ \nabla q ‖_{L^{2}} ‖ \nabla {\tilde{ϕ}}_{f} ‖_{L^{2}} \end{array}$ whence $\begin{array}{l} (3.22) & | \int_{Ω} q f | ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} + ϵ_{0} ‖ \nabla q ‖_{L^{2}}) ‖ f ‖_{L^{2}} . \end{array}$ Since this inequality holds for all $f \in L^{2} (Ω)$ such that $\int_{Ω} f = 0$ , we obtain $\begin{matrix} (3.23) & ‖ q - \bar{q} ‖_{L^{2}} ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} + ϵ_{0} ‖ \nabla q ‖_{L^{2}}) . \end{matrix}$ On the other hand, by (3.16), we also have $\begin{matrix} (3.24) & {‖ \nabla (q - \bar{q}) ‖}_{L^{2}} ⩽ C (‖ π ‖_{L^{2}} + ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} + ‖ q - \bar{q} ‖_{L^{2}} + ‖ \bar{q} ‖_{L^{2}}) . \end{matrix}$ Combining (3.23) and (3.24) and then choosing $ϵ_{0}$ 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 $| \bar{q} |$ , let H be a solution of the Dirichlet/Neumann problem $\begin{array}{l} (3.25) & Δ H = 1 in Ω \\ (3.26) & H = 0 on Γ_{1} \\ (3.27) & \frac{\partial H}{\partial N} = 0 on Γ_{0} . \end{array}$ Using $\begin{matrix} (3.28) & \int_{Ω} q Δ H + \int_{Ω} \nabla q \cdot \nabla H = \int_{\partial Ω} \frac{\partial H}{\partial N} q \end{matrix}$ we obtain $\begin{matrix} (3.29) & | \int_{Ω} q d x | ⩽ C {‖ \nabla (q - \bar{q}) ‖}_{L^{2}} + C ‖ q ‖_{L^{2} (Γ_{1})} \end{matrix}$ which combined with (3.12) leads to $\begin{matrix} (3.30) & ‖ q ‖_{H^{1}} ⩽ C ‖ π ‖_{L^{2}} + C ‖ g - π \cdot N ‖_{H^{- 1 / 2} (\partial Ω)} + C ‖ q ‖_{L^{2} (Γ_{1})} \end{matrix}$ 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 $a \in H^{2.5} (Ω)$ satisfying (i)–(viii) from Lemma 3.1 , with a sufficiently small constant $ϵ = 1 / C$ . Then the estimate $\begin{array}{l} (3.31) & ‖ q ‖_{H^{3.5}} ⩽ C ‖ \nabla v ‖_{H^{2.5}} ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2} (Γ_{1})} + C \end{array}$ holds for all $t \in (0, T)$ . Moreover, the time derivatives $q_{t}$ and $q_{t t}$ satisfy $\begin{array}{l} ‖ q_{t} ‖_{H^{2.5}} ⩽ & C ‖ \nabla v ‖_{H^{1.5 + ϵ}} (‖ q ‖_{H^{2.5}} + ‖ v_{t} ‖_{H^{1.5}}) \\ + C (‖ \nabla v ‖_{H^{1.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ \nabla v_{t} ‖_{H^{1.5}}) ‖ v ‖_{H^{1.5 + ϵ}} \\ (3.32) & + C ‖ v_{t t} ‖_{H^{1} (Γ_{1})} + C ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2.5}} \end{array}$ and $\begin{array}{l} ‖ q_{t t} ‖_{H^{1}} ⩽ & C (‖ v ‖_{H^{1.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ v_{t} ‖_{H^{1.5}}) (‖ q ‖_{H^{2}} + ‖ v_{t} ‖_{H^{1}}) \\ + C ‖ \nabla v ‖_{L^{\infty}} (‖ q_{t} ‖_{H^{1}} + ‖ v_{t t} ‖_{L^{2}}) \\ + C (‖ v ‖_{H^{1.5}} ‖ \nabla v ‖_{L^{\infty}}^{2} + ‖ v_{t} ‖_{H^{1.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ v_{t t} ‖_{H^{1.5}}) ‖ v ‖_{H^{1}} + C ‖ v_{t t t} ‖_{L^{2}} \\ (3.33) & + C ‖ v ‖_{H^{2}}^{3 / 2} ‖ v ‖_{H^{3}}^{1 / 2} + C ‖ \nabla v ‖_{L^{\infty}} ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2.5}} \end{array}$ for all $t \in (0, T)$ , where $T ⩽ 1 / C M$ for a sufficiently large constant C.
Proof.
Applying the Lagrangian divergence to the evolution equation (2.4) leads to $\begin{matrix} (3.34) & Δ q = \partial_{m} ((δ_{k m} - a_{i}^{m} a_{i}^{k}) \partial_{k} q) + \partial_{t} a_{i}^{m} \partial_{m} v^{i} . \end{matrix}$ In order to obtain the boundary condition for q, we multiply the equation (2.4) with $a_{i}^{m} N_{m}$ and sum. We get $\begin{matrix} (3.35) & \frac{\partial q}{\partial N} = (δ_{k m} - a_{i}^{m} a_{i}^{k}) \partial_{k} q N_{m} - a_{i}^{m} \partial_{t} v^{i} N_{m} \end{matrix}$ which holds on $Γ_{0} \cup Γ_{1}$ . As in [17, Lemma 12.1, p. 866], we have a regularity estimate for $\begin{array}{l} Δ q = f in Ω \\ (3.36) & \nabla q \cdot N = g on \partial Ω \end{array}$ which reads $\begin{array}{l} (3.37) & ‖ q ‖_{H^{s}} ⩽ C ‖ f ‖_{H^{s - 2}} + C ‖ g ‖_{H^{s - 1.5} (\partial Ω)} + C ‖ q ‖_{L^{2}} \end{array}$ and is valid for $s ⩾ 2$ , with the constant C depending on s. Using (3.29), we then get $\begin{array}{l} (3.38) & ‖ q ‖_{H^{s}} ⩽ C ‖ f ‖_{H^{s - 2}} + C ‖ g ‖_{H^{s - 1.5} (\partial Ω)} + C ‖ q ‖_{L^{2} (Γ_{1})} \end{array}$ for any $s ⩾ 2$ . We use this estimate with $\begin{array}{l} (3.39) & f = \partial_{m} ((δ_{k m} - a_{i}^{m} a_{i}^{k}) \partial_{k} q) + \partial_{m} (\partial_{t} a_{i}^{m} v^{i}) \end{array}$ and $\begin{array}{l} (3.40) & g & = (δ_{k m} - a_{i}^{m} a_{i}^{k}) \partial_{k} q N_{m} - a_{i}^{m} \partial_{t} v^{i} N_{m} on Γ_{0} \cup Γ_{1} . \end{array}$ In order to obtain (3.31), we apply the estimate (3.38) with $s = 3.5$ . We thus have $\begin{array}{l} ‖ f ‖_{H^{1.5}} & ⩽ {‖ (I - a^{T} a) \nabla q ‖}_{H^{2.5}} + ‖ a_{t} v ‖_{H^{2.5}} \\ ⩽ {‖ I - a^{T} a ‖}_{H^{2.5}} ‖ q ‖_{H^{3.5}} + ‖ a_{t} ‖_{H^{2.5}} ‖ v ‖_{H^{2.5}} \\ (3.41) & ⩽ ϵ ‖ q ‖_{H^{3.5}} + C ‖ \nabla v ‖_{H^{2.5}} ‖ v ‖_{H^{2.5}} \end{array}$ and, similarly, $\begin{array}{l} ‖ g ‖_{H^{2} (Γ_{1})} & ⩽ {‖ I - a^{T} a ‖}_{H^{2.5}} ‖ q ‖_{H^{3.5}} + ‖ a v_{t} ‖_{H^{2} (Γ_{1})} \\ (3.42) & ⩽ ϵ ‖ q ‖_{H^{3.5}} + C ‖ v_{t} ‖_{H^{2} (Γ_{1})} \end{array}$ by using (3.1), (3.2) and part (v) from Lemma 3.1. Also, $\begin{matrix} (3.43) & ‖ q ‖_{L^{2} (Γ_{1})} ⩽ C ‖ η ‖_{H^{2} (Γ_{1})} ⩽ C ‖ η ‖_{H^{2.5}} ⩽ C . \end{matrix}$ Next, for $q_{t}$ we apply (3.38) for the time differentiated problem (3.36) with $s = 2.5$ . We get $\begin{array}{l} ‖ f_{t} ‖_{H^{0.5}} ⩽ & {‖ {(I - a^{T} a)}_{t} \nabla q ‖}_{H^{1.5}} + {‖ (I - a^{T} a) \nabla q_{t} ‖}_{H^{1.5}} + ‖ a_{t t} v ‖_{H^{1.5}} + ‖ a_{t} v_{t} ‖_{H^{1.5}} \\ ⩽ & C ‖ a_{t} ‖_{H^{1.5 + ϵ}} ‖ q ‖_{H^{2.5}} + {‖ I - a^{T} a ‖}_{H^{1.5 + ϵ}} ‖ q_{t} ‖_{H^{2.5}} + ‖ a_{t t} ‖_{H^{1.5}} ‖ v ‖_{H^{1.5 + ϵ}} \\ + ‖ a_{t} ‖_{H^{1.5 + ϵ}} ‖ v_{t} ‖_{H^{1.5}} \\ ⩽ & C ‖ \nabla v ‖_{H^{1.5 + ϵ}} ‖ q ‖_{H^{2.5}} + ϵ ‖ q_{t} ‖_{H^{2.5}} \\ (3.44) & + C (‖ \nabla v ‖_{H^{1.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ \nabla v_{t} ‖_{H^{1.5}}) ‖ v ‖_{H^{1.5 + ϵ}} + ‖ \nabla v ‖_{H^{1.5 + ϵ}} ‖ v_{t} ‖_{H^{1.5}} \end{array}$ where we utilized the multiplicative Sobolev inequality and the parts (ii), (vi), and (viii) from Lemma 3.1. As in (3.43), we have $\begin{matrix} (3.45) & ‖ q_{t} ‖_{L^{2} (Γ_{1})} ⩽ C ‖ a_{t} ‖_{L^{4} (\partial Ω)} ‖ η ‖_{H^{2.5} (\partial Ω)} + C ‖ a ‖_{L^{\infty}} ‖ η_{t} ‖_{H^{2} (\partial Ω)} ⩽ C ‖ v ‖_{H^{2.5}} . \end{matrix}$ Lastly, we consider the twice differentiated in time system (3.34). First, we rewrite it as $\begin{matrix} (3.46) & \partial_{m} ((a_{i}^{m} a_{i}^{k}) \partial_{k} q) = \partial_{t} a_{i}^{m} \partial_{m} v^{i} \end{matrix}$ while the boundary condition (3.35) is $\begin{matrix} (3.47) & a_{i}^{m} a_{i}^{k} \partial_{k} q N_{m} = - a_{i}^{m} \partial_{t} v^{i} N_{m} . \end{matrix}$ The twice differentiated system then reads $\begin{matrix} (3.48) & \partial_{m} ((a_{i}^{m} a_{i}^{k}) \partial_{k} q_{t t}) = - \partial_{m} (\partial_{t t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q) - 2 \partial_{m} (\partial_{t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q_{t}) + \partial_{m} (\partial_{t t} (\partial_{t} a_{i}^{m} v^{i})) \end{matrix}$ with the boundary condition $\begin{matrix} (3.49) & a_{i}^{m} a_{i}^{k} \partial_{k} q_{t t} N_{m} = - \partial_{t t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q N_{m} - 2 \partial_{t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q_{t} N_{m} - \partial_{t t} (a_{i}^{m} \partial_{t} v^{i} N_{m}) . \end{matrix}$ Applying the inequality (3.30), we obtain $\begin{array}{l} ‖ q_{t t} ‖_{H^{1}} ⩽ & C \sum_{m} {‖ \partial_{t t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q ‖}_{L^{2}} + C \sum_{m} {‖ \partial_{t} (a_{i}^{m} a_{i}^{k}) \partial_{k} q_{t} ‖}_{L^{2}} + C \sum_{m} {‖ \partial_{t t} (\partial_{t} a_{i}^{m} v^{i}) ‖}_{L^{2}} \\ + C {‖ \partial_{t t} (a_{i}^{m} \partial_{t} v^{i} N_{m} + \partial_{t} a_{i}^{m} v^{i} N_{m}) ‖}_{H^{- 1 / 2} (\partial Ω)} + C ‖ q_{t t} ‖_{L^{2} (Γ_{1})} \\ (3.50) & = & I_{1} + I_{2} + I_{3} + I_{4} + I_{5} . \end{array}$ In order to estimate the last term $I_{5}$ in (3.50), we use (2.11), which, when rewritten as $\begin{matrix} (3.51) & N_{i} q = (δ_{i}^{k} - a_{i}^{k}) q N_{k} - Δ_{2} η^{i} \end{matrix}$ on $Γ_{1}$ , leads to $\begin{matrix} (3.52) & q = (1 - a_{3}^{3}) q - Δ_{2} η^{3} on Γ_{1} . \end{matrix}$ Therefore, $\begin{matrix} (3.53) & ‖ q ‖_{L^{4} (Γ_{1})} ⩽ C ‖ η ‖_{H^{3}} ⩽ C \end{matrix}$ by Lemma 3.1. Using (3.45) and (3.52) $\begin{array}{l} ‖ q_{t t} ‖_{L^{2} (Γ_{1})} & ⩽ C {‖ \partial_{t t} a_{3}^{3} q ‖}_{L^{2} (Γ_{1})} + C {‖ \partial_{t} a_{3}^{3} \partial_{t} q ‖}_{L^{2} (Γ_{1})} + C ‖ v_{t} ‖_{H^{2} (Γ_{1})} \\ (3.54) & ⩽ C ‖ a_{t t} ‖_{L^{4} (Γ_{1})} ‖ q ‖_{L^{4} (Γ_{1})} + C {‖ \partial_{t} a_{3}^{3} \partial_{t} q ‖}_{L^{2} (Γ_{1})} + C ‖ v_{t} ‖_{H^{2} (Γ_{1})} . \end{array}$ In order to estimate the first term on the far right side, we use $\begin{array}{l} (3.55) & ‖ a_{t t} ‖_{L^{4} (Γ_{1})} ⩽ C ‖ a_{t t} ‖_{H^{1} (Ω)} ⩽ C ‖ \nabla v ‖_{H^{5 / 4}}^{2} + C ‖ \nabla v_{t} ‖_{H^{1}} . \end{array}$ Replacing this inequality in (3.54), we get $\begin{array}{l} ‖ q_{t t} ‖_{L^{2} (Γ_{1})} & ⩽ C ‖ \nabla v ‖_{H^{5 / 4}}^{2} + C ‖ \nabla v_{t} ‖_{H^{1}} + C ‖ a_{t} ‖_{L^{\infty}} ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2.5}} \\ ⩽ C ‖ \nabla v ‖_{H^{5 / 4}}^{2} + C ‖ \nabla v_{t} ‖_{H^{1}} + C ‖ a_{t} ‖_{L^{\infty}} ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2.5}} \\ (3.56) & ⩽ C ‖ v ‖_{H^{2}}^{3 / 2} ‖ v ‖_{H^{3}}^{1 / 2} + C ‖ \nabla v_{t} ‖_{H^{1}} + C ‖ a_{t} ‖_{L^{\infty}} ‖ v ‖_{H^{2.5}} + C ‖ v_{t} ‖_{H^{2.5}} . \end{array}$ In order to bound $I_{4}$ , we write $\begin{matrix} (3.57) & I_{4} = C {‖ \partial_{t t t} (a_{i}^{m} v^{i}) N_{m} ‖}_{H^{- 1 / 2} (\partial Ω)} ⩽ C \sum_{m} {‖ \partial_{t t t} (a_{i}^{m} v^{i}) ‖}_{L^{2} (Ω)} \end{matrix}$ the last inequality following from $\partial_{m} (a_{i}^{m} v^{i}) = 0$ . Therefore, $\begin{array}{l} I_{1} + I_{2} + I_{3} + I_{4} \\ ⩽ C {‖ {(a^{T} a)}_{t t} \nabla q ‖}_{L^{2}} + C {‖ {(a^{T} a)}_{t} \nabla q_{t} ‖}_{L^{2}} + C ‖ a_{t t t} v ‖_{L^{2}} \\ + C ‖ a_{t t} v_{t} ‖_{L^{2}} + C ‖ a_{t} v_{t t} ‖_{L^{2}} + C ‖ a v_{t t t} ‖_{L^{2}} \\ ⩽ C (‖ \nabla v ‖_{H^{0.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ \nabla v_{t} ‖_{H^{0.5}}) ‖ \nabla q ‖_{H^{1}} + C ‖ \nabla v ‖_{L^{\infty}} ‖ \nabla q_{t} ‖_{L^{2}} \\ + ‖ a_{t t t} ‖_{L^{3}} ‖ v ‖_{L^{6}} + C (‖ \nabla v ‖_{H^{0.5}} ‖ \nabla v ‖_{L^{\infty}} + ‖ \nabla v_{t} ‖_{H^{0.5}}) ‖ v_{t} ‖_{H^{1}} \\ (3.58) & + C ‖ \nabla v ‖_{L^{\infty}} ‖ v_{t t} ‖_{L^{2}} + C ‖ v_{t t t} ‖_{L^{2}} . \end{array}$ The third term on the far right side is then estimated as $\begin{array}{l} (3.59) & ‖ a_{t t t} ‖_{L^{3}} ‖ v ‖_{L^{6}} ⩽ C (‖ \nabla v ‖_{L^{3}} ‖ \nabla v ‖_{L^{\infty}}^{2} + ‖ \nabla v_{t} ‖_{L^{3}} ‖ \nabla v ‖_{L^{\infty}} + ‖ \nabla v_{t t} ‖_{L^{3}}) ‖ v ‖_{H^{1}} \end{array}$ and (3.33) follows. □

4. Local in time solutions

4.1. $L^{2}$ estimate on $v_{t t t}$

Applying $\partial_{t}^{3}$ to (2.4), multiplying the resulting equation by $v_{t t t}$ , and integrating in space and time gives $\begin{array}{l} (4.1) & \frac{1}{2} {‖ v_{t t t} (t) ‖}_{L^{2}}^{2} = \frac{1}{2} {‖ v_{t t t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int {(a_{i}^{k} \partial_{k} q)}_{t t t} v_{t t t}^{i} = \frac{1}{2} {‖ v_{t t t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int \partial_{k} {(a_{i}^{k} q)}_{t t t} v_{t t t}^{i} \end{array}$ where we utilized the Piola identity $\begin{matrix} (4.2) & \partial_{k} a_{i}^{k} = 0, i = 1, 2, 3 . \end{matrix}$ In order to bound the integral on the right side, we integrate by parts, $\begin{array}{l} (4.3) & - \int_{0}^{t} \int \partial_{k} {(a_{i}^{k} q)}_{t t t} v_{t t t}^{i} = - \int_{0}^{t} \int_{\partial Ω} {(a_{i}^{k} q)}_{t t t} v_{t t t}^{i} N_{k} + \int_{0}^{t} \int {(a_{i}^{k} q)}_{t t t} \partial_{k} v_{t t t}^{i} = I_{1} + I_{2} . \end{array}$ Since $v^{3} = 0$ on $Γ_{0}$ , we have $\bar{\partial} v^{3} = 0$ , where $\begin{matrix} (4.4) & \bar{\partial} = (\partial_{1}, \partial_{2}) \end{matrix}$ and $v_{t t t}^{3} = 0$ on $Γ_{0}$ . Also, $\bar{\partial} η^{3} = 0$ on $Γ_{0}$ , which implies that $a_{1}^{3} = a_{2}^{3} = 0$ on $Γ_{0}$ . As a consequence, $\begin{array}{l} (4.5) & \int_{0}^{t} \int_{Γ_{0}} {(a_{i}^{k} q)}_{t t t} v_{t t t}^{i} N_{k} = \int_{0}^{t} \int_{Γ_{0}} {(a_{i}^{3} q)}_{t t t} v_{t t t}^{i} = 0 . \end{array}$ Thus, for the boundary term in (4.3), we obtain $\begin{array}{l} (4.6) & I_{1} = - \int_{0}^{t} \int_{Γ_{1}} {(a_{i}^{k} q N_{k})}_{t t t} v_{t t t}^{i} = \int_{0}^{t} \int_{Γ_{1}} Δ_{2} η_{t t t}^{i} v_{t t t}^{i} = - \frac{1}{2} {‖ \bar{\partial} v_{t t} (t) ‖}_{L^{2} (Γ_{1})}^{2} + \frac{1}{2} {‖ \bar{\partial} v_{t t} (0) ‖}_{L^{2} (Γ_{1})}^{2} \end{array}$ by using (2.7), (2.11), and integrating by parts in the tangential direction.

Now, we bound the second integral $\begin{array}{l} I_{2} & = \int_{0}^{t} \int a_{i}^{k} q_{t t t} \partial_{k} v_{t t t}^{i} + 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} q_{t t} \partial_{k} v_{t t t}^{i} + 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t t} q_{t} \partial_{k} v_{t t t}^{i} + \int_{0}^{t} \int {(a_{i}^{k})}_{t t t} q \partial_{k} v_{t t t}^{i} \\ (4.7) & = I_{21} + I_{22} + I_{23} + I_{24} . \end{array}$ Using the incompressibility condition to write $\begin{array}{l} a_{i}^{k} \partial_{k} v_{t t t}^{i} & = {(a_{i}^{k} \partial_{k} v^{i})}_{t t t} - 3 {(a_{i}^{k})}_{t} \partial_{k} v_{t t}^{i} - 3 {(a_{i}^{k})}_{t t} \partial_{k} v_{t}^{i} - {(a_{i}^{k})}_{t t t} \partial_{k} v^{i} \\ (4.8) & = - 3 {(a_{i}^{k})}_{t} \partial_{k} v_{t t}^{i} - 3 {(a_{i}^{k})}_{t t} \partial_{k} v_{t}^{i} - {(a_{i}^{k})}_{t t t} \partial_{k} v^{i}, \end{array}$ we get $\begin{array}{l} I_{21} & = - \int_{0}^{t} \int 3 {(a_{i}^{k})}_{t t} \partial_{k} v_{t}^{i} q_{t t t} - \int_{0}^{t} \int 3 {(a_{i}^{k})}_{t} \partial_{k} v_{t t}^{i} q_{t t t} - \int_{0}^{t} \int {(a_{i}^{k})}_{t t t} \partial_{k} v^{i} q_{t t t} \\ (4.9) & = I_{211} + I_{212} + I_{213} . \end{array}$ For $I_{211}$ we integrate by parts in time: $\begin{array}{l} I_{211} = & - 3 \int {(a_{i}^{k})}_{t t} \partial_{k} v_{t}^{i} q_{t t} |_{0}^{t} + 3 \int_{0}^{t} \int \partial_{t} ({(a_{i}^{k})}_{t t} \partial_{k} v_{t}^{i}) q_{t t} \\ ⩽ & C {‖ a_{t t} (0) ‖}_{L^{2}} {‖ \nabla v_{t} (0) ‖}_{L^{3}} {‖ q_{t t} (0) ‖}_{L^{6}} + C {‖ a_{t t} (t) ‖}_{L^{2}} {‖ \nabla v_{t} (t) ‖}_{L^{3}} {‖ q_{t t} (t) ‖}_{L^{6}} \\ + C \int_{0}^{t} ‖ a_{t t t} ‖_{L^{3}} ‖ \nabla v_{t} ‖_{L^{6}} ‖ q_{t t} ‖_{L^{2}} + C \int_{0}^{t} ‖ a_{t t} ‖_{L^{6}} ‖ \nabla v_{t t} ‖_{L^{3}} ‖ q_{t t} ‖_{L^{2}} \\ ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ a_{t t} (t) ‖}_{L^{2}} {‖ \nabla v_{t} (t) ‖}_{L^{3}} {‖ q_{t t} (t) ‖}_{L^{6}} \\ (4.10) & + \int_{0}^{t} P (‖ q_{t t} ‖_{L^{2}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2}}, ‖ v ‖_{H^{3.5}}) . \end{array}$ Integrating by parts $I_{212} = - 3 \int {(a_{i}^{k})}_{t} \partial_{k} v_{t t}^{i} q_{t t t}$ in space, we have $\begin{array}{l} (4.11) & I_{212} = - 3 \int_{0}^{t} \int_{\partial Ω} {(a_{i}^{k})}_{t} v_{t t}^{i} q_{t t t} N_{k} + 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} v_{t t}^{i} \partial_{k} q_{t t t} = I_{2121} + I_{2122} \end{array}$ where we used (4.2). Observe that $\begin{array}{l} I_{2121} = & 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v^{l} a_{l}^{k} v_{t t}^{i} q_{t t t} N_{k} \\ = & - 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v^{l} Δ_{2} η_{t t t}^{l} v_{t t}^{i} \\ (4.12) & - \int_{0}^{t} \int_{Γ_{1}} (3 {(a_{l}^{k})}_{t} q_{t t} N_{k} + 3 {(a_{l}^{k})}_{t t} q_{t} N_{k} + {(a_{l}^{k})}_{t t t} q N_{k}) a_{i}^{j} \partial_{j} v^{l} v_{t t}^{i} \end{array}$ by using (2.6) in the first and (2.11) in the second equality; also note that the integral over $Γ_{0}$ vanishes. Integrating by parts in the tangential directions, we obtain $\begin{array}{l} - 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v^{l} Δ_{2} η_{t t t}^{l} v_{t t}^{i} & = 3 \sum_{k = 1}^{2} \int_{0}^{t} \int \partial_{k} η_{t t t}^{l} \partial_{k} (a_{i}^{j} \partial_{j} v^{l} v_{t t}^{i}) \\ (4.13) & ⩽ C \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ v ‖_{H^{3}}), \end{array}$ while the lower order terms are bounded as $\begin{array}{l} - \int_{0}^{t} \int_{Γ_{1}} (3 {(a_{l}^{k})}_{t} q_{t t} N_{k} + 3 {(a_{l}^{k})}_{t t} q_{t} N_{k} + {(a_{l}^{k})}_{t t t} q N_{k}) a_{i}^{j} \partial_{j} v^{l} v_{t t}^{i} \\ (4.14) & ⩽ \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v ‖_{H^{3.5}}, ‖ q_{t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2.5}}) . \end{array}$ Next, integrating by parts in time gives $\begin{array}{l} I_{2122} = & 3 \int {(a_{i}^{k})}_{t} v_{t t}^{i} \partial_{k} q_{t t} |_{0}^{t} - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t t} v_{t t}^{i} \partial_{k} q_{t t} - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} v_{t t t}^{i} \partial_{k} q_{t t} \\ ⩽ & P (‖ v_{0} ‖_{H^{3.5}}) + C {‖ q_{t t} (t) ‖}_{H^{1}} {‖ v_{t t} (t) ‖}_{L^{3}} {‖ \nabla v (t) ‖}_{L^{6}} \\ (4.15) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{1.5}}, ‖ v ‖_{H^{3}}) . \end{array}$ By (2.6) and integrating by parts in time we have $\begin{array}{l} (4.16) & I_{213} & = \int_{0}^{t} \int a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i} q_{t t t} + R . \end{array}$ Until the end of this paper, we denote by R the remainder terms. In (4.16), the lower order terms are of the form $\begin{array}{l} (4.17) & \int_{0}^{t} \int (a_{t t} \nabla v a + a_{t} \nabla v a_{t} + a_{t} \nabla v_{t} a) \nabla v q_{t t t}, \end{array}$ (written in a symbolic way, omitting all the indices) which can be bounded by $\begin{array}{l} R ⩽ & P (‖ v_{0} ‖_{H^{3.5}}) + P (‖ v (t) ‖ e_{H^{3}}, {‖ q_{t t} (t) ‖}_{H^{1}}) {‖ v_{t} (t) ‖}_{H^{1.5}} \\ (4.18) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v ‖_{H^{3}}, ‖ v_{t} ‖_{H^{1.5}}, ‖ v_{t t} ‖_{H^{1.5}}) d t \end{array}$ after integrating by parts in time. Integrating by parts in space, the leading term of (4.16) becomes $\begin{array}{l} \int_{0}^{t} \int a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i} q_{t t t} \\ = - \int_{0}^{t} \int a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{j k} v^{i} q_{t t t} - \int_{0}^{t} \int a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i} \partial_{j} q_{t t t} \\ - \int_{0}^{t} \int a_{i}^{j} v_{t t}^{l} \partial_{j} a_{l}^{k} \partial_{k} v^{i} q_{t t t} + \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i} q_{t t t} N_{j} \\ (4.19) & = I_{2131} + I_{2132} + I_{2133} + I_{2134}, \end{array}$ where we have omitted the term when the j-th derivatives fall on $a_{i}^{j}$ which equals to zero by (4.2). First, observe that the boundary term $I_{2134}$ can be treated exactly as $I_{2121}$ above. Now, using $a_{i}^{j} \partial_{j k} v^{i} = - \partial_{k} a_{i}^{j} \partial_{j} v^{i}$ for $k = 1, 2, 3$ , we write $\begin{array}{l} I_{2131} = & - \int_{0}^{t} \int \partial_{k} a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{j} v^{i} q_{t t t} \\ = & - \int \partial_{k} a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{j} v^{i} q_{t t} |_{0}^{t} + \int_{0}^{t} \int {(\partial_{k} a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{j} v^{i})}_{t} q_{t t} \\ ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ q_{t t} (t) ‖}_{L^{6}} {‖ v_{t t} (t) ‖}_{L^{2}} {‖ \nabla v (t) ‖}_{L^{3}}, \\ (4.20) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{1.5}}, ‖ v ‖_{H^{3}}), \end{array}$ while $\begin{array}{l} I_{2132} = & - \int a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i} \partial_{j} q_{t t} |_{0}^{t} + \int_{0}^{t} \int {(a_{i}^{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v^{i})}_{t} \partial_{j} q_{t t} \\ ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ q_{t t} (t) ‖}_{H^{1}} {‖ v_{t t} (t) ‖}_{L^{3}} {‖ \nabla v (t) ‖}_{L^{6}} \\ (4.21) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{1.5}}, ‖ v ‖_{H^{3}}) . \end{array}$ Note that the lower order term $I_{2133}$ is also bounded by the right side of (4.21). For $I_{22}$ we integrate by parts in space $\begin{array}{l} I_{22} & = 3 \int_{0}^{t} \int_{Γ_{1}} \partial_{t} a_{i}^{k} q_{t t} v_{t t t}^{i} N_{k} - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} \partial_{k} q_{t t} v_{t t t}^{i} \\ = - 3 \int_{0}^{t} \int_{Γ_{1}} (a_{i}^{j} \partial_{j} v^{l} a_{l}^{k}) q_{t t} v_{t t t}^{i} N_{k} - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} \partial_{k} q_{t t} v_{t t t}^{i} \\ = 3 \int_{0}^{t} \int_{Γ_{1}} (a_{i}^{j} \partial_{j} v^{l} Δ_{2} η_{t t}^{l} v_{t t t}^{i} + a_{i}^{j} \partial_{j} v^{l} ({(a_{l}^{k})}_{t t} q + 2 {(a_{l}^{k})}_{t} q_{t}) N_{k} v_{t t t}^{i}) - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t} \partial_{k} q_{t t} v_{t t t}^{i} \\ (4.22) & = I_{221} + I_{222} . \end{array}$ We denote the first boundary term in $I_{221}$ by $I_{2211}$ . The other two terms in $I_{221}$ are easy to bound. Integrating by parts in the tangential directions, we get $\begin{array}{l} (4.23) & I_{2211} = - 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v^{l} \bar{\partial} v_{t}^{l} \bar{\partial} v_{t t t}^{i} - 3 \int_{0}^{t} \int_{Γ_{1}} \bar{\partial} (a_{i}^{j} \partial_{j} v^{l}) \bar{\partial} v_{t}^{l} v_{t t t}^{i} \end{array}$ which after an additional integration by parts in time leads to $\begin{array}{l} I_{2211} = & - 3 \int_{Γ_{1}} (a_{i}^{j} \partial_{j} v^{l} \bar{\partial} v_{t}^{l} \bar{\partial} v_{t t}^{i} + \bar{\partial} (a_{i}^{j} \partial_{j} v^{l}) \bar{\partial} v_{t}^{l} v_{t t}^{i}) |_{0}^{t} \\ (4.24) & + 3 \int_{0}^{t} \int_{Γ_{1}} {(a_{i}^{j} \partial_{j} v^{l} \bar{\partial} v_{t}^{l})}_{t} \bar{\partial} v_{t t}^{i} + {(\bar{\partial} (a_{i}^{j} \partial_{j} v^{l}) \bar{\partial} v_{t}^{l})}_{t} v_{t t}^{i} . \end{array}$ Thus, $\begin{array}{l} I_{2211} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ v_{t t} (t) ‖}_{H^{1.5}} {‖ v_{t} (t) ‖}_{H^{2}} {‖ v (t) ‖}_{H^{2.5}} \\ (4.25) & + \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{3}}) . \end{array}$

Next, for $I_{23}$ we proceed as in $I_{22}$ by first integrating by parts in space $\begin{array}{l} I_{23} & = - 3 \int_{0}^{t} \int_{Γ_{1}} {(a_{i}^{j} \partial_{j} v^{l} a_{l}^{k})}_{t} q_{t} v_{t t t}^{i} N_{k} - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t t} \partial_{k} q_{t} v_{t t t}^{i} \\ (4.26) & = 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v_{t}^{l} Δ_{2} η_{t}^{l} v_{t t t}^{i} + R - 3 \int_{0}^{t} \int {(a_{i}^{k})}_{t t} \partial_{k} q_{t} v_{t t t}^{i}, \end{array}$ where the remainder term $\begin{array}{l} R = & - 3 \int_{0}^{t} \int_{Γ_{1}} {(a_{i}^{j})}_{t} \partial_{j} v^{l} a_{l}^{k} q_{t} v_{t t t}^{i} N_{k} - 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v^{l} {(a_{l}^{k})}_{t} q_{t} v_{t t t}^{i} N_{k} \\ (4.27) & + 3 \int_{0}^{t} \int_{Γ_{1}} a_{i}^{j} \partial_{j} v_{t}^{l} {(a_{l}^{k})}_{t} q v_{t t t}^{i} N_{k} \end{array}$ is bounded by $\begin{array}{l} R ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ q_{t} (t) ‖}_{H^{2.5}} {‖ v_{t t} (t) ‖}_{H^{1.5}} {‖ v (t) ‖}_{H^{2.5}}^{2} \\ + C {‖ q (t) ‖}_{H^{1.5}} {‖ v_{t t} (t) ‖}_{H^{1.5}} {‖ v_{t} (t) ‖}_{H^{1.5}} {‖ v (t) ‖}_{H^{2.5}} \\ (4.28) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{1.5}}, ‖ q ‖_{H^{2.5}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{3}}) . \end{array}$ The first boundary term on the far right sides in (4.26) can be bounded similarly as $I_{2211}$ above, by integrating by parts in time. We omit further details.

Lastly, we consider $I_{24}$ . We use that ${(a_{i}^{k})}_{t t t} = {(a_{i}^{j} \partial_{j} v^{l} a_{l}^{k})}_{t t} = a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} + l . o . t .$ , where the lower order terms are of the form $a_{t t} \nabla v a$ , $a_{t} \nabla v_{t} a$ , $a_{t} \nabla v a_{t}$ (and the resulting integrals are clearly easy to bound). Thus, we estimate only the leading term in $I_{24}$ . We have $\begin{array}{l} (4.29) & I_{24} & = \int_{0}^{t} \int a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t t}^{i} q + R \end{array}$ and observe that $\begin{array}{l} \partial_{t} (a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i}) = & a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t t}^{i} + a_{i}^{j} \partial_{j} v_{t t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} \\ + {(a_{i}^{j})}_{t} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} + a_{i}^{j} \partial_{j} v_{t t}^{l} {(a_{l}^{k})}_{t} \partial_{k} v_{t t}^{i} \\ (4.30) & = & 2 a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t t}^{i} + 2 {(a_{i}^{j})}_{t} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} . \end{array}$ Hence, $\begin{array}{l} I_{24} = & \frac{1}{2} \int a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} q |_{0}^{t} - \frac{1}{2} \int_{0}^{t} \int a_{i}^{j} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} q_{t} - \int_{0}^{t} \int {(a_{i}^{j})}_{t} \partial_{j} v_{t t}^{l} a_{l}^{k} \partial_{k} v_{t t}^{i} q + l . o . t . \\ ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C {‖ v_{t t} (t) ‖}_{H^{1.5}} {‖ v_{t t} (t) ‖}_{H^{1}} {‖ q (t) ‖}_{H^{1}} \\ (4.31) & + \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{2}}, ‖ v ‖_{H^{3}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2}}) . \end{array}$ Therefore, we conclude $\begin{array}{l} {‖ v_{t t t} (t) ‖}_{L^{2}}^{2} + {‖ \bar{\partial} v_{t t} (t) ‖}_{L^{2} (Γ_{1})}^{2} \\ ⩽ P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + ϵ ({‖ v_{t t} (t) ‖}_{H^{1.5}}^{2} + {‖ q_{t t} (t) ‖}_{H^{1}}^{2}) \\ (4.32) & + \int_{0}^{t} P (‖ v_{t t t} ‖_{L^{2}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{3}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2}}) . \end{array}$

4.2. Tangential $H^{1}$ estimate on $v_{t t}$

Applying $\partial_{m} \partial_{t}^{2}$ to the equation (2.4), multiplying by $\partial_{m} v_{t t}$ , summing for $m = 1, 2$ , and integrating in space and time, we get $\begin{matrix} (4.33) & \begin{matrix} \frac{1}{2} {‖ \bar{\partial} v_{t t} (t) ‖}_{L^{2}}^{2} & = \frac{1}{2} {‖ \bar{\partial} v_{t t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int \partial_{m} {(a_{i}^{k} \partial_{k} q)}_{t t} \partial_{m} v_{t t}^{i} \\ = \frac{1}{2} {‖ \bar{\partial} v_{t t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int a_{i}^{k} \partial_{k m} q_{t t} \partial_{m} v_{t t}^{i} + R . \end{matrix} \end{matrix}$ 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 $\begin{array}{l} R & = - \int_{0}^{t} \int (\partial_{m} {(a_{i}^{k})}_{t t} \partial_{k} q + {(a_{i}^{k})}_{t t} \partial_{k m} q + 2 \partial_{m} {(a_{i}^{k})}_{t} \partial_{k} q_{t} + 2 {(a_{i}^{k})}_{t} \partial_{k m} q_{t} + \partial_{m} a_{i}^{k} \partial_{k} q_{t t}) \partial_{m} v_{t t}^{i} \\ (4.34) & ⩽ \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{3}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2.5}}) . \end{array}$ Now, we integrate by parts in the higher order term $\begin{array}{l} (4.35) & - \int_{0}^{t} \int a_{i}^{k} \partial_{k m} q_{t t} \partial_{m} v_{t t}^{i} = - \int_{0}^{t} \int_{\partial Ω} a_{i}^{k} \partial_{m} q_{t t} \partial_{m} v_{t t}^{i} N_{k} + \int_{0}^{t} \int a_{i}^{k} \partial_{m} q_{t t} \partial_{k m} v_{t t}^{i} = I_{1} + I_{2} . \end{array}$ For $I_{1}$ , the integral over $Γ_{0}$ vanishes, while on $Γ_{1}$ we use $\begin{array}{l} (4.36) & a_{i}^{k} \partial_{m} q_{t t} N_{k} = \partial_{m} \partial_{t t} (a_{i}^{k} q N_{k}) - \partial_{m} ({(a_{i}^{k})}_{t t} q N_{k}) - 2 \partial_{m} ({(a_{i}^{k})}_{t} q_{t} N_{k}) - \partial_{m} a_{i}^{k} q_{t t} N_{k} \end{array}$ (to check this, write $\partial_{m} (a_{i}^{k} q_{t t} N_{k}) = a_{i}^{k} \partial_{m} q_{t t} N_{k} + \partial_{m} a_{i}^{k} q_{t t} N_{k}$ and rewrite the second term) and get $\begin{array}{l} I_{1} = & \int_{0}^{t} \int_{Γ_{1}} Δ_{2} \partial_{m} η_{t t} \partial_{m} v_{t t}^{i} + \int_{0}^{t} \int_{Γ_{1}} (\partial_{m} ({(a_{i}^{k})}_{t t} q) + 2 \partial_{m} ({(a_{i}^{k})}_{t} q_{t}) + \partial_{m} a_{i}^{k} q_{t t}) \partial_{m} v_{t t}^{i} N_{k} \\ = & - \frac{1}{2} {‖ {\bar{\partial}}^{2} v_{t} (t) ‖}_{L^{2} (Γ_{1})}^{2} + \frac{1}{2} {‖ {\bar{\partial}}^{2} v_{t} (0) ‖}_{L^{2} (Γ_{1})}^{2} \\ (4.37) & + \int_{0}^{t} \int_{Γ_{1}} (\partial_{m} ({(a_{i}^{k})}_{t t} q) + 2 \partial_{m} ({(a_{i}^{k})}_{t} q_{t}) + \partial_{m} a_{i}^{k} q_{t t}) \partial_{m} v_{t t}^{i} N_{k}, \end{array}$ and the last term on the right side can be bounded by $\begin{array}{l} (4.38) & \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q ‖_{H^{2.5}}, ‖ v ‖_{H^{3.5}}) . \end{array}$ For $I_{2}$ , we use the divergence free condition to write $\begin{array}{l} (4.39) & a_{i}^{k} \partial_{k m} v_{t t}^{i} = - \partial_{m} a_{i}^{k} \partial_{k} v_{t t}^{i} - \partial_{m} (2 {(a_{i}^{k})}_{t} \partial_{k} v_{t}^{i} + {(a_{i}^{k})}_{t t} \partial_{k} v^{i}) . \end{array}$ Thus, we obtain $\begin{array}{l} I_{2} & = - \int_{0}^{t} \int (\partial_{m} a_{i}^{k} \partial_{k} v_{t t}^{i} + \partial_{m} (2 {(a_{i}^{k})}_{t} \partial_{k} v_{t}^{i} + {(a_{i}^{k})}_{t t} \partial_{k} v^{i})) \partial_{m} q_{t t} \\ (4.40) & ⩽ \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2}}, ‖ v ‖_{H^{3}}) . \end{array}$ We conclude $\begin{array}{l} {‖ \bar{\partial} v_{t t} (t) ‖}_{L^{2}}^{2} + {‖ {\bar{\partial}}^{2} v_{t} (t) ‖}_{L^{2} (Γ_{1})}^{2} \\ (4.41) & ⩽ {‖ {\bar{\partial}}^{2} v_{t} (0) ‖}_{L^{2} (Γ_{1})}^{2} + \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{2}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q ‖_{H^{2.5}}, ‖ v ‖_{H^{3.5}}) . \end{array}$

4.3. Tangential $H^{2}$ estimate on $v_{t}$

Applying $\partial_{l m} \partial_{t}$ to (2.4), multiplying by $\partial_{l m} v_{t}$ , summing for $l, m = 1, 2$ , and integrating in space and time, we get $\begin{array}{l} \frac{1}{2} {‖ {\bar{\partial}}^{2} v_{t} (t) ‖}_{L^{2}}^{2} & = \frac{1}{2} {‖ {\bar{\partial}}^{2} v_{t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int \partial_{l m} {(a_{i}^{k} \partial_{k} q)}_{t} \partial_{l m} v_{t}^{i} \\ (4.42) & = \frac{1}{2} {‖ {\bar{\partial}}^{2} v_{t} (0) ‖}_{L^{2}}^{2} - \int_{0}^{t} \int a_{i}^{k} \partial_{k l m} q_{t} \partial_{l m} v_{t}^{i} - l . o . t ., \end{array}$ where the lower order terms on the right are bounded by $\int_{0}^{t} P (‖ v_{t} ‖_{H^{2}}, ‖ q ‖_{H^{3}}, ‖ v ‖_{H^{3.5}}, ‖ q_{t} ‖_{H^{2.5}})$ . Next, integrating by parts, we get similarly as in the previous section $\begin{array}{l} (4.43) & - \int_{0}^{t} \int a_{i}^{k} \partial_{k l m} q_{t} \partial_{l m} v_{t}^{i} = - \int_{0}^{t} \int_{\partial Ω} a_{i}^{k} \partial_{l m} q_{t} \partial_{l m} v_{t}^{i} N_{k} + \int_{0}^{t} \int a_{i}^{k} \partial_{l m} q_{t} \partial_{k l m} v_{t}^{i} = I_{1} + I_{2}, \end{array}$ where $\begin{array}{l} I_{1} & = \int_{0}^{t} \int_{\partial Ω} Δ_{2} \partial_{l m} η_{t} \partial_{l m} v_{t}^{i} - l . o . t . \\ (4.44) & = - \frac{1}{2} {‖ {\bar{\partial}}^{3} v (t) ‖}_{L^{2} (Γ_{1})}^{2} + \frac{1}{2} {‖ {\bar{\partial}}^{3} v (0) ‖}_{L^{2} (Γ_{1})}^{2} - l . o . t . \end{array}$ and, by using the divergence free condition, $\begin{array}{l} (4.45) & I_{2} ⩽ \int_{0}^{t} P (‖ q_{t} ‖_{H^{2}}, ‖ v_{t} ‖_{H^{2}}, ‖ v ‖_{H^{3}}) . \end{array}$ Therefore, we conclude $\begin{matrix} (4.46) & \begin{matrix} {‖ {\bar{\partial}}^{2} v_{t} (t) ‖}_{L^{2}}^{2} + {‖ {\bar{\partial}}^{3} v (t) ‖}_{L^{2} (Γ_{1})}^{2} \\ ⩽ {‖ {\bar{\partial}}^{2} v_{t} (0) ‖}_{L^{2}}^{2} + {‖ {\bar{\partial}}^{3} v (0) ‖}_{L^{2} (Γ_{1})}^{2} + \int_{0}^{t} P (‖ v_{t} ‖_{H^{2}}, ‖ q ‖_{H^{3}}, ‖ v ‖_{H^{3.5}}, ‖ q_{t} ‖_{H^{2.5}}) . \end{matrix} \end{matrix}$

4.4. Div-curl estimates

We use the elliptic estimate (cf. [12,17]) $\begin{array}{l} (4.47) & ‖ f ‖_{H^{s}} ⩽ C ‖ f ‖_{L^{2}} + C ‖ curl f ‖_{H^{s - 1}} + C ‖ div f ‖_{H^{s - 1}} + C ‖ f \cdot N ‖_{H^{s - 0.5} (\partial Ω)} \end{array}$ for $s ⩾ 1$ .

We recall that $\bar{\partial} v_{t t} \in L^{2} (Γ_{1})$ , so in particular $v_{t t}^{3} \in H^{1} (Γ_{1})$ . By (4.47) with $s = 1.5$ , we have $\begin{array}{l} (4.48) & ‖ v_{t t} ‖_{H^{1.5}} ⩽ C ‖ v_{t t} ‖_{L^{2}} + C ‖ curl v_{t t} ‖_{H^{0.5}} + C ‖ div v_{t t} ‖_{H^{0.5}} + C ‖ v_{t t}^{3} ‖_{H^{1} (Γ_{1})}, \end{array}$ where we also used $v_{t t}^{3} = 0$ on $Γ_{0}$ . Similarly, applying (4.47) with $s = 2.5$ and $s = 3.5$ respectively, we have $\begin{array}{l} (4.49) & ‖ v_{t} ‖_{H^{2.5}} ⩽ C ‖ v_{t} ‖_{L^{2}} + C ‖ curl v_{t} ‖_{H^{1.5}} + C ‖ div v_{t} ‖_{H^{1.5}} + C {‖ v_{t}^{3} ‖}_{H^{2} (Γ_{1})} \end{array}$ and $\begin{array}{l} (4.50) & ‖ v ‖_{H^{3.5}} ⩽ C ‖ v ‖_{L^{2}} + C ‖ curl v ‖_{H^{2.5}} + C ‖ div v ‖_{H^{2.5}} + C {‖ v^{3} ‖}_{H^{3} (Γ_{1})} . \end{array}$ 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 $\begin{array}{l} (4.51) & {‖ v_{t t} (t) ‖}_{L^{2}} = {‖ v_{t t} (0) ‖}_{L^{2}} + t^{1 / 2} \int_{0}^{t} ‖ v_{t t t} ‖_{L^{2}} . \end{array}$ By the multiplicative Sobolev inequality, for $div v$ we have $\begin{array}{l} (4.52) & ‖ div v ‖_{H^{2.5}} = {‖ (δ_{i k} - a_{i}^{k}) \partial_{k} v^{i} ‖}_{H^{2.5}} ⩽ ϵ ‖ v ‖_{H^{3.5}}, \end{array}$ as well as $\begin{array}{l} (4.53) & ‖ div v_{t} ‖_{H^{1.5}} = {‖ (δ_{i k} - a_{i}^{k}) \partial_{k} v_{t}^{i} - {(a_{i}^{k})}_{t} \partial_{k} v^{i} ‖}_{H^{1.5}} ⩽ ϵ ‖ v_{t} ‖_{H^{2.5}} + C ‖ a_{t} ‖_{H^{1.5 + δ}} ‖ v ‖_{H^{2.5}} \end{array}$ and $\begin{array}{l} ‖ div v_{t t} ‖_{H^{0.5}} & = {‖ (δ_{i k} - a_{i}^{k}) \partial_{k} v_{t t}^{i} - 2 {(a_{i}^{k})}_{t} \partial_{k} v_{t}^{i} - {(a_{i}^{k})}_{t t} \partial_{k} v^{i} ‖}_{H^{0.5}} \\ (4.54) & ⩽ ϵ ‖ v_{t t} ‖_{H^{1.5}} + C ‖ a_{t} ‖_{H^{1.5 + δ}} ‖ v_{t} ‖_{H^{1.5}} + C ‖ a_{t t} ‖_{H^{0.5}} ‖ v ‖_{H^{2.5 + δ}} . \end{array}$

Recall the Cauchy invariance (cf. [32] for instance) $\begin{array}{l} (4.55) & ϵ_{i j k} \partial_{j} v^{l} \partial_{k} η^{l} = curl v_{0}^{i}, i = 1, 2, 3, \end{array}$ for $t ⩾ 0$ , where $ϵ_{i j k}$ is the antisymmetric tensor defined by $ϵ_{123} = 1$ with $ϵ_{i j k} = - ϵ_{j i k}$ and $ϵ_{i j k} = ϵ_{j k i}$ . Thus, we have $\begin{array}{l} (4.56) & {(curl v)}^{i} = ϵ_{i j k} \partial_{j} v^{k} = ϵ_{i j k} \partial_{j} v^{l} (δ_{l k} - \partial_{k} η^{l}) + curl v_{0}^{i}, \end{array}$ where $\begin{array}{l} (4.57) & δ_{l k} - \partial_{k} η^{l} = - \int_{0}^{t} \partial_{k} η_{t}^{l} = - \int_{0}^{t} \partial_{k} v^{l}, k, l = 1, 2, 3, \end{array}$ which implies $\begin{array}{l} (4.58) & ‖ curl v ‖_{H^{2.5}} ⩽ C ‖ v ‖_{H^{3.5}} \int_{0}^{t} ‖ v ‖_{H^{3.5}} + ‖ curl v_{0} ‖_{H^{2.5}} . \end{array}$ Differentiating (4.55) in time, we have $\begin{array}{l} (4.59) & 0 = {(ϵ_{i j k} \partial_{j} v^{l} \partial_{k} η^{l})}_{t} = ϵ_{i j k} \partial_{j} v_{t}^{l} \partial_{k} η^{l} + ϵ_{i j k} \partial_{j} v^{l} \partial_{k} η_{t}^{l}, \end{array}$ where the second term on the right vanishes because it is equal to $ϵ_{i j k} \partial_{j} v^{l} \partial_{k} v^{l}$ and $ϵ_{i j k} = - ϵ_{i k j}$ . Thus, we also get $\begin{matrix} (4.60) & ϵ_{i j k} \partial_{j} v_{t}^{l} \partial_{k} η^{l} = 0 \end{matrix}$ from where $\begin{array}{l} (4.61) & {(curl v_{t})}^{i} = ϵ_{i j k} \partial_{j} v_{t}^{k} = ϵ_{i j k} \partial_{j} v_{t}^{l} (δ_{l k} - \partial_{k} η^{l}) . \end{array}$ Therefore, $\begin{array}{l} (4.62) & ‖ curl v_{t} ‖_{H^{1.5}} ⩽ C ‖ \nabla v_{t} ‖_{H^{1.5}} ‖ I - \nabla η ‖_{H^{1.5 + δ}} ⩽ C ‖ v_{t} ‖_{H^{2.5}} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}} . \end{array}$ Differentiating (4.60), using (2.7), and rearranging the terms in the equality, we obtain $\begin{array}{l} (4.63) & ϵ_{i j k} \partial_{j} v_{t t}^{l} \partial_{k} η^{l} = - ϵ_{i j k} \partial_{j} v_{t}^{l} \partial_{k} v^{l} . \end{array}$ Then, we may write $\begin{array}{l} (4.64) & {(curl v_{t t})}^{i} = ϵ_{i j k} \partial_{j} v_{t t}^{k} = ϵ_{i j k} \partial_{j} v_{t t}^{l} (δ_{l k} - \partial_{k} η^{l}) - ϵ_{i j k} \partial_{j} v_{t}^{l} \partial_{k} v^{l}, \end{array}$ from where $\begin{array}{l} (4.65) & ‖ curl v_{t t} ‖_{H^{0.5}} ⩽ C ‖ v_{t t} ‖_{H^{1.5}} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}} + C ‖ v_{t} ‖_{H^{1.5}} ‖ v ‖_{H^{2.5 + δ}} . \end{array}$ Now, we gather the div-curl inequalities to obtain Sobolev estimates on v, $v_{t}$ , and $v_{t t}$ . Namely, we have $\begin{array}{l} ‖ v ‖_{H^{3.5}} ⩽ & C ({‖ v (0) ‖}_{L^{2}} + ‖ curl v_{0} ‖_{H^{2.5}}) + t^{1 / 2} \int_{0}^{t} ‖ v_{t} ‖_{L^{2}} \\ (4.66) & + C ‖ v ‖_{H^{3.5}} \int_{0}^{t} ‖ v ‖_{H^{3.5}} + C {‖ v^{3} ‖}_{H^{3} (Γ_{1})} \end{array}$ and $\begin{array}{l} ‖ v_{t} ‖_{H^{2.5}} ⩽ & C {‖ v_{t} (0) ‖}_{L^{2}} + C t^{1 / 2} \int_{0}^{t} ‖ v_{t t} ‖_{L^{2}} \\ (4.67) & + C ‖ v_{t} ‖_{H^{2.5}} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}} + C ‖ a_{t} ‖_{H^{1.5 + δ}} ‖ v ‖_{H^{2.5}} + C {‖ v_{t}^{3} ‖}_{H^{2} (Γ_{1})} . \end{array}$ Finally, $\begin{array}{l} ‖ v_{t t} ‖_{H^{1.5}} ⩽ & C {‖ v_{t t} (0) ‖}_{L^{2}} + C t^{1 / 2} \int ‖ v_{t t t} ‖_{L^{2}} + C ‖ v_{t t} ‖_{H^{1.5}} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}} + C ‖ v_{t} ‖_{H^{1.5}} ‖ v ‖_{H^{2.5 + δ}} \\ (4.68) & + C ‖ a_{t} ‖_{H^{1.5 + δ}} ‖ v_{t} ‖_{H^{1.5}} + C ‖ a_{t t} ‖_{H^{0.5}} ‖ v ‖_{H^{2.5 + δ}} + C {‖ v_{t t}^{3} ‖}_{H^{1} (Γ_{1})} . \end{array}$

5. Closing the estimates

Squaring the estimate (4.66) and using (4.46) for the bound of $‖ v^{3} ‖_{H^{3} (Γ_{1})}$ , we have $\begin{array}{l} ‖ v ‖_{H^{3.5}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C ‖ v ‖_{H^{3.5}}^{2} \int_{0}^{t} ‖ v ‖_{H^{3.5}}^{2} \\ (5.1) & + \int_{0}^{t} P (‖ v_{t} ‖_{H^{2}}, ‖ q ‖_{H^{3}}, ‖ v ‖_{H^{3.5}}, ‖ q_{t} ‖_{H^{2.5}}) . \end{array}$ By the pressure estimate (3.31), $\begin{array}{l} (5.2) & ‖ q ‖_{H^{3.5}} ⩽ C (‖ v ‖_{H^{3.5}} + 1) (‖ v_{0} ‖_{H^{2.5}} + C \int_{0}^{t} ‖ v_{t} ‖_{H^{2.5}}) + C ‖ v_{t} ‖_{H^{2} (Γ_{1})} . \end{array}$ This, combined with (4.41) for the bound of $‖ v_{t} ‖_{H^{2} (Γ_{1})}$ , gives $\begin{array}{l} ‖ q ‖_{H^{3.5}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C ‖ v ‖_{H^{3.5}}^{2} (‖ v_{0} ‖_{H^{2.5}}^{2} + C \int_{0}^{t} ‖ v_{t} ‖_{H^{2.5}}^{2}) \\ (5.3) & + \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{2}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q ‖_{H^{2.5}}, ‖ v ‖_{H^{3.5}}) . \end{array}$ Similarly, squaring (4.67) and using (4.41), $\begin{array}{l} ‖ v_{t} ‖_{H^{2.5}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) \\ + C ‖ v_{t} ‖_{H^{2.5}}^{2} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}}^{2} + C ‖ v ‖_{H^{2.5 + δ}}^{2} (‖ v_{0} ‖_{H^{2.5}}^{2} + \int_{0}^{t} ‖ v_{t} ‖_{H^{2.5}}^{2}) \\ (5.4) & + \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ q_{t t} ‖_{H^{1}}, ‖ q_{t} ‖_{H^{2}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q ‖_{H^{2.5}}, ‖ v ‖_{H^{3.5}}), \end{array}$ while combining the square of the estimate (3.32), $\begin{array}{l} ‖ q_{t} ‖_{H^{2.5}}^{2} ⩽ & C ‖ v ‖_{H^{2.5 + δ}}^{2} (P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C \int_{0}^{t} (‖ v_{t t} ‖_{H^{1.5}}^{2} + ‖ q_{t} ‖_{H^{2.5}}^{2})) \\ + C ‖ v ‖_{H^{2.5}}^{2} + C ‖ v_{t} ‖_{H^{2.5}}^{2} \\ (5.5) & + C (‖ v ‖_{H^{2.5 + δ}}^{4} + ‖ v_{t} ‖_{H^{2.5}}^{2}) (‖ v_{0} ‖_{H^{1.5 + δ}}^{2} + \int_{0}^{t} ‖ v_{t} ‖_{H^{1.5 + δ}}^{2}) + C ‖ v_{t t} ‖_{H^{1} (Γ_{1})}^{2} \end{array}$ (4.32) and (5.4), we obtain $\begin{array}{l} ‖ q_{t} ‖_{H^{2.5}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C ϵ (‖ v_{t t} ‖_{H^{1.5}}^{2} + ‖ q_{t t} ‖_{H^{1}}^{2} + ‖ v ‖_{H^{3.5}}^{2}) \\ + P (‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{2.5 + δ}}) \int_{0}^{t} P (‖ v ‖_{H^{2.5 + δ}}, ‖ v_{t} ‖_{H^{2.5}}) \\ (5.6) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v ‖_{H^{3.5}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2}}) . \end{array}$ Lastly, squaring (4.68) and using (4.32), $\begin{array}{l} ‖ v_{t t} ‖_{H^{1.5}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C ϵ (‖ v_{t t} ‖_{H^{1.5}}^{2} + ‖ q_{t t} ‖_{H^{1}}^{2} + ‖ v ‖_{H^{3.5}}^{2}) \\ + C ‖ v_{t t} ‖_{H^{1.5}}^{2} \int_{0}^{t} ‖ v ‖_{H^{2.5 + δ}}^{2} + C ‖ v ‖_{H^{2.5 + δ}}^{2} \int_{0}^{t} P (‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}) \\ (5.7) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v ‖_{H^{3.5}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2}}), \end{array}$ while squaring (3.33) and (4.32) give $\begin{array}{l} ‖ q_{t t} ‖_{H^{1}}^{2} ⩽ & P (‖ v_{0} ‖_{H^{3.5}}, ‖ v_{0} ‖_{H^{3.5} (Γ_{1})}) + C ϵ (‖ v_{t t} ‖_{H^{1.5}}^{2} + ‖ q_{t t} ‖_{H^{1}}^{2} + ‖ v ‖_{H^{3.5}}^{2}) \\ + C ‖ v_{t t} ‖_{H^{1.5}}^{2} (‖ v_{0} ‖_{H^{1}}^{2} + \int_{0}^{t} ‖ v_{t} ‖_{H^{1}}^{2}) \\ + P (‖ v ‖_{H^{3.5}}, ‖ v_{t} ‖_{H^{2.5}}) \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ q_{t} ‖_{H^{2}}, ‖ v_{t t} ‖_{H^{1}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ v ‖_{H^{3.5}}) \\ (5.8) & + \int_{0}^{t} P (‖ q_{t t} ‖_{H^{1}}, ‖ v_{t t t} ‖_{L^{2}}, ‖ v ‖_{H^{3.5}}, ‖ v_{t t} ‖_{H^{1.5}}, ‖ v_{t} ‖_{H^{2.5}}, ‖ q_{t} ‖_{H^{2}}, ‖ q ‖_{H^{2}}) . \end{array}$ Combining all the estimates, we obtain a Gronwall type inequality yielding the a priori estimates for the local in time existence.

Footnotes

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 (). We also thank Marcelo Disconzi and the referee for useful suggestions.

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

Abstract

Keywords

1. Introduction

2. The main result

4.1. L 2 estimate on v t t t

4.2. Tangential H 1 estimate on v t t

4.3. Tangential H 2 estimate on v t

4.4. Div-curl estimates

5. Closing the estimates

Footnotes

Acknowledgements

References

4.1. $L^{2}$ estimate on $v_{t t t}$

4.2. Tangential $H^{1}$ estimate on $v_{t t}$

4.3. Tangential $H^{2}$ estimate on $v_{t}$