A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to $H^3$, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature, and a new compressible Cauchy invariance.


Introduction
In this paper we derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three space dimensions (Theorem 1.3 below) in the case of a liquid. Our a priori estimates provide bounds for the Lagrangian velocity and Lagrangian density in H 3 , an improvement in regularity as compared to [27].
The compressible free-boundary Euler equations in a domain of R 3 are given by ∂u ∂t Above, the quantities u = u(t, x), p = p(t, x), ̺ = ̺(t, x) are the velocity, pressure, and density of the fluid; Ω(t) ⊂ R 3 is the moving (i.e., changing over time) domain, which may be written as Ω(t) = η(t)(Ω 0 ), where η is the flow of u; σ is a non-negative constant known as the coefficient of surface tension. Equation (1.1c) is the equation of state, indicating that the pressure is a given function of the density. In (1.1d), H is the mean curvature of the moving (time-dependent) boundary ∂Ω(t); and T ∂D is the tangent bundle of ∂D. The equation (1.1e) means that the boundary ∂Ω(t) moves at a speed equal to the normal component of u. The quantity u 0 is the velocity at time zero, ̺ 0 is the density at time zero, and Ω 0 is the domain at the initial time. The symbol ∇ u is the derivative in the direction of u, often written as u · ∇. The unknowns in (1.1) are u, ̺, and Ω(t). Note that H, T ∂D, and p are functions of the unknowns and, therefore, are not known a priori, and have to be determined alongside a solution to the problem. We focus on the case when σ > 0 and consider the model case when Denoting coordinates on Ω by (x 1 , x 2 , x 3 ), set and Γ 0 = T 2 × {x 3 = 0}, so that ∂Ω = Γ 0 ∪ Γ 1 . The general domain can then be handled as in [68,Remark 4.2]. We assume that the lower boundary does not move, and thus η(t)(Γ 0 ) = Γ 0 , where η is the flow of the vector field u. We introduce the Lagrangian velocity, pressure, and density, respectively, by v(t, x) = u(t, η(t, x)), q(t, x) = p(t, η(t, x)), and R(t, x) = ̺(t, η(t, x)), or more simply v = u • η, q = p • η, and R = ̺ • η. Therefore, Denoting by ∇ the derivative with respect to the spatial variables x, introduce the matrix a = (∇η) −1 , which is well defined for η near the identity. Equation (1.1c) gives q = q(R), i.e., the equation of state written in Lagrangian variables. From a we obtain the cofactor matrix where J = det(∇η). (1.4) As a consequence of these definitions, we have the Piola identity ∂ β A βα = ∂ β (Ja βα ) = 0.
(1.5) (The identity (1.5) can be verified by direct computation using the explicit form of a given in (2.13) below, or cf. [46, p. 462].) Above and throughout we adopt the following agreement.
In terms of v, q, R, and a, the system (1.1) becomes where id is the identity diffeomorphism on Ω, N is the unit outer normal to ∂Ω, a T is the transpose of a, | · | is the Euclidean norm, and ∆ g is the Laplacian of the metric g ij induced on ∂Ω(t) by the embedding η. Explicitly, where · is the Euclidean inner product, and ∆ g (·) = 1 √ g ∂ i ( √ gg ij ∂ j (·)), (1.8) with g the determinant of the matrix (g ij ). In (1.6e), ∆ g η α simply means ∆ g acting on the scalar function η α , for each α = 1, 2, 3; see Lemma 2.4 below for some important identities used to obtain (1.6e).
Since η(0, ·) = id, the initial Lagrangian and Eulerian velocities agree, i.e., v 0 = u 0 . Clearly, v 0 is orthogonal to Γ 0 in view of (1.6f). Note that a(0, ·) = I, (1.9) where I is the identity matrix, in light of (1.6g). It also follows from the above definitions that J satisfies Physically, the equation of state has to satisfy q ′ (R) > 0 (pressure cannot decrease with an increase in density). Mathematically, this assumption guarantees the coercivity of the kinetic term for R in the energy. Here, we shall adopt a slightly more restrictive equation of state that allows us to simplify the estimates. We assume there exists a constant A q > 0 such that for all R in a certain interval [a, b], we have By Lemma 2.1(x), the first condition follows from the second if we allow A q to be decreased if necessary. Importantly, the condition (1.12) is satisfied for equations of state of the form q(R) = αR 1+γ , where α > 0 and γ > 0 are constants (with further assumptions on the constants and the range of R, (1.12) is also satisfied by q(R) = αR 1+γ + β, β > 0). Notation 1.2. Sobolev spaces are denoted by H s (Ω) (or simply by H s when no confusion can arise), with the corresponding norm denoted by · s ; note that · 0 refers to the L 2 norm. We denote by H s (∂Ω) the Sobolev space of maps defined on ∂Ω, with the corresponding norm · s,∂ , and similarly the space H s (Γ 1 ) with the norm · s,Γ 1 . The L p norms on Ω and Γ 1 are denoted by · L p (Ω) and · L p (Γ 1 ) or · L p when no confusion can arise. We use ↾ to denote restriction, and ∆ is the Euclidean Laplacian in Ω.
We now state our main result. Theorem 1.3. Let Ω be as described above and let σ > 0 in (1.6). Let v 0 be a smooth vector field on Ω, and ̺ 0 a smooth positive function on Ω bounded away from zero from below. Let q : (0, ∞) → (0, ∞) be a smooth function satisfying (1.12), in a neighborhood of ̺ 0 . Then, there exist a T * > 0 and a constant C * , depending only on such that any smooth solution (v, R) to (1.6) with initial condition (v 0 , ̺ 0 ) and defined on the time The dependence of T * and C * on a higher norm on the boundary Γ 1 comes from the usual problems caused by the moving boundary in free-boundary problems. The technical difficulties leading to the necessity of including such higher norm are similar to those in [56] (see Section 3.3 and Remark 3.7 below). The assumption on (∆ div v 0 )↾Γ 1 is technical. It can be understood as a consequence of the fact that our techniques generalize methods previously applied to incompressible fluids in [42], where of course the condition is immediately satisfied as div v 0 = 0 then. A regularity condition on the normal derivatives of the normal component of v 0 would suffice, but the assumption on (∆ div v 0 )↾Γ 1 is simpler to state.
Without attempting to be exhaustive, we now briefly review the literature on problem (1.6), and it is instructive to first recall some results for the incompressible free-boundary Euler equations.
Although we are concerned here with σ > 0, it is worth mentioning that the free-boundary Euler equations behave differently for σ = 0 and σ > 0. In view of a counter-example to well-posedness by Ebin [45], an extra condition (known as Taylor sign condition in the incompressible case), has to be imposed when σ = 0. However, it seems more difficult to obtain local existence in lower regularity spaces when σ > 0 compared to σ = 0 due to the presence of two space derivatives of η on the free boundary.
In this work we restricted ourselves to derive a priori estimates, hence a solution is assumed to be given. Therefore, there is no need to state compatibility conditions for the initial data. But we remind the reader that such conditions are necessary for construction of solutions. We also note that in our setting, compatibility conditions will be different on Γ 1 and on Γ 0 (see, e.g., [27], for the compatibility conditions on Γ 1 , and [41] for those on Γ 0 ). Assumption 1.4. For the rest of the paper, we work under the assumptions of Theorem 1.3 and denote by (v, q) a smooth solution to (1.6). We also assume that Ω, Γ 1 , and Γ 0 are as described above.
1.1. Strategy and organization of the paper. The paper is organized as follows. Theorem 1.3 states the main result. Section 2 contains the preliminary estimates of the coefficients and the Lagrangian map. We also introduce the notation used in the rest of the paper. Section 3 contains the energy estimates. First, we start with the energy equality for the third time derivatives (cf. (3.2) below). Special care is required for the boundary integral, which is treated with complete details in Subsection 3.1.4. Two time derivative energy equality is written in (3.37) below, with the estimates given in Section 3.2. We emphasize that the obtained terms are not of lower order as they contain one more space derivative. We also point out that we can not use the H 3 energy equality with no time derivatives, since there is an interior term which can not be treated by the methods from the rest of the paper; instead, we need to rely on the div-curl estimates to obtain control of the H 3 norms of the velocity and the density. Section 4 contains estimates for the curl of the velocity; the main building block is a new Cauchy invariance formula, generalizing the incompressible version from [56,68]. The conclusion of the proof, where all the bounds are suitably combined, is provided in the last section.
Several of the terms that appear in our energy identities, especially in the case of some boundary integrals, cannot be bounded directly. To control them, we explore the structure of the equations and make frequent use of several geometric identities. These lead to a cancellation of top-order terms, allowing us to close the estimates.

Auxiliary results
In this section we state some preliminary results that are employed in the proof of Theorem 1.3 below.

Notation 2.2.
In the rest of the paper, the symbol C denotes a positive sufficiently large constant. It can vary from expression to expression, but it is always independent of the (v, R). We also write X Y to mean X ≤ CY . The a priori estimates require for T to be sufficiently small so that it satisfies T M ≤ 1/C, where M is an upper bound on the norm of the solution (cf. Lemma 2.1 below). In several estimates it suffices to keep track of the number of derivatives so we write ∂ ℓ to denote any derivative of order ℓ and ∂ ℓ to denote any derivative of order ℓ on the boundary, i.e., with respect to x i . We use upper-case Latin indices to denote x i or t, so ∂ A means ∂ t or ∂ i . Remark 2.3. (Simple lower order estimates and symbolic notation) In the subsequent sections, we use the following consequence of Lemma 2.1. Let Q be a rational function of derivatives of η with respect to x i , More precisely, we are given a map Q : D → R, where D is a domain in R 6 , and consider the composition of Q with D(η↾Γ 1 ), where D means the derivative. Assume that 0 / ∈ D and that (1, 0, 0, 1, 0, 0) ∈ D. Assume that the derivatives of Q belong to H s (D ′ ), where 1 < s ≤ 1.5 and D ′ is some small neighborhood of (1, 0, 0, 1, 0, 0). The application we have in mind is when Q is a combination of the terms √ g and g ij . It is not difficult to check that such terms satisfy the assumptions just stated on Q. In this regard, note that at time zero g is the Euclidean metric on Γ 1 , and that (1, 0, 0, 1, 0, 0) corresponds to D(η(0)↾Γ 1 ).
In what follows it suffices to keep track of the generic form of some expressions so we write Q symbolically as where the terms Q i α (∂η) are also rational function of derivatives of η with respect to x i . Note that Q i α (∂η) are simply the partial derivatives of Q evaluated at ∂η. We write the last equality symbolically as (2.1) For s > 1, we have the estimate where C 1 depends only on s and on the domain Γ 1 . The term Q(∂η) s,Γ 1 can be estimated in terms of the Sobolev norm of the map Q, i.e., Q H s (D) , and the Sobolev norm of ∂η, i.e., ∂η s,Γ 1 . Under the conditions of Lemma 2.1, we have where C 2 depends only on the domain Γ 1 and we used that H 1.5 (Γ 1 ) embeds into C 0 (Γ 1 ). Therefore, if t is very small, we can guarantee that and thus, shrinking D if necessary, we can assume that the derivatives of Q are in H s (D) for 1 < s ≤ 1.5, and, therefore, that Q H s (D) is bounded for s ≤ 1.5. Since Lemma 2.1 also provides a bound for ∂η s,Γ 1 , s ≤ 1.5, we conclude that where C depends only on M , s, and Γ 1 , and provided that t is small enough. The above also shows that We also need some geometric identities that may be known to specialists, but we state them below and provide some of the corresponding proofs for the reader's convenience.
To prove (2.5), we use (2.4) to write Contracting g kl ∂ l η λ a µλ N µ with g mk gives (2.14) Above, the first equality follows because N = (0, 0, 1) (and g mk g kl = δ l m ), the second equality uses (2.13), and the third equality follows upon setting m = 1 and then m = 2 and observing that in each case all the terms cancel out. Thus, contracting (2.14) with g mn , which implies (2.5). Identity (2.6) follows from the fact that Π is a projection operator or, alternatively, by direct computation using (2.5). Identity (2.7) follows from (2.4), (2.12), and the standard formula (see e.g. [51]) In order to prove (2.8), recall that (see e.g. [51]) where Γ k ij are the Christoffel symbols. Recalling (1.7), a direct computation using the definition of the Christoffel symbols gives and (2.8) follows from (2.15) and (2.16). Identity (2.9) is a standard formula for the mean curvature of an embedding into R 3 (see e.g. [51] or [84]).
Identities (2.10) and (2.11) are well-known, but we provide their proofs for the reader's convenience. Denoten = n • η. Since {∂ 1 η, ∂ 2 η,n} are linearly independent, we can write (2.17) Taking the dot product withn we see that b = 0, since ∂ An ·n = 0 in view ofn ·n = 1, and the fact that ∂ i η is tangent to the embedding. Taking the dot product with ∂ 1 η and ∂ 2 η, and using the definition (1.7), we obtain Using ∂ l η · ∂ An = −∂ A ∂ l η ·n (which follows from ∂ l η ·n = 0) to eliminate ∂ An on the right-hand side, solving for a 1 and a 2 , and using the result into (2.17), produces (2.10) when ∂ A = ∂ t and (2.11) when ∂ A = ∂ i .
For future reference, we record the identity which follows from the well-known identities (see e.g. [84]), We will also need the following result about a gain or regularity of the moving boundary.
Notation 2.5. From here on, we use P (·), with indices attached when appropriate, to denote a general polynomial expression of its arguments.
Proposition 2.6. Assume that that conditions of Lemma 2.1 are valid. We have the estimate Proof. We would like to apply elliptic estimates to (1.6e). While we do not know a priori that the coefficients g ij have enough regularity for an application of standard elliptic estimates, we can use improved estimates for coefficients with lower regularity as in [43]. For this, it suffices to check that g ij has small oscillation, in the following sense. Given r > 0 and x ∈ Γ 1 , set We need to verify that there exists R ≤ 1 such that where ρ is sufficiently small. Since g ij ∈ H 1.5 (Γ 1 ), we have g ij ∈ C 0,α (Γ 1 ) with 0 < α < 0.5 fixed. Thus, for y ∈ B r (x), Hence, and we can ensure (2.19). Therefore, the results of [43] imply that . We remark that [43] deals only with Sobolev spaces of integer order, but since the estimates are linear on the norms we can extend them to fractional order Sobolev spaces as well.
Corollary 2.7. Under the same assumptions of Proposition 2.6, Proof. Since g ij involves only tangential derivatives of η, by Proposition 2.6 we have an estimate for g ij in H 2.5 (Γ 1 ). We can thus use elliptic regularity to bootstrap the estimate on η restricted to Γ 1 to H 4.5 (Γ 1 ).
We conclude this section with a compressible version of the Cauchy invariance (see, e.g., [68] for the incompressible case).

20)
for 0 ≤ t < T . Here, ε αβγ is the totally anti-symmetric symbol with ε 123 = 1 and ω 0 is the vorticity at time zero.
Proof. Compute where we used the anti-symmetry of ε αβγ and (1.6a). From a∇η = I, we have where we used again the anti-symmetry of ε αβγ and that a λµ ∂ γ η µ = δ λ γ . Integrating in time yields the result.

Energy estimates
In this section we derive estimates for v, R, v · N , and their time derivatives.
Assumption 3.1. Throughout this section, we suppose that the hypotheses of Lemma 2.1 hold. Therefore, we make frequent use of the conclusions of this lemma without mentioning it every time. The reader is also reminded of (1.2), which is often going to be used without mention as well. We assume further that T is as in part (ix) of that lemma, and that (v, q) are defined on [0, T ).
Notation 3.2. We use ǫ to denote a small positive constant which may vary from expression to expression. Typically, ǫ comes from choosing the time sufficiently small, from Lemma 2.1, or from the Cauchy inequality with epsilon. The important point to keep in mind, which can be easily verified in the expressions containing ǫ, is that once all estimates are obtained, we can fix ǫ to be sufficiently small in order to close the estimates.
where we abbreviate Notation 3.4. We shall use the following abbreviated notation: Three time derivatives. In this section we derive the estimate where we recall that Π is given by (2.5).
3.1.1. Energy identity. We begin by establishing the identity To obtain it, we first multiply (1.6a) by J (replacing α with β), differentiate three times in t, contract with ∂ 3 t v β , and integrate. We obtain Using the Piola identity (1.5) and integrating by parts in ∂ α , we get where we also used (1.11), that R(0) = ̺ 0 , and the fact that the boundary integral vanishes on Γ 0 . Now we write The terms I 2 and I 3 correspond to the first and second terms on the right side of (3.2) respectively. To handle I 1 , we use the density equation (1.6b) to eliminate the spatial derivative: Since The terms I 12 and I 13 give the third and the fourth terms on the right side of (3.2). For I 11 , we write 3) The first term on the right side leads to the second term on the left side of (3.2), while the second term on the right side of (3.3) gives the last term in (3.2). Denote the terms on the right side of (3.2) by J 1 -J 5 .
3.1.2. Estimate of J 1 , J 3 , J 4 , and J 5 . In this section we estimate J 1 , J 3 , J 4 , and J 5 . We begin with First observe that When the expression in parentheses in (3.4) involving three time derivatives is expanded and one of them canceled, we obtain eight terms, which are all bounded in a similar way. For instance, we have After estimating all the terms in this manner, we obtain Next, we treat the term For the first term in (3.5), we have Using Lemma 2.1(x) as well as the Sobolev and Young's inequalities, we get where we also used and Jensen's inequality. Also, The second term in (3.5), J 32 , is simpler, as we just apply Hölder's inequality and write The term J 4 is treated similarly to J 3 by differentiating by parts in time. Namely, we have (3.6) The pointwise terms are estimated using Hölder and Sobolev inequalities as For the second term J 42 in (3.6), we use Hölder's inequality, yielding Finally, the last term J 5 can be bounded using Hölder's inequality Remark 3.5. (Recurrent estimates of lower order terms) Ideas similar to the above, relying on a combination of Sobolev embedding, Young and Jensen's inequalities, and interpolation, will be used throughout the paper to estimate lower order terms, many times without explicit mention. Before proceeding further, we illustrate in detail how a typical lower order is bounded.
, and using the Cauchy inequality with ǫ, we find Next, choosing p = 2/(1 + 2δ) and q = 2/(1 − 2δ), we apply Young's inequality with ǫ to get where in the second step we chose ǫ ′ so small that C( ǫ)ǫ ′ ≤ ǫ. The fundamental theorem of calculus and Jensen's inequality provide We conclude that for t less than a certain fixed T , we have There is a part of the integral which can not be estimated using integration by parts and Hölder estimates and involves a special cancellation, namely the "tricky" term where, recall, A = Ja. From (2.13), we may write Expanding the index µ in (3.8), we have where L 1 denotes lower order terms, which are all of the form We group the leading terms in (3.9) as T 1 + T 3 , T 4 + T 6 , and T 2 + T 5 . Integrating by parts in time in T 3 , we find where from the first to the second line we relabeled the indices α ↔ λ in the second integral, from the second to the third we used that ǫ λατ = −ǫ αλτ , and from the third to the fourth we observed that the first two integrals cancel each other. The symbol L 2 denotes the lower order terms, which are treated below. We now analyze the term We have where the last integral may be bounded by for small time; we also used q ≤ C by Lemma 2.1(x). For the first integral in (3.11), again by the initial condition, we have that ∂ 3 η 3 = 1 + O( ǫ) and thus where the last integral is also bounded by ǫ ∂ 2 t v 2 1 . For the remaining integral, we expand ǫ αλ3 : − Ω qǫ αλ3 ∂ 1 ∂ 2 after using ǫ 123 = 1 = −ǫ 213 . We integrate by parts the ∂ 2 in the first term and the ∂ 1 in the second term to find where the last integral obeys The symbol L 2 in (3.10), denotes the sum of For the sum of T 1 and T 3 , we conclude The terms T 4 + T 6 and T 2 + T 5 are handled in the same way, with one extra step. In the last step above, we integrated ∂ 1 and ∂ 2 by parts. For T 4 + T 6 we integrate by parts the derivatives ∂ 2 and ∂ 3 ; this last one produces the boundary term Using that ∂η 3 = 0 at t = 0, we may write ∂η 3 = t 0 ∂v 3 to conclude

Thus we have established
Now, we complete the treatment of J 2 by estimating the rest of the terms appearing in (3.7), i.e., to bound the expression which we may rewrite as After time integration, the first integral in (3.13) equals The second term is bounded by t 0 P, while the pointwise term at t = 0 by P 0 . It is easy to check that the pointwise term at t is bounded by (3.14) The second integral in (3.13) is treated the same way, resulting in the bound as in (3.14) but with an additional term ǫ ∂ 3 t R 2 0 .

Estimate of the boundary integral.
We now estimate the boundary integral on the left-hand side of (3.2) or, rather, its time integral, which in view of (1.6e) and (2.7) can be written as where We shall repeatedly use the identity The identity (3.17) follows from (2.15) and (2.16) since and the term inside the parentheses equals Π α µ by (2.5). Using (3.17) and applying the Leibniz rule, we may split 3.1.4.1. Estimate of I 11 . To bound I 11 , integrate by parts in ∂ i and then in t to obtain The first term on the right produces a coercive term, as we may write the second term is absorbed in the first provided T ≤ 1/CM for a sufficiently large C. Thus The term I 112 is rewritten as where we used Π α µ = Π σ µ Π α σ . We have and since by (2.18) we have The term I 1122 is more delicate. First, by Π α µ = Π σ µ Π α σ , we have Therefore, I 1122 may be rewritten as For the second term, we usen t v µ and thus I 11222 is controlled by the right side of (3.18). For I 11221 , we use (recall (2.10)), which gives From the equation (1.6a) for the velocity and the definition of a, we have which we replace in (3.22). The commutators are easily controlled, so we only need to consider the main term where we henceforth adopt: Notation 3.6. We use L = to denote equality modulo lower order terms that can be controlled. Thus, L = in (3.25) indicates the leading term of I 11221 . Now, we integrate by parts in x j , leading to At this point, we use the identity which follows from (1.6e) and (3.17), which after applying ∂ 3 t gives After replacing the first term in I 112211 and I 112212 , resulting terms may be controlled using H 1/2 (Γ 1 )-H −1/2 (Γ 1 ) duality. We illustrate this on the term where both time derivatives hit q, i.e., −(1/σ)(J/ √ g)a µα N µ ∂ 2 t q. After replacing this in I 112212 , we get the term of the form which is estimated by where δ > 0 is a small parameter. Before continuing, it is worthwhile to formalize the (3.21), (3.23), and (3.24) into the identity Also, similarly to (3.21), we have (recall (2.11)) ∂ inµ = −g kl ∂ ik η τn τ ∂ l η µ , whence, as for (3.26), we have Next, we consider where we used Π α µ =n αn µ . The first term I 1131 is of high order and can not be treated directly. It cancels with a term resulting from I 14 further below; cf. (3.34). Using (3.27), we have The first term is easily controlled sincê For the second term in (3.28), we use q = −σ∆ g η αn α , which follows fromn α q = −σ∆ g η α and consequently and we obtain Integrating by parts in x l and then in x i , we get (3.30) The last two integrals cancel by the symmetry property 2 i,j,k,l g ji g kl − g ij g lj = 0 (3.31) (which is true for any matrix); this identity can be proved by writing out eight terms for i, k, l = 1, 2 (keeping j), when some terms are outright zero ((i, k, l) = (1, 2, 3), (1, 2, 3), (2, 1, 2, ), (2, 2, 2)) while (i, k, l) = (1, 1, 2) cancels with (i, k, l) = (2, 1, 1) and (i, k, l) = (1, 2, 2) cancels with (2, 2, 1). Thus we only need to treat the first term in (3.30). Integrating by parts in x i , x j , and then in t, we get It is easy to check that both terms can be controlled. For the first term on the far right, we use that ∂ 3 kl η vanishes at t = 0. This completes the treatment of the term I 11 . 3.1.4.2. Estimate of I 12 and I 13 . The term I 12 is split as = I 121 + I 122 + I 123 + I 124 + I 125 .
All the terms except I 123 are estimated as above. For I 123 , we use (3.20) and obtain The terms are treated as I 11221 and I 11222 respectively. This concludes the treatment of I 12 .
The term I 13 is handled analogously to I 12 , so we omit the details. 3.1.4.3. Estimate of I 14 . For I 14 , we have where we used (3.19) in the last step. The terms I 142 and I 143 are treated with similar methods; here we focus on the high order term I 141 . Since, by (3.21), we have At this point we need the identity which we prove next. First, by (2.18), we have In the second term on the far right side, we relabel m and n and then factor out ∂ 2 im η µ ∂ n η µ . We get im (g ij g mn − g in g jm = 0) due to anti-symmetry in i and m, the identity (3.33) follows. Using (3.33) in (3.32), we get As pointed out above, this term cancels with I 1131 above. As said, the terms I 142 and I 143 are treated with similar ideas as above. We illustrate this by estimating I 143 . Integrating by parts in time where I 143,0 is controlled by P 0 . Let us handle I 1431 . Using (2.18) to write We split I 1431 accordingly, and note I 14311 that can be directly estimated producing For I 14312 , we time differentiate (3.23) and integrate by parts with respect to x k to obtain This produces an estimate for I 1431 and I 1433 is handled along the same lines.
Let us now investigate I 1432 . Taking one further time derivative of (3.35) and using the resulting expression into I 1432 , we see that the top term is With the help of (3.23), we have ij η µ , we observe that the first term cancels by (3.31). Writing now Π α µ =n αn µ and invoking (3.29), we see that the resulting integral is estimated as the integral I 1132 (see what follows (3.31)).
3.1.5. Finalizing the three time derivatives estimate. Combining the energy identity (3.2) with the estimates for J i , i = 1, . . . , 5 from Sections 3.1.2 and 3.1.3, and with the boundary estimates of Section 3.1.4 produces (3.1). In doing so, we use assumption (1.12) to bound the integral due to the boundary integral vanishing on Γ 0 . For the term on the right side, we have from where, using (1.6b), The terms I 2 and I 3 give the first and second terms on the right side of (3.37) respectively. In order to treat we write and thus (3.39) The terms I 12 , I 13 , and I 14 give the third, fourth, and fifth terms on the right side of (3.37) respectively. For I 11 , we write The first term on the right side leads to the second term on the left side of (3.37), while the second term on the right side of (3.3) gives the last term in (3.37).
3.2.1. Treatment of the terms involving two time derivatives. The estimates for the right side of (3.37) is the same as the estimates of the corresponding terms in (3.2) and we thus do not provide full details. However, we still show how to treat the most involved term As in (3.9), we have We group the leading terms as before; the analog for (3.10) is The symbol L 4 denotes the lower order terms, which are bounded below. The first term on the far right side is treated as The last integral is bounded by The last integral is bounded by ǫ ∂ t∂ v 2 1 q L ∞ (Ω) . For the remaining integral, we write We integrate by parts in both terms obtaining where the last integral obeys The symbol L 4 above consists of the sum of the terms We thus conclude As above, when treating S 4 + S 6 and S 2 + S 5 we obtain an extra boundary term of the type which is bounded analogously to (3.12). In summary, we obtain

3.3.
Estimates at t = 0. As we have seen, in the above estimates we had several expressions involving time derivatives of v and R evaluated at zero. Here, we show that these quantities can all be estimated in terms of P 0 . More precisely, we show that and The estimate (3.42) is straightforward. In light of (1.11), the equation (1.6f) can be written as From (3.44) and (1.6b) we get ∂ t v(0) 2 ≤ P 0 and ∂ t R(0) 2 ≤ P 0 . Differentiating (3.44) and (1.6b) in time and evaluating at zero gives ∂ 2 t v(0) 1 ≤ P 0 and ∂ 2 t R(0) 1 ≤ P 0 . Taking another time derivative of (3.44) and (1.6b) and evaluating at zero produces (3.42).
To obtain (3.43), we use (1.6a) to estimate terms in v i (0) and (1.6e) to estimate terms in v 3 (0). Evaluating (1.6a) at t = 0 with α = i and recalling (1.9) gives . Note that the conclusion would not be true if we had a ∂ 3 R term, that is why α = 3 has to be treated differently.
Remark 3.7. The estimate (3.45) illustrates why we require higher regularity for the initial data on the boundary. We want ∂ t v ∈ H 2 (Γ 1 ) in order to apply div-curl estimates, as explained in Section 1.1. But this would not hold even at time zero without the regularity assumption on the boundary.
The estimate for ∂ 2 t v is obtained in a similar way, upon differentiating one more time in time and proceeding as above. We omit the details, but explain where the assumption on (∆ div v 0 )↾Γ 1 is used. Proceeding as just explained, we find (writing ∼ to mean "up to lower order") t R(0). But from (1.6a) and (1.6b) we obtain ∂ 3 t R(0) ∼ ∆ div v(0), which requires (∆ div v(0))↾Γ 1 in H −1 (Γ 1 ) in order to produce ∂ 2 t v 3 (0) in H 1 (Γ 1 ) from elliptic estimates.

Estimates for the curl
In this section, we obtain estimates for the curl of v and its time derivatives. First, write (2.20) as from which we obtain (where we used that ε αβγ ∂ β v µ ∂ γ v µ = 0) and the term δ γµ − ∂ γ η µ can be made arbitrarily small for small time. Hence, the relevant norm of the terms proportional to δ γµ − ∂ γ η µ on the right-hand side of (4.1), (4.2), and (4.3) can be absorbed into the left-hand side. We then have to estimate the remaining terms on the right-hand side. From (4.1) we immediately get where we used Jensen's inequality.
In what follows, let ε > 0 be a small number. Moving to (4.2), we estimate We may write where F is a smooth function and we have the estimate F (R) 2 R 2 + R For the first term on the right-hand side of (4.5), we use Using again that the first term in between parentheses is bounded by C(P 0 + t 0 ∂ t R 2 ) 2 , we obtain (4.6) Interpolating, Choosing p = 6/(5 + 2ε) (which is greater than one for small ε) and invoking Young's inequality with epsilon, we obtain Combining (4.5), (4.6), (4.7) with (4.2), and invoking Jensen's inequality, we conclude that curl ∂ t v 2 1 ǫ R 2 3 + P 0 + P t 0 P. (4.8) Moving on to curl ∂ 2 t v, we compute and estimate each term. We have Squaring and using (4.7), we conclude that aq ′ (R)∂η ∂R∂ t ∂R R 2 2 0 ǫ R 2 3 + P 0 + P t 0 P.
Next we look at the term ∂v(∂R) 2 0 ∂v L 6 (Ω) ∂R 2 L 6 (Ω) Squaring, writing v = v(0) + t 0 v and similarly for R, using (4.7) and proceeding as above we find that The other terms in (4.9) are handled in a similar fashion. Finally, we have We also need an estimate for v 3 2.5,Γ 1 . This follows directly from the boundary condition, as we now show. Differentiating (1.6e) in time and setting α = 3 yields where we also used (2.15). In light of Proposition 2.6, we have g ij 2.5,Γ 1 ≤ C and Γ k ij 1.5,Γ 1 ≤ C.
Thus, by the elliptic estimates for operators with coefficients bounded in Sobolev norms (see [29,44] where C depends on the bounds for g ij 2.5,Γ 1 and Γ k ij 1.5,Γ 1 stated above. The right-hand side is now estimated in a routine fashion, and we conclude where P is now a fixed polynomial and C 0 is a fixed positive constant. The inequality (5.5) implies, via a routine continuity argument that we now sketch for the reader's convenience, the boundedness of N (t) on a positive interval of time (cf. [67,Section 8] where a similar inequality was treated). Assume, without loss of generality, that P is strictly positive and non-decreasing, and denote M = N (0). Let where the lower order term ∂ 2 t v 0 was estimated in a standard fashion. We now move to estimate ∂ 2 t R. First, write (1.6a) as R∂ t v α + q ′ (R)a µα ∂ µ R = 0. (5.9) Taking ∂ 2 t of (5.9) gives where we recall Notation 3.6. Taking α = 1, 2, 3 and invoking (3.1) produces where we also used (1.12). Next we estimate div ∂ t v 1 . From (1.6b) we have leading to div ∂ t v 2 1 ≤ ǫ ∂ t v 2 2 + C ∂ 2 t R 2 1 + P 0 + Finally, to bound div v 2 , note that In the same spirit as above, choosing now X = v, s = 2 and squaring (5.6), invoking (4.4), (5.14), Similarly to the foregoing, (5.9) gives an estimate for R in light of the estimate (5.12) for ∂ t v, so Using successive applications of Young's inequality, we can trade the polynomial expressions P by polynomials in N ; choosing ǫ small enough finally produces (5.5). This concludes the proof of Theorem 1.3.