A regularity result for the free boundary compressible Euler equations of a liquid

We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density.


Introduction
In this paper, we study the vacuum free boundary problem for compressible fluids in R 3 described by the compressible Euler equations ρ(∂ t u + u • ∇u) + ∇p = 0 in Ω(t), (1.1) where Ω(t) represents an open subset of R 3 which is occupied by the fluid at the time t, whose boundary is denoted by ∂Ω(t).In (1.1)-(1.2),u = u(t, x), ρ = ρ(t, x), and p = p(t, x) denote the velocity field, density, and pressure, respectively.To close the system, we consider the equation of state for polytropic fluids, where K 1 , K 2 > 0 are constants and γ > 1 is the adiabatic constant (see [CF]).The system (1.1)-(1.3) is supplemented with the initial, kinematic, and vacuum boundary conditions Ω(0) = Ω (1.4) (u(0, x), ρ(0, x)) = (u 0 (x), ρ 0 (x)), x ∈ Ω (1.5) where Γ(t) ⊂ ∂Ω(t) is the moving free boundary, n(t) is the outward unit normal vector to Γ(t), and V(∂Ω(t)) is the normal velocity of Γ(t).In order to model a compressible liquid, we assume that the initial density is strictly positive everywhere in the domain.This problem is known to be ill-posed (see [E]) unless the Rayleigh-Taylor sign condition holds for some constant ν > 0. The condition (1.8) is a natural physical condition which indicates that the pressure, and thus also the density, is greater in the interior than on the boundary.In recent years, there have been a lot of effort made toward understanding the well-posedness of this free-boundary problem.For the incompressible Euler equations, the divergence of the velocity field is zero and the density is a fixed constant.Additionally, the pressure is not determined by the equation of state but rather a Lagrange multiplier enforcing the divergence free condition.For the existence theory, we refer the reader to [BHL,C,KN,N,RS1,RS2,S,W1,W2,Y1,Y2] in various settings including irrotationality (i.e., curl u = 0) and [CS1,CS2,CL,DE,L3,SZ1,SZ2] for rotational fluids.
Concerning the compressible fluids, we distinguish the cases between a liquid and a gas.The gas problem refers to the equation of state (1.3) with the constant K 2 = 0, which combined with the boundary condition (1.7) leads to the vanishing of the density on the moving boundary.As a consequence, the system becomes degenerate along the vacuum boundary ( [CLS,JM2]) and the standard method of symmetrizable hyperbolic equations cannot be applied.We refer to [CS3,IT,JM1,JM2,LXZ] and reference therein for the well-posedness results.In the case of a liquid, a commonly used equation of state is (1.3) with a positive constant K 2 > 0, which is the pressure law we treat in this paper.As opposed to the vanishing of the density on the boundary in the case of a gas, the density remains strictly positive everywhere in the domain and thus the Euler equations are uniformly hyperbolic.The existence theory was established by Lindblad in [L2] using the Nash-Moser construction.Trakhinin provided a different proof for the local well-posedness in [T], using the theory of symmetric hyperbolic systems.Then, Coutand, Hole, and Shkoller in [CHS] proved the well-posedness using the vanishing viscosity method and the time-differentiated a priori estimates.A recent work [LZ] by Luo and Zhang established the local well-posedness of the compressible gravity water wave problem using the Alinhac's good unknowns.For other local well-posedness results in various settings, we refer to [DK1,DK2,DL,GLL,L1,LL].
In this paper, we derive a priori estimates for the compressible free boundary Euler equations without surface tension in the case of a liquid.Using a decomposition of the initial density (see (2.14)-(2.15)),we provide a new weighted functional framework for the liquid problem.An application of the Hardy inequality then leads to the improved regularity of the flow map and hence Jacobian, which in turn closes the energy estimates.Note that in [CHS], the higher-order regularity of the Jacobian is obtained using the wave equation of the density, which requires a high time derivative regularity of the initial data.Here we provide a direct method and a different functional framework with a low time-differentiated regularity.In addition, we assume that the mixed tangential-time derivative of the initial data is in some weighted L 2 space as opposed to inhomogeneoues Sobolev space of the time derivative used in [CHS].As a consequence, our regularity result and the regularity in [CHS] do not imply each other.
Under the Rayleigh-Taylor sign condition, (2.12) below, the initial density grows as the distance function in the inward normal direction.Also, the initial density is strictly bounded from below.In light of this, we decompose the initial density into two parts (see (2.14)-(2.15)):one is degenerate along the boundary with the growth rate of the distance function and the other is uniformly bounded from below.A careful analysis of each part leads to weighted and homogeneous structure of the functional space.For the degenerate part, we deploy the Hardy inequality, Lemma A.1,and obtain an improved H 0.5 regularity of the flow map which in turn leads to the improved regularity of the Jacobian (see Section 6).The part of the initial density which is uniformly bounded from below produces a positive energy contribution on the boundary, allowing us to close the estimate for the improved regularity using the elliptic estimate.
In the pure tangential energy estimates, the Jacobian and cofactor matrix enter as a highest order term which requires improved regularity of the Jacobian and flow map.To overcome this difficulty, we uncover the improved regularity by using the Euler momentum equation and a div-curl elliptic estimate.Namely, we introduce a new fractional H 1.5 Sobolev space so that the highest order term involving the Jacobian and the cofactor matrix can be controlled at the expense of two time derivatives and the above-mentioned improved regularity of the flow map.As a result, the scaling between space and time is necessarily one to two at the pure tangential level.For the energy estimates with at least two time derivatives, the top order term involving the Jacobian and cofactor matrix can be handled by using integration by parts in space and time.The fact that DJ and ∂ t v are of the same regularity is essential to close the estimates.Consequently, the scaling between space and time becomes one to one when at least two time derivatives are involved.The normal derivative estimate of the solution is obtained by the elliptic regularity, which requires divergence, curl, time, and tangential components of the solution.
Another major difficulty lies in the energy estimate with pure time derivatives.The boundary integrals of the tangential component of the flow map cannot be controlled directly by the normal component or the trace lemma.Instead, we resort to the momentum equation restricted to the boundary (see (4.34)) and show that the normal component dominates the tangential component on the free boundary, at least for a short time interval due to the closeness of the cofactor matrix to its initial state.
The paper is organized as follows.In Section 2, we introduce the Lagrangian formulation of the free boundary problem and the assumptions on the domain and initial data.Also, we state the main result, Theorem 2.1.The a priori estimate needed for the boundedness of the solution, stated in Proposition 2.2, is proven in Section 8.In Sections 3-8, we restrict ourselves to the case γ = 2 in (1.3) for the equation of state, while in Section 9 we explain the modifications needed for the general case γ > 1.In Section 3, we provide curl estimates which shall be needed when using the div-curl lemma.In Section 4, we derive energy estimates of the tangential, time, and divergence components of the solution, while in Section 5, we provide the normal derivative estimates of the solution using the elliptic regularity.In order to close the estimate, we establish the improved regularity of the Lagrangian flow map and the Jacobian in Section 6 and the improved regularity of the curl in Section 7. At last, in Appendix A, we recall some auxiliary lemmas used in the proof.
2. Lagrangian formulation and the main result 2.1.Lagrangian formulation.The flow map η(t, •) : x → η(t, x) ∈ Ω(t) associated with the fluid velocity, defined as a solution to denotes the location of a particle at time t that is initially placed at a Lagrangian label x.We define the following Lagrangian quantities: The notation F, k is used to represent ∂F/∂x k , i.e., the partial derivative of F with respect to the Lagrangian variable x k .We adopt the Einstein's summation convention for repeated indices.The Latin indices i, j, k, l, m, r, s are summed from 1 to 3, while the Greek indices α, β are summed from 1 to 2. Without loss of generality, we set the constants K 1 = K 2 = 1 in (1.3).Namely, the equation of state reads Using the Lagrangian quantities, the equations (1.1)-(1.7)can be rewritten in a fixed domain Ω as where Γ = Γ(0) denote the initial vacuum free boundary.Since which indicates that the initial density can be viewed as a parameter in the Euler equations.Thus, using (2.6), the system (2.2)-(2.5)becomes (2.9) From (2.7) we write equivalently and (2.11) The three equivalent equations (2.7) and (2.10)-(2.11)are being used for different purposes: (2.7) is used for the energy estimates, (2.10) is used for estimates of the vorticity, while (2.11) is used for the Jacobian and the normal derivative estimates.

The reference domain.
To avoid the use of local coordinates charts, we assume the initial domain Ω ⊂ R 3 at time t = 0 is given by where T 2 denotes the 2-dimensional torus with length 1.At time t = 0, the reference vacuum boundary is the top boundary Γ = {x 3 = 1}, while N = (0, 0, 1) denotes the outward unit normal vector to Γ.The reference bottom boundary {x 3 = 0} is fixed with the boundary condition The moving vacuum free boundary is given by Γ(t) = {η(t, x 1 , x 2 , 1) : (x 1 , x 2 ) ∈ T 2 }.
2.3.Notations.Given a vector field F over Ω, we use DF , div F , and curl F to denote its full gradient, divergence, and curl.Namely, we denote by where ǫ ijk represents the Levi-Civita symbol.We denote the Lie derivatives along the trajectory map η by which correspond to Eulerian full gradient, divergence, and curl represented in Lagrangian coordinates.The symbol ∂ is used for tangential derivative For integers k ≥ 0, we denote by H k (Ω) the standard Sobolev spaces of order k with corresponding norms • k .For real number s ≥ 0, the Sobolev spaces H s (Ω) and the norms • s are defined by interpolation.The L p norms on Ω are denoted by • L p (Ω) or simply • L p when no confusion can arise.The negative-order Sobolev spaces H −s (Ω) are defined via duality, namely, Note that in our configuration the boundary ∂Ω has two parts: the reference free boundary Γ = {x 3 = 1} and the fixed bottom boundary {x 3 = 0}.We shall work on the Sobolev spaces on the reference free boundary Γ and use the notations H s (Γ) and | • | s for the Sobolev spaces and norms, with s ≥ 0. The negative-order Sobolev spaces associated to the boundary are defined analogously as the domain Ω, while the L p norms on Γ are denoted by • L p (Γ) .
From (2.1) the above condition is equivalent to (2.12) Since γ > 1 and ρ 0 = 1 on Γ, the condition (2.12) implies that ρ 0 − 1 grows like the distance function to the reference boundary Γ.Throughout this paper, we assume that where w(x) = 1 − x 3 is the distance function to the reference free boundary.It is clear that the Rayleigh-Taylor sign condition (2.12) is satisfied for any γ > 1.We introduce the decomposition ρ γ 0 (x) = ρ γ d (x) + ρ γ u (x), where )) and It is readily checked that ρ d (x) : Ω → [0, ∞) is well-defined and ρ d = 0 on Γ.Moreover, we have that ρ u ≥ 1 for x ∈ Ω.The ρ d part of the density is degenerate along the boundary, while the ρ u part is uniformly bounded from below.
2.5.The Jacobian and cofactor matrix.We recall the following differentiation identities: (2.17) Using (2.16)-(2.17)and the identity a = JA, we obtain (2.18) Note that the identities (2.16)-(2.18)become spatial-differentiation formulas by replacing v r , s with Dη r , s on the right hand side, where D = ∂ l with l = 1, 2, 3. Namely, we have (2.21) The cofactor matrix satisfies the Piola identity which reads (2.22) for i = 1, 2, 3.By the definition of the cofactor matrix, we have that (2.23) Note that only tangential derivatives appear in each entry on the right hand side, which shall play an essential role in our analysis.
2.6.Main result.The system (2.7)-(2.9)admits a conserved physical energy A direct computation shows that formally d dt E(t) = 0, assuming sufficient differentiability of the solution.However, it is too weak to control the regularity of the evolving free boundary.Instead, we introduce the higher order energy functional (2.24) Note that the time derivatives at t = 0 are defined iteratively by differentiating (2.10) and evaluating at t = 0.
The following theorem is our main result.
In this paper we restrict ourselves to derive a priori estimates and thus a smooth solution is assumed to be given.As a result, there is no need to state the compatibility conditions for the initial data.Nevertheless, we refer the reader to [CHS] where the solution is constructed using the tangentially smoothed approximate system.
We denote by M 0 = P (E(0)) some constant depending on the initial data, which may vary from line to line.Throughout this paper, the notation A B means that A ≤ CB for some constant C > 0 which depends on M 0 and γ.The notation c l,m ∈ R stands for some constant that depends on l, m ∈ N, which may vary from line to line.In fact, the true value of c l,m is not essential in our analysis.
The a priori estimate needed to prove Theorem 2.1 is the following.
PROPOSITION 2.2.Let δ ∈ (0, 1).There exists a constant T > 0 and a nonnegative continuous function P such that (2.25) for all t ∈ [0, T ], where C δ > 0 is a constant that depends on δ.We emphasize that the implicit constant used in (2.25) is independent of δ.
For the proof of Theorem 2.1 given Proposition 2.2, we take δ > 0 sufficiently small and use the Gronwall inequality and the standard continuity argument.Throughout Sections 3-8, we set γ = 2 in (2.7), (2.10)-(2.11),(2.14)-(2.15),and (2.24).We shall prove that the a priori bound (2.25) in Proposition 2.2, thus completing the proof of Theorem 2.1.In proving Proposition 2.2, we assume the following: × Ω for some constant C > 0, where I 3 is the three-dimensional identity matrix.The above assumptions can be justified by using the fundamental theorem of calculus and taking T > 0 sufficiently small, after we establish the a priori bounds.

Curl estimates
The following lemma provides the curl estimates of the solutions η and v.
PROOF OF LEMMA 3.1.We start with the estimates of ∂ l t D 4−l curl v 2 0 for l = 2, 3, 4. Applying the Lagrangian curl to (2.10), we get From (2.18) and (3.2) it follows that where Q 0 is a quadratic function of A and Dv.In (3.3) and below, we use [F ] k to denote the k-th component of a vector field F .Using the fundamental theorem of calculus, we obtain Fix l ∈ {2, 3, 4}.Applying ∂ l t D 4−l to the above equation, we obtain Inserting the identity from where The term K 1 is estimated using the Hölder and Sobolev inequalities as for l = 2, 3, 4 and A − I 3 2 T .The highest order term (in terms of derivative counts on η) in K 2 can be written as Using the fundamental theorem of calculus, we obtain where we used the Jensen's inequality.From the Hölder and Sobolev inequalities it follows that where we drop the indices for simplicity.The rest of the terms in K 2 are of lower order which can be treated in a similar fashion using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we have Similarly, the term K 3 is estimated as The term K 4 consists of essentially lower order terms which can be estimated analogously using the Hölder and Sobolev inequalities and the fundamental theorem of calculus, and we obtain Consequently, we conclude The term D 3 curl v 2 0 is treated in a similar fashion using the above arguments, and we arrive at Now, we derive the estimates of ρ d ∂5−l ∂ l t curl η 2 0 , where l = 2, 3, 4, 5.We rewrite (3.4) as to the above equation, we arrive at The highest order term in J 1 scales like Dη which can be estimated using the fundamental theorem of calculus as , for l = 2, 3, 4, 5.The rest of the terms in J 1 are of lower order which can be treated in a similar fashion using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we have The highest order term in J 2 is estimated using the Hölder and Sobolev inequalities as The rest of the terms in J 2 , as well as the term J 3 , are treated analogously using the above arguments.As a result, we get Combining the above estimates, we conclude For the term I 1 , from the Leibniz rule it follows that (3.9) Recall the multiplicative Sobolev inequality for 0 ≤ r ≤ 1.5 and ǫ > 0. From (3.10) it follows that where we used the fundamental theorem of calculus and ∂4−l , for l = 2, 3, 4. Similarly, we have For the term I 13 , using (3.10), we get where we used the Sobolev interpolation inequality and the fundamental theorem of calculus.For the term I 2 , from the Leibniz rule it follows that We bound the term I 21 using (3.10) as The terms I 22 , I 23 , and I 3 are estimated analogously using the above arguments.
Thus, we have Consequently, we conclude (3.12) Finally, we derive the estimate of ∂3 curl η 2 0.5 .An application of the fundamental theorem of calculus to (3.7) leads to Applying ∂3 to the above equation, we arrive at The term L 1 is bounded by T M 0 since ∂3 curl u 0 0.5 = ∂3 curl η v(0) 0.5 .For the term L 2 , we use the Leibniz rule, obtaining For the term L 21 , we integrate by parts in time, leading to For the term L 211 , we appeal to (3.10), obtaining The term L 212 is treated in a similar fashion as in (3.11), and we arrive at Similarly, the term L 22 is estimated as The terms L 23 and L 3 are treated using similar arguments as in (3.13)-(3.15).Therefore, we conclude the estimate Collecting the estimates in (3.5)-(3.6),(3.8), (3.12), and (3.16), we complete the proof of the lemma.

Energy estimates
The following lemma provides ∂4 energy estimates of the solutions v, J, and η.
LEMMA 4.1.We have PROOF OF LEMMA 4.1.Applying ∂4 to (2.7) and taking the inner product with ∂4 v, we obtain 1 2 Estimate of J 1 in (4.2):From (2.20) and the Leibniz rule it follows that For the term J 11 , we integrate by parts in ∂ s , obtaining where we note that Note that from (2.13) we may assume that which can be justified by using the fundamental theorem of calculus and taking T > 0 sufficiently small after we establish the a priori bounds.From (4.4) it follows that For the term J 1112 , using the Hölder and Sobolev inequalities and Lemma A.3, we get Similarly, the term J 1113 is estimated as For the term J 112 , we integrate by parts in time, obtaining For the pointwise in time term J 1122 , using the Hölder and Sobolev inequalities and the fundamental theorem of calculus, we arrive at For the term J 1123 , we appeal to Lemma A.5, obtaining where we used the Hölder and Sobolev inequalities.The term J 113 is estimated using the Hölder and Sobolev inequalities as The term J 13 consists of essentially lower-order terms which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as Collecting (4.5) and the above estimates, we conclude that, after integrating in time of (4.2), where we used the Hölder and Sobolev inequalities and Lemma A.3. Combining (4.6)-(4.7),we arrive at Estimate of J 2 in (4.2):We integrate by parts in ∂ k and use the Piola identity (2.22) to get Note that from (2.16) and the Leibniz rule it follows Inserting (4.10) to (4.9), we arrive at The term J 21 can be rewritten as Using the Hölder and Sobolev inequalities, we obtain The highest order term in J 22 is of the form Ω ρ 2 0 J −3 ∂4 J ∂4 DηDv, which can be treated using the Hölder and Sobolev inequalities and Lemma A.5 as The rest of the terms in J 22 are of lower order which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Therefore, we obtain The term J 23 consists of lower-order terms which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as For the term J 24 , we integrate by parts in ∂ k and use the Piola identity (2.22), obtaining Thus, using the Hölder and Sobolev inequalities, the term J 24 is estimated as Combining (4.8), (4.11), and the above estimates, we conclude Estimate of J 3 in (4.2):We proceed analogously as in (4.12)-(4.13),obtaining Concluding the proof : Combining (4.14)-(4.15),we arrive at The proof of the lemma is thus completed.
The following lemma provides ∂3 ∂ 2 t energy estimates of the solutions v, J, and η.
PROOF OF LEMMA 4.2.Applying ∂3 ∂ 2 t to (2.7) and taking the inner product with ∂3 ∂ 2 t v i , we arrive at 1 2 (4.17) Estimate of J 1 in (4.17):Using the decomposition )), we split the term J 1 as First we estimate the term G in (4.18).Using integration by parts in ∂ k , we arrive at where we used the Piola identity (2.22).The first term on the right hand side vanishes since The second term on the right hand side of (4.19) also vanishes since Therefore, from (2.17) and the Leibniz rule it follows that . (4.20) It is clear that The first term on the right hand side can be rewritten as while the second term on the right hand side of (4.21) can be rewritten as (4.23) Combining (4.21)-(4.23),we arrive at where Q is a quadratic function with L ∞ ([0, T ] × Ω) coefficients which may vary from line to line.Integrating G 1 in time from 0 to T , we get For the term G 2 , we have which leads to From (2.16) and the Leibniz rule it follows that Inserting (4.25) to the first term on the right hand side of (4.24), we arrive at where we drop the indices for simplicity.The highest order terms in G 21 can be treated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as and The rest of the terms in G 21 are of lower order which can be treated in a similar fashion using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we arrive at (4.26) For the term G 22 , using the Young inequality, we obtain where the last inequality follows from (4.26).For the term G 3 in (4.20), we integrate by parts in time, leading to We bound the term G 31 using the Young and Sobolev inequalities and the fundamental theorem of calculus as For the term G 32 , we use the Hölder and Sobolev inequalities and the fundamental theorem of calculus, obtaining Collecting the above estimates, we conclude where G ′ consists of the terms satisfying (4.28) Next we estimate the term L in (4.18).From (2.17) and the Leibniz rule it follows that (4.29) For the term L 1 , we integrate by parts in ∂ s , obtaining where we note that a k r (ρ The term L 11 can be rewritten as Note that from (2.15) we may assume that (4.31) from where Using the Hölder and Sobolev inequalities, we get (4.33)where α = 1, 2. To prove the claim, we resort to the boundary dynamics for the flow map η.In fact, from the momentum equation (2.11), we see that η satisfies we deduce the following algebraic relation between ∂ 2 t η α and ∂ 2 t η 3 : Applying ∂3 to the above equation, we get The term I 1 is estimated using the Hölder and Sobolev inequalities as where we used a − I 3 2 T .For the term I 2 , using the Leibniz rule, we get Using the Hölder and Sobolev inequalities and the fundamental theorem of calculus, we arrive at where we used | ∂4 η| 2 0 ∂3 η 2 1.5 and (2.23) in the last inequality.For the term I 22 , from (2.23) and the Leibniz rule it follows that The term I 221 is estimated using the Hölder and Sobolev inequalities as where the last inequality follows from Lemma 4.1.For the term I 222 , we use the Hölder inequality to get T which holds by the fundamental theorem of calculus.The term I 223 is estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as Similarly, we bound the term I 23 as For the term I 3 , using the Hölder and Sobolev inequalities, we obtain , where we used the fundamental theorem of calculus in the last inequality.The highest order term in I 4 is scales like | ∂3 η ∂∂ 2 t η 3 | 2 0 which is treated using the Hölder and Sobolev inequalities as where we used the Sobolev embedding H 0.5 (Γ) ֒→ L 4 (Γ), Lemma A.3, and the fundamental theorem of calculus.The rest of the terms in I 4 are of lower order which can treated in a similar fashion as above.Thus, we have Collecting the estimates (4.35)-( 4.45), we complete the proof of the claim (4.33).
Next we estimate the term L 12 in (4.30).Using integration by parts in time, we obtain It is clear that −L 121 + T 0 L 2 = 0. Note that from (2.16) and the Leibniz rule it follows that Inserting (4.46) into the term L 122 and using the Young, Hölder, Sobolev inequalities and the fundamental theorem of calculus, we arrive at where we recall that For the term L 123 , we appeal to the Hölder and Sobolev inequalities and Lemma A.5, obtaining The term L 13 in (4.30) is estimated using the Hölder and Sobolev inequalities as The term L 3 in (4.29) consists of essentially lower order terms which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as Collecting (4.17), (4.27)-(4.28),and (4.32) and the above estimates, we conclude that where we used (4.33) in the last inequality.Estimate of J 2 in (4.17):We integrate by parts in ∂ k and use the Piola identity (2.22) to get Note that from (2.16) and the Leibniz rule it follows Combing (4.48)-(4.49), we arrive at The term J 21 can be rewritten as where the term J ′ 21 satisfies For the term J 22 , we integrate by parts in time, obtaining For the term J 221 , we integrate by parts in ∂ to get since integration by parts in ∂ does not produce any boundary terms.Therefore, using the Hölder and Sobolev inequalities and the fundamental theorem of calculus, we obtain For the term J 222 , we integrate by parts in ∂, obtaining (4.54) From the Young, Hölder, and Sobolev inequalities and the fundamental theorem of calculus it follows that For the term J 23 , we proceed analogously as in (4.51)-(4.55),obtaining The term J 24 consists of essentially lower order terms which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as For the term J 25 , we integrate by parts in ∂ k , obtaining from where Combining (4.47), (4.50), and the above estimates, we arrive at since ρ 2 d ≤ ρ 2 0 .Estimate of J 3 in (4.17):The term J 3 consists of essentially lower order terms which can be treated in a similar fashion as in (4.56), and we obtain Concluding the proof : Combining (4.57)-(4.58),we get The proof of the lemma is thus completed by the curl estimate in Lemma 3.1.
The following lemma provides ∂2 ∂ 3 t , ∂∂ 4 t , and ∂ 5 t energy estimates of the solutions η, J, and v.
PROOF OF LEMMA 4.3.Similar arguments as in Lemma 4.2 lead to It remains to establish the ∂ 5 t energy estimates.Applying ∂ 5 t to (2.7) and taking the inner product with ∂ 5 t v, we obtain )), we split the term J 1 as For the term G, we proceed in a similar fashion as in Lemma 4.2, obtaining (4.63)where G ′ consists of the terms satisfying Next we estimate the term L in (4.62).Using (2.17) and the Leibniz rule, we get For the term L 1 , we integrate by parts in ∂ s , leading to The term L 11 can be rewritten as From (4.31) it follows that Using the Hölder and Sobolev inequalities, we get Note that from (2.16) and the Leibniz rule it follows that Inserting (4.66) to the term L 121 and using the Young, Hölder, and Sobolev inequalities and the fundamental theorem of calculus, we obtain where we recall that For the term L 123 , we use the Hölder and Sobolev inequalities, obtaining Similarly, the terms L 13 and L 3 are estimated as Combining (4.61), (4.63)-(4.65),and the above estimates, we conclude The term I 1 is estimated using the Hölder and Sobolev inequalities as where we used a − I 3 2 T and |∂ 5 t η| 2 0 ≤ E(t).For the term I 2 , using the Leibniz rule, we get We bound the term I 21 using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as The term J 21 can be rewritten as where the term J ′ 21 satisfies The highest order term in J 22 is of the form Ω ρ 2 0 J −3 ∂ 5 t JDv∂ 5 t Dη, which can be treated using the Hölder and Sobolev inequalities as The rest of the terms in J 22 are of lower order which can be treated in a similar fashion using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we obtain Similarly, the term J 23 is estimated as For the term J 24 , we integrate by parts in ∂ k , obtaining From the Hölder and Sobolev inequalities and the fundamental theorem of calculus it follows that T 0 J 24 T P ( sup Combining (4.77)-(4.78)and the above estimates, we conclude (4.82) Concluding the proof : Combining (4.81)-(4.82),we arrive at (4.83) From the curl estimates in Lemma 3.1 we complete the ∂ 5 t energy estimates.Combining (4.60) and (4.83), we conclude the proof of the lemma.

Normal derivative estimates
In the following three lemmas, we derive the normal derivative estimates of J and v.
where we also used the Sobolev interpolation in the second inequality and (5.9) in the last inequality.We bound the term K 4 using the fundamental theorem of calculus as (5.12) It remains to estimate ∂ 3 t ∂ 33 J 2 0 .Applying ∂ 3 t to (5.4), we obtain Applying ∂ 3 to the above equation, while noting that ρ 0 , 3 = −1, we obtain (5.13) Now we estimate the terms on the right hand side of (5.13) in L 2 .Using the Hölder and Sobolev inequalities and the fundamental theorem of calculus, we obtain Similarly, we have The term J 3 ∂ 5 t η, 3 2 0 is already estimated in Lemma 5.1.The highest order term in J 3 (ρ 0 ∂ 3 t (a α 3 J −2 , α )), 3 scales like ∂ 3 t ∂J, 3 which is estimated in (5.12).The rest of the terms in J 3 (ρ 0 ∂ 3 t (a α 3 J −2 , α )), 3 , as well as the terms on the right hand side of (5.13) are of lower order which can estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Collecting the above estimates, we infer from (5.13) that (5.14) The highest order term in I 1 scales like ρ 0 J, 3 D 3 ∂v, which is estimated using Hölder and Sobolev inequalities and (5.17).The rest of the terms in I 1 are of lower order which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we have For the term I 2 , we have since a − I 3 2 T .The highest order term in I 3 is of the form ρ 0 a α 3 J, α ∂ t D 3 J which can be estimated using the fundamental theorem of calculus, since ∂ 2 t J 2 3 ≤ E(t).The rest of the terms in I 3 are of lower order which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Thus, we have

E(t)).
The term I 4 is already estimated in Lemma 5.2, while the term I 5 is estimated using Hölder and Sobolev inequalities and (5.17).The term I 6 is estimated using the fundamental theorem of calculus, since ∂ 2 t J 2 3 ≤ E(t).The terms I 7 , I 8 , and I 9 are of lower order which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as

E(t)).
The estimate of ∂ t ∂J 2 3 follows analogously using the above arguments.The proof of the lemma is thus completed by combining (5.17).

The improved Lagrangian flow map and Jacobian estimates
The next two lemmas provide the estimates of the improved regularity of solutions J and η.LEMMA 6.1.For δ ∈ (0, 1), we have

E(t)).
Therefore, we complete the proof of the Lemma by combining (6.2).
The proof of Lemma 6.2 is analogously to the proof of Lemma 6.1 by using Lemma 4.3 and thus we omit the details.The term I 1 is bounded by M 0 since ∂3 curl v(0) 0.5 = ∂3 curl u 0 0.5 .From (3.3), we infer that the highest order term in I 2 can be written as

E(t)),
where we appealed to Lemma 6.1 in the last inequality.The rest of the terms in I 2 are of lower order which can be estimated using the Hölder and Sobolev inequalities and the fundamental theorem of calculus.Consequently, we conclude the proof of the lemma.

Concluding the proof of Proposition 2.2
We sum the estimates in Lemmas 3. 1, 4.1-4.3, 5.1-5.3, 6.1-6.2, and 7.1, thus completing the proof of Proposition 2.2.The proof of Theorem 2.1 is thus concluded.9.The general case γ > 1 In this section we prove Theorem 2.1 for general γ > 1.The proof is similar to those in Sections 3-8, thus we only outline the necessary modifications and omit further details.
The Lagrangian curl of (2.10) equals zero for any γ > 1.Consequently, similar estimates in Section 3 can be carried out.From (2.15), we may assume that on [0, T ] × Γ.Thus, using the decomposition ρ γ 0 = ρ γ d + ρ γ u (see (2.14)-(2.15)),similar energy estimates in Section 4 hold for γ > 1.The normal derivative estimates in Section 5 relies on the equation (2.11).With the additional term γ(ρ 0 J −1 ) γ−2 , we get the same normal derivative estimates using the energy estimates.From (2.14) it is readily checked that from where we obtain Therefore, the improved regularity in Section 6 can also be carried out using the above Hardy-type inequality.In Section 7, the improved curl estimates hold for γ > 1.Consequently, we conclude the proof of Proposition 2.2 and thus completing the proof of Theorem 2.1.
LEMMA A.2 (Hodge type estimate).Let F be a vector field over Ω and s ≥ 1.Then we have F s ≤ C F 0 + div F s−1 + curl F s−1 + ∂F • N H s−1.5 (∂Ω) , (A.1) for some constant C > 0, where N denotes the outward unit normal to ∂Ω.
The estimate (A.1) follows from the well-known identity −∆F = curl curl F − ∇ div F and the analysis of second order elliptic operators.
We recall from [A] the following Sobolev trace theorem.
The next lemma provide the normal trace estimate which is needed in the elliptic estimates (cf.[A, CCS]).LEMMA A.4 (Normal trace).Let F ∈ L 2 (Ω) and div F ∈ L 2 (Ω).
for some constant C > 0, where N denotes the outward unit normal vector to ∂Ω.
We recall from [CS1] the following duality inequality.

.
]. Using integration by parts in time, where we drop the indices for simplicity, we obtain For the term I 21 , we appeal to the multiplicative Sobolev inequality (3
For the term L 12 , we integrate by parts in time, obtaining The term I 23 consists of lower-order terms which can be estimated in a similar fashion using the Hölder and Sobolev inequalities and the fundamental theorem of calculus as .77) Estimate of J 2 in (4.61):We integrate by parts in ∂ k and use the Piola identity (2.22) to get