Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\ri \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

Throughout the paper, we will use Einstein's summation convention. The k th -partial derivative of F will be denoted by F ,k = ∂ F ∂ x k . Then we have (div (ρu ⊗ u)) j =(ρu i u j ) ,i = div (ρu)u j + ρu · ∇u j , which yields that (1.1b) can be rewritten, in view of (1.1a), as ρ(∂ t u + u · ∇u) + ∇p = 0. Thus, the system (1.1) can be rewritten as ∂ t ρ + div (ρu) = 0 in Ω(t) × (0, T], (1.4a) ρ(∂ t u + u · ∇u) + ∇p = 0 in Ω(t) × (0, T], (1.4b) in Ω × {t = 0}, (1.4f) Ω(0) = Ω, Γ 1 (0) = Γ 1 . (1.4g) With the sound speed given by c := ∂ p/∂ ρ and N denoting the outward unit normal to Γ 1 , the satisfaction of the condition in a small neighborhood of the boundary defines a physical vacuum boundary (cf. [12]), where c 0 = c| t=0 denotes the initial sound speed of the gas. In other words, the pressure accelerates the boundary in the normal direction. The physical vacuum condition (1.5) for γ = 2 is equivalent to the requirement ∂ ρ 0 ∂ N < 0 on Γ 1 . (1.6) Since ρ 0 > 0 in Ω, (1.6) implies that for some positive constant C and x ∈ Ω near the vacuum boundary Γ 1 , where dist (x, Γ 1 ) denotes the distance of x away from Γ 1 . The moving boundary is characteristic because of the evolution law (1.1e), and the system of conservation laws is degenerate because of the appearance of the density function as a coefficient in the nonlinear wave equation which governs the dynamics of the divergence of the velocity of the gas. In turn, weighted estimates show that this wave equation indeed loses derivatives with respect to the uniformly hyperbolic non-degenerate case of a compressible liquid, wherein the density takes the value of a strictly positive constant on the moving boundary [2]. The condition (1.7) violates the uniform Kreiss-Lopatinskii condition [9] because of resonant wave speeds at the vacuum boundary for the linearized problem. The methods developed for symmetric hyperbolic conservation laws would be extremely difficult to implement for this problem, wherein the degeneracy of the vacuum creates further difficulties for the linearized estimates. Now, we transform the system (1.4) in terms of Lagrangian variables. Let η(x,t) denote the "position" of the gas particle x at time t. Thus, ∂ t η = u • η for t > 0, and η(x, 0) = x, (1.8) where • denotes the composition, i.e., (u • η)(x,t) := u(η(x,t),t).
From the derivative formula of determinants, we have It follows from (1.10a) and (1.11) that thus, the initial density function ρ 0 can be viewed as a parameter in compressible Euler equations.

Using the identity
in Ω × {t = 0}, (1.13d) To understand the behavior of vacuum states is an important problem in gas and fluid dynamics. In particular, the physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water [8]. Despite its importance, there are only few mathematical results available near vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. The existence and uniqueness for the three-dimensional compressible Euler equations modeling a liquid rather than a gas was established in [11] where the density is positive on the vacuum boundary. Trakhinin provided an alternative proof for the existence of a compressible liquid, employing a solution strategy based on symmetric hyperbolic systems combined with the Nash-Moser iteration in [14].
The local existence for the physical vacuum singularity can be found in the recent papers by Jang and Masmoudi [7,8] and by Coutand and Shkoller [5,6] for the one-dimensional and three-dimensional compressible gases. Coutand, Lindblad and Shkoller [3] established a priori estimates based on time differentiated energy estimates and elliptic estimates for normal derivatives for γ = 2 with ρ 0 ∈ H 4 (Ω) where the energy function was given bȳ We will not attempt to address exhaustive references in this paper. For more related references, we refer the interested reader to [6,8] and references therein for a nice history of the analysis of compressible Euler equations.
In the present paper, we will improve the results given in [3,6] by using a similar argument therein and fully exploiting the degeneracy of the initial density ρ 0 and the physical vacuum condition (1.7), especially focusing mainly on the following improvement comparing with [3]. Firstly, the a priori estimates were given in [3] based on a proof for a special case ρ 0 = 1 − x 3 of the density, instead of general densities, from which one can not easily obtain the exact Sobolev space that the density should belong to from this special density. Because it will appear a term ∂ r ρ 0 v i t evidently for general ρ 0 in the r-th order horizontal energy estimates, we must assume ρ 0 ∈ H r+1 (Ω) at least in view of the higher order Hardy inequality given in Section 2.2, for instance, ρ 0 ∈ H 5 (Ω) if r = 4, see more details in Step 4 of the proof of Proposition 6.1. Secondly, we will prove, in details, a mixed space-time interpolation inequality under the framework of Lebesgue spaces which will play a vital role in our energy estimates, rather than the framework of Lebesgue spaces for time but Sobolev spaces for spatial variables in bounded domain used in [3]. Finally, it is not enough to be controlled for the fourth order energy estimates, which can not close themselves, for general ρ 0 at least, especially in dealing with some lower order terms in the argument for the time derivatives. Thus, we have to investigate the fifth order energy estimates which are closed with themselves.
We now derive the physical energy of the system (1.13). From (1.11), the Piola identity (2.16) given in Section 2.3, we get Since ρ 0 = 0 on the boundary Γ 1 and a 3 i v i = 0 on Γ 0 , integrating over Ω yields, with the help of Gauss' theorem, that conserves for all t 0. Although the physical energy is a conserved quantity, it is far too weak for the purposes of constructing solutions. Instead, we introduce the following fifth order energy function Now, we state our main result as follows.
and that the initial density ρ 0 > 0 in Ω and ρ 0 ∈ H 6 (Ω) satisfies the physical vacuum condition (1.7). Then there exists a T > 0 so small that the energy function E(t) constructed from the solution (η(t), v(t)) satisfies the a priori estimate where both M 0 and T are functions of E(0) and V .
Remark 1.2. The same arguments and the results hold true if the bottom boundary Γ 0 is also a moving vacuum boundary, i.e., by changing the boundary condition (1.1d) into p = 0 on Γ 0 (t) × (0, T ], which will not cause any additional difficulties except for the transformation of coordinates.
Remark 1.3. For the general cases γ > 1 with general densities, we give some further remarks. We think that they are much more different from the special case γ = 2. They need to reform the energy function in order to get a priori estimates. For the cases γ > 2, it seems to be similar to the case γ = 2 due to γ − 1 > γ/2 in view of the exponent of the weight ρ γ/2 0 and ρ γ−1 0 and weighted Sobolev embedding relations given in Section 2.1, but it is not easy to deal with the weight ρ γ/2 0 in energy estimates in view of the higher order Hardy inequality. For the cases 1 < γ < 2, one have to use the weight ρ γ−1 0 instead of ρ γ/2 0 in constructing the energy function according to the physical vacuum condition, the higher order Hardy inequality and weighted Sobolev embedding relations, especially for the cases 3/2 γ < 2, however one must deal with many extra, important and difficult remainder integrals in the estimates of every horizontal, time or mixed derivatives. For the cases 1 < γ < 3/2, it might be different from and difficult than the above cases. We will discuss the above general cases in forthcoming papers if possible.
The rest of this paper is organized as follows. We will give some preliminaries in Section 2. Precisely, we introduce some notations and weighted Sobolev spaces in Section 2.1; we recall the higher-order Hardy-type inequality and Hodge's decomposition elliptic estimates in Section 2.2; we give the properties of the determinant J, the inverse of the deformation tensor A and the transpose of the cofactor matrix a in Section 2.3. Then, we give a mixed space-time interpolation inequality in Section 3, and derive the zero-th order energy estimates in Section 4 and the curl estimates in Section 5. Since the standard energy method is very problematic due to the degeneracy of ρ 0 , we first derive the estimates of the horizontal and time derivatives in Sections 6-8 and then obtain the estimates of normal or full derivatives through the elliptictype estimates in Section 9. Finally, we will complete the proof of the a priori estimates in Section 10.

PRELIMINARIES
2.1. Notations and weighted Sobolev spaces. Throughout the paper, we will use the following notation: two-dimensional gradient vector or horizontal derivative ∂ = (∂ 1 , ∂ 2 ), the H s (Ω) interior norm · s , and H s (Γ) boundary norm | · | s . The component of a matrix M at the i th row and the j th column will be denoted by M i j . Sometimes, we will use " " to stand for " C" with a generic constant C.
We make use of the Levi-Civita permutation symbol where it means that we have taken the sum with respect to the repeated scripts j and k. Since The chain rule shows that (curl u(η)) i = (curl η v) i := ε i jk v k ,s A s j , where the right-hand side defines the Lagrangian curl operator curl η . Similarly, we have divu(η) = div η v := v j ,s A s j , and the right-hand side defines the Lagrangian divergence operator div η . We also use the notation for any vector field F For any vector field F, we have and then where the superscript ⊤ denotes the transpose of the matrix. As a generalization of the standard nonlinear Gronwall inequality, we introduce a polynomialtype inequality. For a constant M 0 0, suppose that f (t) 0, t → f (t) is continuous, and where P denotes a polynomial function, and C is a generic constant. Then for t taken sufficiently small, we have the bound (cf. [3,4]) For integers k 0 and a smooth, open domain Ω of R 3 , we define the Sobolev space H k (Ω) (H k (Ω; R 3 )) to be the completion of C ∞ (Ω) (C ∞ (Ω; R 3 )) in the norm , for a multi-index α ∈ Z 3 + , with the standard convention |α| = α 1 + α 2 + α 3 . For real numbers s 0, the Sobolev spaces H s (Ω) and the norms · s are defined by interpolation. We will write H s (Ω) instead of H s (Ω; R 3 ) for vector-valued functions. In the case that s 3, the above definition also holds for domains Ω of class H s .
Our analysis will often make use of the following subspace of H 1 (Ω): where, as usual, the vanishing of u on Γ is understood in the sense of trace. For functions u ∈ H k (Γ), k 0, we set , for a multi-index α ∈ Z 2 + . For real s 0, the Hilbert space H s (Γ) and the boundary norm | · | s is defined by interpolation. The negative-order Sobolev spaces H −s (Γ) are defined via duality: for real s 0, The derivative loss inherent to this degenerate problem is a consequence of the weighted embedding we now describe.
Using d to denote the distance function to the boundary Γ 1 , i.e., d(x) = dist (x, Γ 1 ), and letting p = 1 or 2, the weighted Sobolev space H 1 d p (Ω), with norm given by , satisfies the following embedding: that is, there is a constant C > 0 depending only on Ω and p, such that See, for example, Section 8.8 in Kufner [10]. From this embedding relation and (1.7), we obtain where ⌈·⌉ is the ceiling function defined by ⌈p⌉ = min{n ∈ Z : n p}.

Lemma 2.3 (Normal trace theorem).
Let w be a vector field defined on Ω such that ∂ w ∈ L 2 (Ω) and div w ∈ L 2 (Ω), and let N denote the outward unit normal vector to Γ. Then the normal for some constant C independent of w.
, and let T 1 , T 2 denote the unit tangent vectors on Γ, so that any vector field u on Γ can be uniquely written as u α T α . Then for some constant C independent of w.
Combining (2.10) and (2.11), we have for some constant C independent of w. The construction of our higher-order energy function is based on the following Hodge-type elliptic estimate (see, e.g., [3]): where N denotes the outward unit normal to Γ and T α are tangent vectors for α = 1, 2.

Properties of J, A and a. From the derivative formula of matrices and determinants, we have
It follows from a = JA, (2.13) and (2.15) that the columns of every adjoint matrix are divergencefree, i.e., the Piola identity, which will play a vital role in our energy estimates. We also have

A MIXED SPACE-TIME INTERPOLATION INEQUALITY
In this section, we prove a useful mixed space-time interpolation inequality which will play a vital role in our energy estimates.

Proposition 3.1 (Mixed interpolation inequality). Let F(t, x) be a scalar or vector-valued func-
where C is a constant independent of T , Ω and F.
Then, by the Fubini theorem, integration by parts with respect to time, the fundamental theorem of calculus, the Hölder inequality and the Minkowski inequality, we have which implies the desired inequality by eliminating F t L 3 ([0,T ]×Ω) from both sides of the inequality.

A PRIORI ASSUMPTION AND THE ZERO-TH ORDER ENERGY ESTIMATES
We assume that we have smooth solutions η on a time interval [0, T ], and that for all such solutions, the time T > 0 is taken sufficiently small so that for t ∈ [0, T ], Once we establish the a priori bounds, we can ensure that our solution verifies the assumption (4.1) by means of the fundamental theorem of calculus. Then, by (1.9), Sobolev's embedding It follows from a = JA and (4.1) that Now, we prove the following zero-th order energy estimates.

THE CURL ESTIMATES
Taking the Lagrangian curl of (1.13a) yields that For the last term, we can interchange the indices k and s, i and j to find that it vanishes due to the fact ε l ji = −ε li j . For the first term, we can use the equation (1.13a). Then using (2.17) for the third term, it follows that Interchanged the indices b and s, i and j, the term ε l ji JA s The first and last terms vanish by interchanging the indices k and s, i and j. The second and third terms eliminate. Therefore, it yields We can obtain the following proposition.
and computing the r-th order horizontal derivatives of this relation yields Noticing and then by the fundamental theorem of calculus Notice that for k = 1, integration by parts with respect to time and the fact ∂ ∇η(0) = 0 imply Since other terms can be estimated easily, thus, it follows that The weighted estimates for the curl of ∂ 2m t ∂ r−m η can be obtained similarly.
From (5.3), we see that and then by the fundamental theorem of calculus, It follows that where we have used integration by parts with respect to time for the integrals involving ∇ r v in order to control them by η r and v r−1 . Thus, we get From (5.3), we get, by the fundamental theorem of calculus, and, by taking the first order time derivative, Since H r−2 (Ω) is a multiplicative algebra for r 4, we can directly estimate the H r−2 (Ω)-norm of curl v t to show that The estimates for curl ∂ 2m t η(t) in H r−1−m (Ω) for 2 m r − 1 follow from the same arguments.

THE ESTIMATES FOR THE HORIZONTAL DERIVATIVES
We have the following estimates.
Proof. Letting ∂ r act on (1.13a), then we have where C l r is the binomial coefficient. Taking the L 2 (Ω)-inner product with ∂ r v i , we obtain Here, Step 1. Analysis of the integral I 1 . We use the identity (2.16) to integrate by parts with respect to x j to find that due to the boundary conditions (1.13b) and (1.13c). It follows from (2.17) that Since v = η t , we get Thus, we have It follows that It is clear that , ). Now, we analyze the integral T 0 (6.4)dt. We will use integration by parts with respect to time for the cases s = 1 and s = 2, while we have to use integration by parts with respect to spatial variables for the case s = r − 1.
Case 1: s = 1. From integration by parts with respect to time, we get It is clear that, by Hölder's inequality and the fundamental theorem of calculus twice, and |(6.10)| CT ρ 0 2 sup We can rewrite (6.9) as, for β ∈ {1, 2} Obviously, we have from the Hölder inequality and Sobolev's embedding theorem that E r (t)).
By integration by parts, it yields It is easy to see that In order to estimate (6.14), we first consider the estimate of the D r−1 v L 3 ([0,T ]×Ω) where D r−1 denotes all the derivatives ∂ θ for the multi-index θ = (θ 1 , θ 2 , θ 3 ) and 0 |θ | r − 1. By Proposition 3.1 with F = D r−1 η and the Sobolev embedding theorem, we have Thus, we obtain By the Hölder inequality, the Sobolev embedding theorem, the Cauchy inequality and (6.16), we easily get E r (t)).

Case 2: s = 2. By integration by parts with respect to time, it yields
Applying the fundamental theorem of calculus yields for a small δ > 0 Similar to (6.14), we can get .
Case 3: s = r − 1. We write the space-time integral as, for β ∈ {1, 2} By integration by parts with respect to x β , we have From (2.17), it follows that We can write It is easy to see that |(6.33)| CT ρ 0 and by an L 6 -L 6 -L 6 -L 2 Hölder inequality and the Sobolev embedding theorem, By using integration by parts with respect to time, we have Obviously, it yields by using the fundamental theorem of calculus twice By the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus, we get Similarly, we have We can deal with (6.31) and (6.32) as the same arguments as for (6.30). Thus, we obtain |(6.25)| M 0 +CT P(sup We write By the Hölder inequality and the Sobolev embedding theorem, we have |(6.45)| + |(6.46)| + |(6.47)| CT ρ 0 Hence, E r (t)). (6.48) By (2.13), we have Hence, E r (t)). (6.52) By integration by parts with respect to time, it holds It is easy to see that by applying the fundamental theorem of calculus three times By (2.17) and (2.13), we have We split (6.57) into two integrals, i.e., Obviously, we see that and |(6.61)| CT ρ 0 2 sup Both (6.58) and (6.59) can be dealt with as the same argument as for (6.57). Thus, we obtain E r (t)).
By (2.18) and (2.14), we have We write Then, by the same arguments as for (6.44) and (6.47), we can estimate (6.65) and (6.66), and then It is easy to see that (6.56) has the same bounds. Thus, we obtain the estimates of (6.24) and then of (6.22), i.e., Case 4: s = r − 2 and r = 5. Integration by parts with respect to time gives which can be controlled by M 0 +CT P(sup [0,T ] E 5 (t)) from the Hölder inequality and the fundamental theorem of calculus. Therefore, we obtain Step 2. Analysis of the integral I 2 . Similar to those of I 1 , by (1.13b), (1.13c) and η t = v, we have Integration by parts shows that which yields, by Hölder's inequality and Sobolev's embedding theorem, that where we require ρ 0 ∈ H max(4,r) (Ω) because, for r 5, by the higher order Hardy inequality.

By integration by parts, we have
We first consider (6.74) and split it into four cases. Case 1: s = 0. By an L 2 -L ∞ -L 2 Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus for the norm ρ 0 ∂ r−1 ∇v 0 , we can easily get Case 2: s = 1. Integration by parts yields E r (t)).
Case 4: s = r − 2 with r = 5. It is easy to see that can be bounded by the desired bound in view of Hölder's inequality and the fundamental theorem of calculus. Next, we consider (6.75). Since for the case s = 0 and for the case s = 1 we have the desired bounds. For the case s = 2, we have, by Hölder's inequality and the Sobolev embedding theorem and the fundamental theorem of calculus, that For the case s = r − 2 and r = 5, For (6.76), by the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus, we can easily obtain the desired bounds. Thus, Now, we turn to the estimates of T 0 (6.69)dt. Case 1: s = 0. By (2.15) and integration by parts, we see that By the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus, we can easily obtain the desired bound for (6.91) and (6.93)-(6.97).
Case 4: s = r − 2 with r = 5. Integration by parts with respect to time yields which can be easily controlled by the desired bound in view of the Hölder inequality and the fundamental theorem of calculus. Therefore, combining with four cases, we obtain T 0 (6.69)dt Step 4. Analysis of the remainder I 4 .
For l = 0, we have from the Hölder inequality, the Sobolev embedding theorem and the Cauchy inequality that where we need the condition ρ 0 ∈ H r+1 (Ω) in view of higher-order Hardy's inequality. For l = 1, we have, at a similar way, that E r (t)).
For l = 2, we get, by the Hölder inequality and the Sobolev embedding theorem, E r (t)).
For l = 3, we obtain, by the Hölder inequality and the Sobolev embedding theorem, that E r (t)).
For l = 4 with r = 5, we have, by the Hölder inequality and the Sobolev embedding theorem, that E r (t)).
Step 5. Analysis of the remainders I 5 and I 6 . By integration by parts, (2.16), (1.13b) and (1.13c) for l = 1, · · · , r − 1, we have which can be written as, by integration by parts with respect to time, Case 1: l = 1. By using the fundamental theorem of calculus twice and (2.13), we get for (6.122) Similarly, we have for (6.123) For (6.124), it is harder to be controlled than (6.122) and (6.123). We write it as It is easy to see that (6.125) and (6.126) are bounded by By integration by parts, Hölder's inequality, (6.16) and Sobolev's embedding theorem, it holds E r (t)).
For (6.128), we can easily get the desired bound by an L 6 -L 3 -L 2 Hölder's inequality and the Sobolev embedding theorem because each component of a 3 i is quadratic in ∂ η due to (1.9).
Case 2: l = 2. By the fundamental theorem of calculus, Hölder's inequality and the Sobolev embedding theorem, we can see that From (2.18), the Hölder inequality, the Sobolev embedding theorem and (6.16), it yields since the remainders can be easily controlled by the desired bound. Similarly, we can get the bound for the last integral E r (t)).
Case 3: l = 3. By using the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus, the spatial integral (6.122) can be bounded by M 0 +δ sup [0,T ] E r (t)+ CT P(sup [0,T ] E r (t)). Similarly, we can get the same bound for the first space-time integral (6.123). Since the norm ρ 0 ∂ 2 t J −2 3 is contained in the energy function E r (t), the last spacetime integral (6.124) have the same bound by the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus.
Case 4: l = r − 1 and r = 5. They can be easily controlled by the desired bound, especially with the help of the fundamental theorem of calculus for (6.124).
We can deal with the integrals in I 6 by using a similar argument and we omit the details for simplicity.

THE ESTIMATES FOR THE TIME DERIVATIVES
We have the following estimates. Proof. Acting ∂ 2r It follows from (2.2) that s = 2, 2, they are controlled by the desired bounds by using the Hölder inequality and the Sobolev embedding theorem. For the case s = 3, we get For the spatial integral (7.41), we can use the same argument as for (7.43) to get the desired bound with the help of the fundamental theorem of calculus. For the double integral (7.44), it is easier to get the bound than either (7.42) or (7.43) and thus we omit the details. Therefore, we obtain the estimates for T 0 (7.4)dt, i.e., E 5 (t)).
Step 5. Analysis of the remainder T 0 I 3 dt. By integration by parts with respect to the spatial variables and the time variable, respectively, we obtain In (7.50), we use an L 3 -L 6 -L 2 Hölder inequality and (7.15) for the higher order terms of the cases s = 1 and s = 2r − 3 and an L 6 -L 6 -L 6 -L 2 Hölder inequality for the other cases to get the desired bounds. For (7.51), since ρ 0 ∂ 4 t J −2 ∈ H r−2 (Ω) ⊂ L ∞ (Ω), we can get the desired bound easily in view of the Hölder inequality, the Sobolev embedding theorem and the fundamental theorem of calculus. For the case l = 4, we have the desired bound as a similar argument as for the case l = 3. For the case l = 2r − 3, we can use an L 6 -L 3 -L 2 Hölder inequality and (7.15) to get the desired bound. For the case l = 2r − 2, we use an L 3 -L 6 -L 2 Hölder inequality and the Sobolev embedding theorem to get the bound due to ρ 0 ∂ 2(r−1) t J −2 ∈ H 1 (Ω) ⊂ L 6 (Ω). For the case l = 2r − 1, it is similar to the case s = 1 in (7.42) and we omit the details. For the other cases, we can easily get the desired bounds by using the Hölder inequality and the Sobolev embedding theorem.
Next, we consider (7.47). Since the cases 1 l 2r − 2 are identical to the cases 2 l 2r − 1 of (7.46) estimated just discussed up to some constant multipliers, we only need to consider the remainder case l = 2r − 1. We can apply (1.11) to split the integral of the case l = 2r − 1 into two integrals. One of them can be used an L 2 -L 2 Hölder inequality to get the estimates, the other one can be dealt with as the same arguments as for the case l = 2r − 1 of (7.46) or the case s = 1 in (7.42). Thus, we omit the details.
For the spatial integral (7.45), it can be estimated as the same arguments as for (7.46) or (7.47) with the help of the fundamental theorem of calculus whose details are omitted.
Step 6. Summing inequalities. As the same argument as in the estimates of the horizontal derivatives, we can obtain the desired result by combining the previous inequalities.

THE ESTIMATES FOR THE MIXED TIME-HORIZONTAL DERIVATIVES
We have the following estimates.

E(t)).
According to the polynomial-type inequality (2.3), by taking T > 0 sufficiently small, we obtain the a priori bound Therefore, we complete the proof of the main theorem.