Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in R+ for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in Tn and Rn respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.


Introduction
We consider the Cauchy problem of the incompressible Navier-Stokes equations in ℝ n + , n ≥ 3: u t + u · ∇u+∇p = Δu for x ∈ ℝ n + , t > 0, ∇·u = 0 for x ∈ ℝ n + , t > 0, uj t=0 = f for x ∈ ℝ n + , uj x n =0 = 0 for t > 0, where u = uðx, tÞ = ðu 1 ðx, tÞ,⋯,u n ðx, tÞÞ and p = pðx, tÞ stand for the unknown velocity vector field of the fluid and its pressure, while f = f ðxÞ = ð f 1 ðxÞ,⋯f n ðxÞÞ is the given initial velocity vector field, with ∇·f = 0 and f j x n =0 = 0. In what follows, we will use the same notations for the space of vectorvalued and scalar functions for convenience in writing.
There is a large literature on the existence and uniqueness of solution of the Navier-Stokes equations in ℝ n . For the given initial data, solutions of (1) have been constructed in various function spaces. For example, if f ∈ L r for some r with 3 ≤ r < ∞, then it is well known that there is a unique classical solution in some maximum interval of time: 0 ≤ t < T f , where 0 < T f ≤ ∞. But, for the uniqueness of the pressure, one requires |pðx, tÞ | ⟶0 as |x | ⟶∞. See [1] and [2] for r = 3 and [3] for 3 < r < ∞. The solution is C ∞ for 0 < T f < ∞.
It is well known that for f ∈ L ∞ ðℝ n Þ, there is a unique, smooth, and local-in-time solution u for the Navier-Stokes equations with where R i = ð−ΔÞ −1/2 ∂ x i is the ith Riesz operator. It is known that in ℝ 2 , this solution can be extended globally in time.
For f ∈ L ∞ ðℝ n + Þ, where n ≥ 3, the existence of a local mild solution is proved by Bae and Jin in [8]. In the same paper, it is also proved that such mild solution is indeed a strong solution of the Navier-Stokes equations (1). Before the result of Bae and Jin, the local-in-time existence of mild (strong) solution of the halfspace problem was provided in [9] by Solonikov for continuous bounded initial data in ℝ n + . In this paper, I am interested in obtaining estimates of the maximum norm of the derivatives of u in terms of the maximum norm of the initial function f , assuming that the solution exists, and it is C ∞ ðℝ n + Þ for 0 < t < T f . The work of this paper is a continuation of the work of my papers [10] and [11] to the halfspace case for nondecaying initial data. Nonempty boundary in the domain in this paper makes this work different, in some aspects, and significantly more challenging in proving the key lemmas than the work in my previous works where the initial functions are in ℝ n or T n .
We begin by transforming the momentum equations of (1) into the abstract ordinary differential equations: where A = −ℙΔ is the Stokes operator and ℙ is the Leray projector, which is given by where f n j x n =0 = 0. Note that where y * = ðy 1 ,⋯y n−1 ,−y n Þ, NðxÞ = ð1/ð2 − nÞω n Þjxj 2−n , if n ≥ 3, and ω n denotes the surface area of the unit sphere in ℝ n which is given by ω n = 2π n/2 /Γðn/2Þ. The solution of (3) is formally expressed in the integral form: Solonikov [9] has expressed the solution operator of the Stokes equations in ℝ n + in the integral form where G = ðG ij Þ i,j=1,⋯n is given by The function Γðx, tÞ is the n-dimensional Gaussian kernel defined by Γðx, tÞ ≡ Γ t ðxÞ ≡ ð1/ð4πtÞ n/2 Þe −jxj 2 /4t . A solution formula of the Stokes equations (3) in ℝ n + has also been provided by Ukai in [12]. Such solution formula has been used in the L q setting, particularly for 1 < q < ∞ (see [13,14]). For L 1 and L ∞ estimates of the Stokes flow or its gradient, see [15,16]. The solution formula provided by Solonikov [9] has mainly been used for L ∞ framework (see [14,17]).
To formulate the main result of this paper, we first introduce some notations as follows: and In what follows, if |α | = j, for any j = 0, 1, ⋯, then we will denote D α = D α 1 ⋯ D α n n by D j . We also set Clearly, jD j uðtÞj ∞ measures all space derivatives of order j in maximum norm. For later purposes, let us also introduce a few other notations: Throughout this paper, D j will be understood as the derivative of order j = |α | = |β| + 1. In addition, 1 ℝ n + denotes the characteristic function which is 1 on ℝ n + and 0 otherwise. τ a is a translation operator defined by τ a f ðxÞ ≔ f ða − xÞ.
The goal of this paper is to prove the following theorem.

Theorem 1.
Consider the Cauchy problem for the Navier-Stokes equations ((1)) where f ∈ L ∞ ðℝ n + Þ and ∇·f = 0 is understood in the sense of distribution. There is a constant c 0 > 0 , and for any α = ðα 1 ,⋯α n−1 , 1Þ with |α | = j where j = 0, 1, ⋯ there is a constant K j so that The constants c 0 and K j are independent of t and f .
One of the important tools in the proof of Theorem 1 is the uniform estimates of the composite operator D j e −At ℙ∇ div. But, obtaining such uniform estimates is complicated because of the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain; hence, D j and e −At ℙ may not be commutative.

2
Abstract and Applied Analysis To overcome this difficulty, we will generalize the techniques of obtaining the uniform estimates on ∇e −At ℙ∇div of the paper [8] by Bae and Jin to obtain our desired uniform estimates on D j e −At ℙ∇div. In their paper, they require the uniform estimates to prove the existence of the local solution of the Navier-Stokes equations in halfspace for bounded initial data.
This paper is organized in the following ways. In "Some Auxiliary Results," we introduce some auxiliary results which will be labelled as propositions. In "Estimate of D j e −At P∇·g," we derive an important estimate on the composite operator D j e −At ℙ∇div. In "Estimates for the Navier-Stokes Equations," we establish some estimates on the solution of the Navier-Stokes equations. In "Estimates for the Navier-Stokes Equations," a proof of Theorem 1 will be provided. Finally, Appendices A, B, and C contain proofs of the propositions which are introduced in "Some Auxiliary Results."

Some Auxiliary Results
Let us consider the Stokes problem in ℝ n + , n ≥ 3: . Here, we note that each g ij is quadratic in components of u. Solonikov in [9] has obtained the solution of (13) which is given by Next, we state the following proposition.
Proof. The proof is given in Appendix A.
Proposition 3. Let x ∈ ℝ n + and f be any Hölder continuous function with the exponent 0 < α < 1: Then, for i, j ≠ n or i, j = n, we have ð18Þ Proof. The proof is given in Appendix B.
Next, we define the Hardy space Next, we state a few well-known results related to the Hardy-norm estimates of the Gaussian kernel Γ t for 0 < α < 1.
We omit the proofs of well-known results of Proposition 4.

Abstract and Applied Analysis
Proposition 5. Let j = 1 ⋯ , n − 1 and i = 1, ⋯n . Then we have Proof. The proof is given in Appendix C.

Estimate of D j e −At ℙ∇·g
Solonikov in [9] and Shimizu in [16] provide the following estimates: where f ∈ L ∞ ðℝ n + Þ, g = ðg ij Þ, 1 ≤ i, j ≤ n, and g ij ∈ L ∞ ðℝ n + Þ for each i, j. Also, f and g vanish on the boundary. In addition, in paper [8] by Bae and Jin, they prove as a critical estimate to prove their desired result. With all the above estimates in hand, we begin to obtain the uniform estimate on the composite operator D j e −At ℙ∇·g. For that purpose, recall where Gðx, y, tÞ is defined by (8). In the following, consider i ≠ n, and denote by H = ðH ijk Þ n i,j,k=1 the kernel tensor of the operator e −At ℙ∇div. For simplicity in computational purpose, we consider g ij as a Schwartz class function in ℝ n + vanishing on the boundary for each i, j. Thus, we begin by writing where With integration by parts, we obtain Use G in = 0 for i ≠ n to write the following: Use ∂ x i G ij ðx, y, tÞ = −∂ y i G ij ðx, y, tÞ for i ≠ n and Proposition 5 to justify the following expression: Therefore, (29) can be rewritten as where Þg jk y ð Þdy, First, we estimate I 1 for k ≠ n. For that purpose, recall ∂ y k G ij ðx, y, tÞ = −∂ x k G ij ðx, y, tÞ. Therefore, x G ij x, y, t ð Þg jk y ð Þdy: Clearly, I 1 ðx, tÞ is the derivative of the ith component of the solution of the Stokes equations. So, using estimate 4 Abstract and Applied Analysis (23), we get the desired estimate on I 1 as below Next, we estimate I 1 for k = n. In [9], G ij is given as where Next, we use the estimate where m′ = ðm 1 , m 2 ,⋯m n−1 Þ, k′ = ðk 1 , k 2 ,⋯k n−1 Þ, provided in [18], to obtain the desired estimate on I 1 for k = n. Therefore, we use modified G ij for k = n and rewrite I 1 as The estimate for I * 1 follows as Applying Proposition 4, we obtain To estimate I * * 1 , we use the estimate of (38) and obtain Finally, we get We obtain Therefore, from (35) and (44), we obtain Next, we estimate I 2 : For that, let us begin by rewriting I 2 after dropping the summation notations and negative signs for convenience in writing.
Equivalently, we write where Using expression for G ij from (8) for i, l ≠ n, we obtain where

Abstract and Applied Analysis
To estimate T 1 , let us proceed by writing It is well known that z Γð x − z * , tÞ are in Hardy space H 1 ðℝ n Þ, for any fixed z ∈ ℝ n . Since the Calderon-Zygmund type transforms are bounded in Hardy space, we obtain that Using the estimates of Proposition 4, we arrive at With exactly the same argument as for T 1 , we also obtain It remains to obtain an estimate for T 3 . We use Proposition 3 for i, l ≠ n by replacing f by Γðw, :Þ and also use G ln = 0 to rewrite T 3 as By the same argument as for T 1 , we can obtain Let us rewrite T * * 3 as Set By Proposition 5, Notice that We also recall that 1 ℝ n + D β+2 w Γðw − z * , tÞ and 1 ℝ n − D β+2 w Γð w − z, tÞ are in the Hardy space H 1 ðℝ n Þ, for any fixed w ∈ ℝ n . Since the Calderon-Zygmund type transforms are bounded in Hardy space, after using D β+2 Let us recall a result of Proposition 4: for any a, b ∈ ℝ n , and is bounded by Ct −ð|β|+2+γÞ/2 e −w 2 n /4t j w − xj γ for 0 < γ < 1. Hence, in similar way as for P jk , we obtain Abstract and Applied Analysis Therefore, Using (56) and (64) leads us to obtain Since T = T 1 + T 2 + T 3 , with the use of (53), (54), and (65), we obtain Finally, using (45) and (66) with fact that e −At commutes with D β x , we have proved the following important lemma.

Lemma 6.
For any g = ðgÞ ij , 1 ≤ i, j ≤ n with g ij ∈ L ∞ ðℝ n + Þ , and g ij ð x, 0Þ = 0 , there exists a constant C independent of t and g such that for 0 < t < T, for some T > 0.

Corollary 7.
Let g be as in the previous lemma, then the solution of for some T > 0.
Proof. The solution of (38) is given by Applying the estimate (24), we obtain Hence, we obtain

Estimates for the Navier-Stokes Equations
Recall the transformed abstract ordinary differential equation (3): Solution of (74) with given initial and boundary condition as in (1) is given by Using the solution (75) along with the use of estimates (23), (24), and (25), we prove the following important lemma.

Lemma 8. Set
There is a constant C > 0, independent of t and f , so that Proof. Using estimate (23) for the solution of the Stokes equations in (75), we obtain From (74), after using estimate (24), with the fact that g is quadratic in u gives us Since Ð t 0 ðt − sÞ −1/2 s −1/2 ds = C > 0, which is independent of t, we arrive at the following estimate Next, apply D i to uðtÞ in the integral form to obtain and 7 Abstract and Applied Analysis Let us estimate the integral in the above expression as below.
We use the estimate (25) again with the fact that g is quadratic in u to obtain Therefore, we have the following estimate for The combination of (80) and (85), proves Lemma 8.

Lemma 9.
Let C and u ∈ L ∞ ðℝ n + × ð0, TÞÞ be same as in Lemma 8for some T > 0 . Set Proof. We prove this lemma by contradiction after recalling the definition of VðtÞ in (76). Suppose that (88) does not hold, then denote by t 0 the smallest time with Vðt 0 Þ = 2C jf j ∞ . Use (77) to obtain Thus Therefore, t 0 ≥ c 0 /jf j 2 ∞ . This contradiction proves (88) and T > c 0 /j f j 2 ∞ .

Proof of Theorem 1
Lemma 9 proves Theorem 1 for j = 0, 1 for 0 < t < c 0 /jf j 2 ∞ . Now, we apply induction on j to prove Theorem 1. Suppose j ≥ 1 and assume The solution of above system can be written as Since ∇·v = 0, we can write Using integral form of vðtÞ from above, we can write Our goal is to prove j∂ x n vðtÞj ∞ ≤ Ct −j/2 j f j ∞ . For that, let us start with the following where i ≠ n.
Using the estimate (23) in the first term of the above expression, we obtain Abstract and Applied Analysis where To estimate I 1 uniformly, we proceed as Using Lemma 6, we obtain We use simple integration, and the fact that g is quadratic in u to arrive at Next, we estimate I 2 . For that, we proceed in the following way: Since the order of the derivatives of jD β ∇·gj ∞ is |β | +1, for convenience in writing, we use jD j gj ∞ to estimate jD β ∇·gj ∞ . Since gðuÞ is quadratic in u; therefore By induction hypothesis (91) we obtain Apply estimate (25) to the integral (103) with the use of (104) to obtain where Since Ð t t/2 ðt − sÞ −1/2 s −j/2 ds = Ct ð1−jÞ/2 , where C is independent of t, and the using the estimate of (105), we obtain jJ 2 ðtÞj ∞ ≤ Cj f j 2 ∞ t ð1−jÞ/2 . For J 1 , let us begin as below. Therefore We use these bounds to bind the integral in (97). We have D j u = ∂ x n D β u. Then, maximizing the resulting estimate for t j/2 jD j uðtÞj ∞ over all derivatives D j of order j and setting and from (98), we obtain the following estimate: Since t 1/2 j f j ∞ ≤ ffiffiffiffi c 0 p , then Ct 1/2 jf j 2 Let us fix C j so that the above estimate holds and set 9 Abstract and Applied Analysis First, let us prove the following: Suppose there is a smallest time t 0 such that 0 < t 0 < c j / jf j 2 ∞ with ϕðt 0 Þ = 2C j jf j ∞ . Then, using (88), we obtain which contradicts the assertion. Therefore, we proved the estimate If then we start the corresponding estimate at t − T j . Using Lemma 9, we have juðt − T j Þj ∞ ≤ 2j f j ∞ and obtain Finally, for any t satisfying (118) and (119) yield This completes the proof of Theorem 1.
In the following appendices, we provide proofs of the propositions that are introduced in "Some Auxiliary Results." However, these proofs have also been provided in [8]. For the reader's convenience, we provide them with more details in this paper as well.