A Navier–Stokes-Type Problem with High-Order Elliptic Operator and Applications

: The existence, uniqueness and uniformly L p estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A , the existence, uniqueness and L p estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly L p estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established.

Boundary value problems (BVPs) for differential-operator equations (DOEs) in classes of functions such as Lebesgue ones have been object of interest of a lots of scientists (see, for example [1][2][3]). Presentations to differential-operator equations have been done by several authors [4][5][6]. Regularity results for differential-operator equations are contained in [7][8][9]. In the present note, authors study degenerate parameter-dependent Boundary Value Problems for arbitrary order differential-operator equations. These kinds of problems have been applied in several fields which are useful in lots of fluid mechanics models.
The focus of our work was to prove uniform existence and uniqueness of the stronger local and global solution of the Navier-Stokes problem with a small parameter (1a)-(1c). This problem is characterized by the presence of an abstract operator A and a small parameter ε k that, respectively, corresponds to the inverse of a Reynolds number Re that is very large for the N-S equations. Regularity results of N-S equations were obtained, for example, by the authors in [4][5][6][10][11][12][13][14][15][16][17]. The N-S equations with small viscosity when the boundary is either characteristic or non-characteristic have been well-studied; see for example in the papers [3,14,16]. In addition, regularity properties of abstract differential equation (ADE) were deeply studied in [2,[7][8][9][18][19][20][21][22]. Here, the authors study abstract N-S equations with a high elliptic part in a Banach space E with operator coefficient A. In [22], we derived the L p −regularity properties of the abstract Stokes problem. For E = C, A = a > 0, m = 1, ε 1 = ε 2 = · · · = ε n = 1 the problem (1a)-(1c) state to be usual N-S problem. In this paper, the authors prove that the corresponding Stokes type problem has a unique solution (u, B 1ε ϕ) for f ∈ L p 0, T; X q = B (p, q) , p, q ∈ (1, ∞) and the following uniform estimate holds ∂u ∂t B(p,q) with C = C (T, p, q) independent of f and ε, where X 2m q (A) , X q 1 p ,p denote the real interpolation space between X 2m q (A) = W 2m,q R n + ; E (A) , E n and X q = L q R n + ; E n defined by the K-method (see e.g., [1], §1.3.2). Then, by following Kato-Fujita [13] method, by using (1e) we derive a local a priori estimates for solutions of (1a)-(1c), i.e., we prove that for γ < 1 and δ ≥ 0 such that n Moreover, the solution of (1a)-(1c) is unique for some β with β > |γ|. For sufficiently small date we show that there exists a global solution of (1a)-(1c). Particularly, we prove that there is a δ > 0 such that if a X q < δ, then there exists a global solution u ε of (1a)-(1c) so that Moreover, the following uniform estimates hold sup t,ε j where Q εq denotes the corresponding Stokes operator and P q is a projection operator in X q . In application, we put E = L p 1 (0, 1) and A to be differential operator in (1a) and (1b), with generalized Wentzell-Robin boundary condition defined by where a, b are complex-valued functions. Then, we obtain the existence, uniqueness and uniformly L p (Ω) estimates for solutions the following Wentzell-Robin type mixed problem for the N-S equation Note that the regularity properties of Wentzell-Robin-type boundary value problems (BVP) for elliptic equations were studied e.g., in [23,24] and the references therein. Here, L p Ω denotes the space of all p-summable complex-valued functions with mixed norm i.e., the space of all measurable functions f defined onΩ, for which By using the general abstract result above, the existence, uniqueness and uniformly L p Ω estimates for solution of the problem (1f)-(1h) is obtained.
Moreover, we choose E = L p 1 (0, b) and A to be degenerated differential operator in L p 1 (0, b) defined by where is a bounded function on y ∈ [0, 1] for a.e. x ∈ R n + , α ki , β ki are complex numbers and W [2],p 1 γ Then, we obtain the existence, uniqueness and uniformly L p (Ω) estimates for solutions of the following mixed problem for degenerate N-S equation Let E be a Banach space and L p (Ω; E) denotes the space of strongly measurable E-valued functions that are defined on the measurable subset Ω ⊂ R n with the norm x−y dy is bounded in L p (R, E) , p ∈ (1, ∞) (see. e.g., [19], § 4). U MD spaces include e.g., L p , l p spaces and Lorentz spaces L pq , p, q ∈ (1, ∞).
Let E 1 and E 2 be two Banach spaces. Let B (E 1 , E 2 ) denote the space of all bounded linear operators from E 1 to E 2 . For E 1 = E 2 = E it will be denoted by B (E) .
Here, N denotes the set of natural numbers. R denotes the set of real numbers. Let C be the set of complex numbers and ≤ M (1 + |λ|) −1 for any λ ∈ S ψ where I is the identity operator in E (see e.g., [1], §1.15.1). The positive operator A is said to be R-positive in a Banach space E if the set λ (A + λ) −1 : λ ∈ S ψ is R-bounded (see [19], § 4). The operator A (s) is said to be positive in E uniformly with respect to parameter s with bound M > 0 if D (A (s)) is independent on s, D (A (s)) is dense in E and (A (s) + λ) −1 ≤ M 1+|λ| for all λ ∈ S ψ , where the constant M does not depend on s and λ.
Assume E 0 and E are two Banach spaces and E 0 is continuously and densely included into E. Here, Ω is a measurable set in R n and m is a positive integer. Let W m,p (Ω; E 0 , E) denote the space of all functions u ∈ L p (Ω; E 0 ) that have the generalized derivatives ∂ m u Let H s,p (R n ; E), −∞ < s < ∞ denotes the E−valued fractional Sobolev space of order s that is defined as: with the norm u H s,p = (I − ∆) .
It clear that H 0,p (R n ; E) = L p (R n ; E) . It is known that if E is a UMD space, then H m,p (R n ; E) = W m,p (R n ; E) for positive integer m (see e.g., [25], § 15). H s,p (R n ; E 0 , E) denote the Fractional Sobolev-Lions type space i.e.,

Regularity Properties of Solutions for System of ADEs with Parameters
In this section, we will derive the maximal regularity properties of the BVP for system of ADE with small parameters in half-space α ki are complex numbers, ε = (ε 1 , ε 2 , . . . , ε n ), ε k are small positive parameters, m is a positive integer with m ≥ 1 and A is a linear operator in a Banach space E.
are represent the E-valued unknown velocity and pressure like functions, respectively and f = ( Let By reasoning as in ( [9], Theorem 2) we have Theorem 1. Let E be a UMD space and A be an R-positive operator in E. Assume m is a nonnegative number, Then for all f ∈ X, λ ∈ S ψ with sufficiently large |λ| > 0 problem (2a) and (2b) has a unique solution u that belongs to X m q (A) and the following coercive uniform estimate holds Consider the operator Q ε generated by problem (2a) and (2b), i.e., From Theorem 1 we obtain the following results: Result 1. Suppose the all conditions of Theorem 1 are satisfied. Then, there exists a resolvent (Q ε + λ) −1 for λ ∈ S ψ satisfying the following uniform estimate It is clear that the solution (2a) and (2b) depend on parameters ε = (ε 1 , ε 2 , . . . , ε n ), i.e., u = u ε (x) . In view of the Theorem 1, we derive the properties of solutions (2a) and (2b).
From Theorem 1 we obtain:

The Stokes System with Small Parameters
In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem are represent the E-valued unknown velocity and pressure like functions, respectively and f = ( Here and hereafter E * will denoted the conjugate of E and (, .) (resp. <, . >)) denotes the duality pairing of functions on R n + (resp. R n−1 ) and X q = X q (E * ) = L q R n + ; E * n will denote the dual (Y m,q (A)) n will be denoted by X m q and X m q (A) , respectively. Let Consider the space Y q (A, B 1 ) becomes a Banach space with this norm. Consider the problem By using Theorem 1, we obtain the following Corollary 1. Let E be a UMD space and A be an R-positive operator in E. Assume m is a nonnegative number, q ∈ (1, ∞). Then for all f ∈ X −m q problem (3b) has a unique solution u ∈ X m q (A) and the following estimate holds It is known that (see e.g., [11,12]) vector field u ∈ L q R n + n has a Helmholtz decomposition.
In the following theorem we generalize this result for E− valued function space X q . By reasoning as in [11,12] and ( [22], Theorem 3.1) we have decomposition result via operatorB generated by problem (3b).

Theorem 2.
Let E be an U MD space and q ∈ (1, ∞). Assume there exists a constant C 0 > 0 such that Then u ∈ X q has a Helmholtz decomposition i.e., there exists a bounded linear projection operator P q from X q onto X σq with null space In particular, all u ∈ X q has a unique decomposition u = u 0 + B 1 ϕ with u 0 = P q u ∈ X qσ so that For proving the Theorem 2 we need the following lemma: By reasoning as in ( [12], Lemma 2) we get: Here, < ,> and (, )denotes the duality pairing of abstract functions defined in R n−1 and R n + , respectively. From [22] we have and the following estimate holds where By virtue of trace theorem in W s,q (0, a; E (A) , E) , the interpolation of intersection and dual spaces (see e.g., ([22], §1.8.2, 1.12.1, 1.11.2)) and by localization argument we obtain that the operator υ → ∂ i ∂x i n υ (x , 0) is a bounded linear and surjective from X m q (A) onto W i,m,q (E). Hence, we can find for each υ i ∈ W i,m,q (E) an element Φ ∈ X 2m−1 q (A) so that Therefore, from (3e) we get This implies the existence of an element Thus, we have proved the existence of the operators u → ∂ i ∂x i n u (x , 0). The uniqueness follows from Lemma 1. Now we are going to construct the projection operator P q . Let f ∈ X q and f = ( f 1 (x) , f 2 (x) , . . . , f n (x)). Consider the boundary value problem Since B 2 f (x) ∈ X −m q , in view of Corollary 1, then for all f ∈ X q problem (3f) has a unique solution u 1 ∈ X m q (A) and the following estimate holds Now consider the problem By Theorem 1, we obtain that for all υ k ∈ W θ k ,m,q (E) problem (3h) has a unique solution u 2 ∈ X m q (A) and the following estimate holds . For any f ∈ X q , we take the solution of (3f), then that of (3h) and put u = u 1 + u 2 . We define P q u = u − B 1 u.
Then by reasoning as in [12,16] we have Lemma 2. Let E be an UDM space and q ∈ (1, ∞). Then, P q X q is a closed subspace of X q .

Lemma 3.
Let E be an U MD space and q ∈ (1, ∞). Then, the operator P q is a linear bounded operator in X q and P q f = f if B 1 f (x) = 0.

Lemma 4.
Assume E is an U MD space, A is an R-positive operator in E and q ∈ (1, ∞). Then the conjugate of P q is defined as P * q = P q , 1 q + 1 q = 1 and this operator is bounded linear in L q R n From Lemmas 3 and 4 we obtain Lemma 5. Assume E is an U MD space and q ∈ (1, ∞). Then P q X q ⊥ = W q , 1 q + 1 q = 1.

Lemma 6.
Assume E is an U MD space and q ∈ (1, ∈). Then Now we are ready to prove the Theorem 2.
Proof of Theorem 2. From Lemmas 5 and 6 we get that X qσ = P q X q ⊥ . Then, by construction of P q we have By Lemmas 2 and 3, we obtain the estimate (3a). Moreover, by Lemma 5, W q is a close subspace of X q . Then, it is known that the dual space of quotient space X q /W q is W ⊥ q . By first assertion we have X q /W q = X σq . Theorem 3. Let E be an U MD space, A is an R-positive operator in E, q ∈ (1, ∞) . Then, problem (3a) and (3b) has a unique solution u ∈ X 2m q (A) for f ∈ X q , ϕ ∈ W m,q R n + ; E , λ ∈ S ψ and the following coercive uniform estimate holds with C = C (q, A) independent of ε 1 , ε 2 , . . . , ε n , λ and f .

Proof. By applying the operator P q to problem (2a) and (2b) we get the Stokes problem (3a) and (3b). It is clear to see that
where O εq is the abstract Stokes operator generated by problem (3a) and (3b) and Q ε is an abstrat elliptic operator in X q defined by (2e).
Then Theorem 2 we obtain the assertion.

Result 3.
From the Theorem 3 we get that Q εq is a positive operator in X q and also generates a bounded holomorphic semigroup S ε (t) = exp −Q εq t for t > 0. In a similar way as in [11] we show Proposition 2. The following estimate holds uniformly in ε = (ε 1 , ε 2 , . . . , ε n ) for α ≥ 0 and t > 0.
Proof. From Theorem 3 we obtain that the operator O εq is uniformly positive in X q , i.e., for λ ∈ S ψ,κ , 0 < ψ < π the following estimate holds where the constant M is independent of λ and ε. Then, by using Danford integral and operator calculus (see e.g., in [10]) we obtain the assertion. Now we can prove the main result of this section Theorem 4. Let 0 < ε k ≤ 1. Then, for f ∈ L p 0, T; X q = B (p, q) and a ∈ X 2m q (A) , X q 1 p ,p , p, q ∈ (1, ∞) there is a unique solution (u, B 1ε ϕ) of the problem (1d) and the following uniform estimate holds ∂u ∂t B(p,q) with C = C (T, p, q) independent of f and ε.
Proof. The problem (1d) can be expressed as the following abstract parabolic problem By Proposition 2, operator O εq is uniform positive and generates holomorphic semigroup in X q . Moreover, by using ( [9], Theorem 3) we get that the operator Q εq is R-positive in X q uniformly with respect to ε = (ε 1 , ε 2 , . . . , ε n ) . Since E is a UMD space, in a similar way as in ( [20], Theorem 4.2) we obtain that for all f ∈ L p 0, T; X q and a ∈ X 2m q (A) , E 1 p ,p there is a unique solution u ∈ W 1,p 0, T, D O εq , E of the problem (4b) so that the following uniform estimate holds du dt L p (0,T;Xq) From the estimates (3k) and (3l) we obtain the assertion.

Result 4.
It should be noted that if ε 1 = ε 2 = . . . ε n = 1 we obtain maximal regularity properties of abstract Stokes problem without any parameters in principal part.

Remark 2.
There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1d) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, by virtue of Theorem 3 and 4 we can obtain the maximal regularity properties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.

Existence and Uniqueness for N-S Equation with Parameters
In this section, we study the N-S problem (1a)-(1c) in X q . The problem (1a)-(1c) can be expressed as du dt We consider the Equation (4a) in integral form For proving the main result we need the following lemma which is obtained from ([11], Theorem 2).

Lemma 8.
Let E be a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. For each k = 1, 2, . . . , n the operator u → Q − 1 2 εq P q ∂ m ∂x m k u extends uniquely to a uniformly bounded linear operator from L q R n + ; E to X q .
Proof. Since Q εq is a positive operator, it has a fractional powers O α εq . From the Lemma 7, It follows that the domain D Q α εq is continuously embedded in H 2α Then by using the duality argument and due to uniform positivity of O 1 2 εq we obtain the following uniform estimate By reasoning as in [10] we obtain the following Lemma 9. Let E be a a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. Let 0 ≤ δ < 1 2 + n 2 1 − 1 q . Then the following uniform estimate holds n ∩ X qσ and L q R n + ; E ∩ X q is the same as X s , by Sobolev embedding theorem we obtain that the operators By duality argument then, we get that the operator u → Q −ν ε,q is bounded from X s to X q , where Consider first the case δ > 1 2 . Since P(u, B 1ε )υ is bilinear in u, υ, it suffices to prove the estimate on a dense subspace. Therefore assume that u and υ are smooth. Since div B 1ε u = 0, we get Taking ν = δ − 1 2 and using the uniform boundedness of Q −ν ε,q , from X s to X q and Lemma 8 for all ε > 0 we obtain the uniform estimate By assumption we can take r and η such that Since D Q α ε,q is continuously embedded in H 2α q R n + ; E (A) , E n ∩ X qσ , then by Sobolev embedding we get i.e., we have the required result for δ > 1 2 . In particular, we get Similarly we obtain ε,q υ η for 1 r + 1 η = 1 q and δ = 0. The above two estimates show that the map υ → P q (u, B 1ε ) υ is a uniform bounded operator from D Q to X q . By using the Lemma 7 and the interpolation theory of Banach spaces for 0 ≤ δ ≤ 1 2 we obtain the uniform estimate By using Lemma 9 and iteration argument, by reasoning as in Fujita and Kato [13] we obtain the following.
Proof. We introduce the following iteration scheme By estimating the term u 0 (t) in (4c) and by using the Lemma 9 for γ ≤ α < 1 − δ we get the uniform estimate where N = sup 0<t≤T t 1−γ−δ Q −δ εq P f (t) and B (β, η) is the beta function. Here we suppose γ + δ > 0. By induction assume that u m (t) satisfies the following We shall estimate Q α εq u m+1 (t) by using (4b). To estimate the term Q −δ εq Fu m (s) we suppose so that the numbers θ, σ, δ satisfy the assumptions of Lemma 9. Using Lemma 9 and (4d), we get Therefore, we obtain We get the uniform estimate. So, the remaining part of proof is obtained the same as in ([10], Theorem 2.3).
By reasoning as in [13] we obtain Lemma 10. Let the operator A ε be uniform positive in a Banach space E and α be a positive number with 0 < α < 1. Then, the following uniform inequality holds Proposition 3. Let E be a space satisfying a multiplier condition, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. Let u be the solution given by Theorem 5 Then Q α εq u for γ < α < 1 − δ is uniform Hölder continuous on every interval [η, T * ], 0 < η < T * for all parameters ε k > 0.
Proof. It suffices to prove the Hölder continuity of Q α ε υ, where Using the Lemma 10 we get the uniform estimate Then by reasoning as in ( [10], Proposition 2.4) we obtain the assertion. Theorem 6. Let E be a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. Assume P q f : (0, T * ] → X q is Hölder continuous on each subinterval [η, T * ]. Then, the solution of (4b) given by Theorem 5 satisfies Equation (4a) for all ε k > 0. Moreover, u ∈ D O εq for t ∈ (0 T * ]. Proof. It suffices to show Hölder continuity of Fu (t) on each interval [η, T * ] . It is clear to see that u (η) ∈ X q and Since P q f is continuous on [η, T * ] we get The uniqueness of u (t) ensured by Theorem 5, implies the following uniform estimates Since we can choose θ, σ so that Lemma 8 implies that Fu (t) is Hölder continuous on every interval [η, T * ] .

Regularity Properties
The purposes of this section is to show that the solutions of (1a) are smooth if the data are smooth. For simplicity, we assume P q f = 0. The proof when P q f = 0 is the same. Consider first all of the Stokes problem (3d) and (3e).
Hence, by Lemma 7 we get the following uniform estimate This estimates together with Lemma 13 shows that Lemmas 11 and 12 now imply that Since D Q Then the proof will be completed as in ( [10], Proposition 3.5) by using the induction.

Now we can state the main result of this section
Theorem 7. Let E be a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. Let E be Banach algebra and a ∈ X q . Suppose that the solution u = u ε (t) of (4b) for P q F = 0 given by Theorem 5 exists Proof. For q > n the assertion is obtained from the Proposition 4. Let us show that the assertion is valid for 1 < q ≤ n. Indeed, the solution u = u ε (t) of (5b) for P q F = 0 given by Theorem 5 satisfies the Equation (5a) on every subinterval [η, T * ] , 0 < η < T. Theorem 6 shows that u ε (η) ∈ D Q εq . Since 0 ≤ n 2q − 1 2 ≤ γ < 1, we have D Q γ εq ⊂ X n so that D Q εq ⊂ X s for some s > n. By (4b) this means that we may assume q > n and a ∈ X q .

Existence of Global Solutions
In this section, we prove the existence and estimate of a global solution of the problem (1a)-(1c). The proofs of these theorems are based on the theory of holomorphic semigroups and fractional powers of generators. We assume for simplicity that f = 0, although it is not difficult to include nonzero f under appropriate conditions. The main result is the following Theorem 8. Let E be a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1 and a ∈ X q .
Theorem 9. Let E be a a UMD space, A an R-positive operator in E, q ∈ (1, ∞) and 0 < ε k ≤ 1. There is a µ > 0 such that if a X q < µ, then there exists a global solution u ε of (1a) − (1c), so that t ; X q for n ≤ q < ∞. Moreover, the following uniform estimates hold Proof. It is clear to see from proof of Theorem 6.1 that M k and M k are bounded by a constant M if M 0 ≤ λ. By (7i) this is true if a X q is sufficiently small. In this case, as in [15] we prove that the sequences t (1−δ)/2 u εk , t 1/2 B 1ε u εk are bounded on (0, ∞) uniformly in k and ε 1 , ε 2 , . . . , ε n i.e., sup t,ε k Then (6k) is obtained from (6j).

Theorem 10.
Let the all conditions of Theorem 9 hold. Then u ε (t) X p → 0 uniformly in ε as t → ∞. More precisely, we have where, u 0ε (t) = S ε (t) a and δ < min 1, n − n q , n q − 1 .
B (p, q) = L q 0, T; X p , p = (p 1 , p) , p 1 , p, q ∈ (1, ∞) denotes the product of Lebesque spaces with corresponding mixed norm and X 2m,2 p , X p 1 p ,p denote real interpolation space between Y 2m,2 p and X p .
Therefore, the problem (1f)-(1h) can be rewritten in the form of (1a)-(1c), where u (x) = u (x, .) , f (x) = f (x, .) are functions with values in E = L p 1 (0, 1) . From [23,24] we get that the operator A generates analytic semigroup in L p 1 (0, 1) . Moreover, we obtain that the operator A is R-positive in L p 1 . Then from Theorem 8 we obtain the assertion.