Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions

Shaoliang Yuan; Lehui Huang; Lin Cheng; Xiaoguang You

doi:10.1515/math-2024-0077

Article Open Access

Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions

Shaoliang Yuan , Lehui Huang , Lin Cheng and Xiaoguang You

Published/Copyright: November 6, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we investigate the Euler- α equations in a three-dimensional bounded domain. On the one hand, we prove in the Euler setting that the equations are locally well-posed with initial data in H s ( s ≥ 3 ) . On the other hand, the relationship between the H s -norm of the velocity field and the parameter α is clarified.

Keywords: non-Newtonian fluids; well-posedness for PDEs; Euler-α equations; Euler equations

MSC 2010: 35A01; 35D35; 35Q35

1 Introduction

In this article, we consider the Euler- α equations

(1) ∂ t ( u − α 2 Δ u ) + u ⋅ ∇ ( u − α 2 Δ u ) + ( u i − α 2 Δ u i ) ∇ u i = − ∇ p , ∇ ⋅ u = 0

in a smooth, simply connected domain Ω ⊂ R 3 with boundary Γ , where the unknowns are the vector field u = ( u 1 , u 2 , u 3 ) and the scalar function p . The parameter α > 0 represents the elastic response.

The Euler- α equations were first introduced by Holm et al. [1,2] as a high-dimensional generalization of the Camassa-Holm equation for modeling and analyzing the nonlinear shallow water waves. In addition, the Euler- α system corresponds to setting viscosity to zero in the second-grade fluid equations, which is a well-known non-Newtonian fluid model, see [3]. Moreover, in the three-dimensional (3D) case the Euler- α system inspired the introduction of the Navier-Stokes- α and Leray- α viscous models, which turned out to be remarkable sub-grid scale models of turbulence; for more details, we refer to [4,5] and references therein.

There has been substantial work on the Euler- α system. Busuioc [6] proved the global existence and uniqueness of solutions in R 2 and the local existence in R 3 . Zang [7] obtained the global existence for two-dimensional (2D) period domains. Existence and uniqueness of solutions for the Euler- α system in a bounded domain with Navier boundary conditions were established in [8]. However, for a bounded domain with no-slip boundary conditions, Lopes et al. proved the global well-posedness for the 2D case, and Shkoller [9] showed that Euler- α equations are locally well-posed for the 3D case by transferring the problem to the Lagrangian setting. However, the global existence of the 3D case is still an open problem.

The main motivation of this article is the vanishing- α limit problem. Indeed, let α = 0 in (1), we formally obtain the incompressible Euler equations. For a 2D bounded domain with no-slip boundary conditions, Lopes Filho et al. [10] verified the convergence, while it is open for the 3D case. Even worse, whether the solutions of (1) exist uniformly for some positive T > 0 with respect to α is unknown. In this article, we aim to estimate the H s -bounds of the velocity field and clarify its relationship with the parameter α . We observe that system (1) is nonlinear and involves high-order partial derivatives; therefore, it is difficult to obtain a priori bounds from (1) directly by using the elementary energy methods. Hence, we in this article consider instead the vorticity formulation of system (1). Indeed, by taking curl of u − α 2 Δ u , we find that u satisfies the following equations:

(2) ∂ t q + u ⋅ ∇ q = q ⋅ ∇ u , ∇ × ( u − α 2 Δ u ) = q , ∇ ⋅ u = 0 , u ∣ Γ = 0 .

Here, q represents the unfiltered vorticity, we also refer to as vorticity for simplicity. Let A be the Stokes operator, we define the vector field v as v ≔ u + α 2 A u . Then, it is ready to check that the pair ( v , q ) obeys the vector potential equation, i.e.,

(3) ∇ × v = q .

To solve (2), the main difficulty is to obtain uniform bounds of v from (3) for given q . In the two dimensional case, this problem is handed by introducing a scalar potential called stream-function. However, in the 3D case, the situation is much more complicated. On the one hand, the “curl curl” operation on a vector field will produce a gradient part, which means that, in general, the stream-function and the vorticity do not satisfy the Poisson equations. On the other hand, the vector potential v satisfies only the slip boundary conditions, so we have to seek new approaches to obtain low bounds of v .

Let us explain briefly the way that we treat these issues. Indeed, we first linearize the vorticity formulation into

(4) ∂ t q + u ⋅ ∇ q = q ⋅ ∇ u in ( 0 , T ] × Ω , ∇ × ( u ˜ − α 2 Δ u ˜ ) = q in [ 0 , T ] × Ω , ∇ ⋅ u ˜ = 0 in [ 0 , T ] × Ω , u ˜ = 0 on [ 0 , T ] × ∂ Ω , u ˜ ∣ t = 0 = u 0 in Ω ,

where u 0 is the initial data. For a given vector field u , we would like to obtain uniform bounds of high-order partial derivatives of u ˜ from (4). With regard to low-order derivatives of u ˜ , we shall show that the pair ( u , u ˜ ) satisfies the following temporary equation:

(5) ∂ t ( u ˜ − α 2 Δ u ˜ ) + u ⋅ ∇ ( u ˜ − α 2 Δ u ˜ ) + ∇ u i ( u ˜ i − α 2 Δ u ˜ i ) + ∇ p ˜ = 0 ,

from which, we may establish uniform bounds of L 2 -norm of u ˜ . Then, we define a mapping G as u ˜ = G [ u ] , and we show that the mapping G is a contraction. Finally, using the Banach fixed point theorem, we are able to show that the fixed point of G is a unique solution of (1). Moreover, the relationship with the parameter α is established. In fact, we have the following result.

Theorem 1.1

Let s ⩾ 3 be an integer. Suppose that the initial data u 0 ∈ X α s ( Ω ) ∩ V ( Ω ) . Then, the Euler- α equations (1) have a unique solution u ∈ L ∞ ( [ 0 , T ] ; X α s ( Ω ) ) , where T is defined as

(6) T ≔ α s K 2 exp K 1 ‖ u 0 ‖ X α s ,

and K 1 , K 2 are two positive constants that are independent of α . Furthermore, the velocity field u satisfies the following estimate:

(7) ‖ u ( t ) ‖ X α s ⩽ K 1 α − s ‖ u 0 ‖ X α s exp K 1 ‖ u 0 ‖ X α s .

Remark 1.2

The dependence on α − s of the estimate (7) comes from the Stokes equations (15), i.e., the inequality (16), which intimates that it is a very challenging task to establish uniform bounds of u with respect to arbitrary α > 0 in the H s -norm ( s ≥ 2 ). More precisely, we consider the following Stokes problem:

(8) u α + α 2 A u α = f , ∇ ⋅ u = 0 , u ∣ Γ = 0 ,

for some f ∈ H 2 ( Ω ) . Suppose that u α are uniformly bounded in H 2 ( Ω ) with respect to α . Let α → 0 , it follows that

(9) u ≔ lim α → 0 u α = f .

Here, u ∈ H 2 ( Ω ) ∩ V ( Ω ) . Observing that f is a vector in H 2 ( Ω ) ; therefore, it is contradiction.

The remainder of this article is organized as follows. In Section 2, we introduce notations, functional spaces, and some preliminary results about transport equations, vector potential theory, and Stokes equations. In Section 3, we prove the well-posedness of the linearized system (4). In addition, some uniform bounds of u ˜ are established. In Section 4, we prove the main result.

2 Some notations and preliminary results

In this article, unless otherwise stated, the symbol Ω represents a bounded, smooth, and simply connected domain in R 3 with boundary Γ . Let n be the unit outward normal to Γ . H s ( Ω ) is the space of vector fields whose three components belong to the Sobolev space H s ( Ω ) equipped with its natural norm, while H s − 1 2 ( Γ ) is the trace space of functions from H s ( Ω ) . We use the standard notation H 0 ( Ω ) ≡ L 2 ( Ω ) . H 0 1 ( Ω ) (respectively H 0 1 ( Ω ) ) is the closure of D of all elements in H 1 ( Ω ) (respectively H 1 ( Ω ) ), where, as usual, D is the space of all indefinitely differential functions with a compact support in Ω .

The standard operators gradient, curl and divergence are, respectively, denoted by ∇ , ∇ × , and ∇ ⋅ . We note that the vector-valued Laplace operator of a vector field v = ( v 1 , v 2 , v 3 ) is equivalently defined by

(10) Δ v = ∇ ( ∇ ⋅ v ) − ∇ × ( ∇ × v ) .

The flows that considered in this article are all incompressible, and therefore, we would like to introduce some Sobolev spaces such that the vector fields are divergence free.

Definition 2.1

The spaces V ( Ω ) , L σ 2 ( Ω ) are defined as

(11) V ( Ω ) ≔ { v ∈ H 1 ( Ω ) ; ∇ ⋅ v = 0 in Ω , v ∣ Γ = 0 } ,

(12) L σ 2 ( Ω ) ≔ { v ∈ L 2 ( Ω ) ; ∇ ⋅ v = 0 in Ω , v ⋅ n = 0 on Γ } .

We denote by P : L 2 ( Ω ) → L σ 2 ( Ω ) the Helmholtz projection and A ≔ − P Δ the Stokes operator.

Noting that the high-order derivatives of u in system (1) are coupled with the coefficient α ; we therefore introduce the following norms for the sake of convenience.

Definition 2.2

Let s be a positive integer. Suppose that u ∈ H s + 2 ( Ω ) , the ‖ ⋅ ‖ X α s -norm of u is defined as

(13) ‖ u ‖ X α s ≔ ‖ u ‖ H s + α 2 ‖ D 2 u ‖ H s .

Definition 2.3

Suppose that u ∈ H 1 ( Ω ) , we define the ‖ ⋅ ‖ H α 1 -norm of u by

(14) ‖ u ‖ H α 1 ≔ ‖ u ‖ H 1 + α ‖ ∇ u ‖ H 1 .

Throughout the article, where we use K as a positive constant with neither any subscript nor superscript, then K is considered a generic constant whose value can change from line to line. Conversely, by K T (respectively K s ), we mean a positive constant that may depend on the parameter T (respectively s ). It should be noted that, unless otherwise stated, these constants are all independent of α . In addition, we will place in bold characters the vector-valued functions and the usual for the scalar functions.

In the sequel of this section, we present some well-known results about Stokes equations, transport equations, and vector potential theory.

We first introduce the existence and uniqueness of solutions for the stationary Stokes equations and establish an estimate that plays a great role in Section 3.

Lemma 2.4

Let s be a non-negative integer and α > 0 . Suppose that v ∈ H s ( Ω ) . Then, the stationary Stokes equations

(15) u − α 2 Δ u + ∇ p = v in Ω , u = 0 o n ∂ Ω

have a unique solution u ∈ X α s ( Ω ) ∩ V ( Ω ) with the estimates

(16) ‖ u ‖ X α s + ‖ ∇ p ‖ H s ⩽ K α − s ‖ v ‖ H s .

Proof

The proof of existence and uniqueness of solutions can be found in [11]. We here verify (16), and let us first rewrite (15) as

(17) α − 2 u − Δ u + ∇ p = α − 2 v .

We infer from Theorem 8 of Chapter 3 in [12] that

(18) ‖ u ‖ L 2 + α ‖ ∇ u ‖ L 2 + α 2 ‖ D 2 u ‖ L 2 ⩽ K ‖ v ‖ L 2 ,

therefore (16) holds for s = 0 . To estimate for s ≥ 1 , we should rewrite (15) as

(19) − Δ u + ∇ p = f ,

with

(20) f = α − 2 ( v − u ) .

It follows from Lemma IV.6.1 in [13] that

(21) ‖ D s u ‖ L 2 + ‖ D s − 1 p ‖ L 2 ⩽ K ‖ f ‖ H s − 2 ⩽ K α − 2 ( ‖ u ‖ H s − 2 + ‖ v ‖ H s − 2 ) ,

which yields (16) by an iterative argument.□

We then state the existence and uniqueness of solutions for the first equation in system (2), which is a transport equation.

Lemma 2.5

Let T > 0 be fixed and suppose that u ∈ L ∞ ( [ 0 , T ] ; H 3 ( Ω ) ∩ V ( Ω ) ) and q 0 ∈ L 2 ( Ω ) . Then, the transport equations

(22) ∂ t q + u ⋅ ∇ q = q ⋅ ∇ u in ( 0 , T ] × Ω , q ∣ t = 0 = q 0 in Ω

have a unique solution q ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) .

For a detailed proof of Lemma 2.5, we refer to [14]. Next, we introduce the Helmholtz decomposition in a 3D bounded domain, which was established in [15].

Lemma 2.6

Let s be a non-negative integer and suppose that q ∈ H s ( Ω ) . Then, there exists a unique v ∈ H s + 1 ( Ω ) ∩ L σ 2 ( Ω ) and p ∈ H s + 1 ( Ω ) ∩ H 0 1 ( Ω ) such that

(23) q = ∇ × v + ∇ p .

Corollary 2.7

Suppose in addition that q is divergence free. Then, the gradient part of the decomposition can be dropped, that is

(24) q = ∇ × v .

Proof of Corollary 2.7

Applying the divergence operation to relation (23), we find that

Δ p = ∇ ⋅ q = 0 in Ω .

Observing that p vanishes on the boundary Γ , it follows that p ≡ 0 in Ω .□

In [16], the estimates with regard to the potential of a vector field were established, and these results play a significant role in the next section.

Lemma 2.8

Let s ⩾ 1 be a integer. Then, the following space

{ v ∈ L 2 ( Ω ) ; ∇ × v ∈ H s − 1 ( Ω ) , ∇ ⋅ v ∈ H s − 1 ( Ω ) a n d v ⋅ n ∈ H s − 1 2 ( Γ ) }

is continuously imbedded in H s ( Ω ) .

It should be noted that the vector space introduced in Lemma 2.8 involves a boundary estimate. Let us denote by γ the trace operator defined on H 1 ( Ω ) . It is great that the operator γ satisfies the following estimate.

Lemma 2.9

Let ε > 0 be fixed arbitrary. Then, for all u ∈ H 1 ( Ω ) , we have that

(25) ‖ γ u ‖ L 2 ( Γ ) ⩽ K ε ‖ ∇ u ‖ L 2 ( Ω ) + ε − 1 ‖ u ‖ L 2 ( Ω ) ,

where K is a constant that is independent of ε .

The proof of Lemma 2.9 can be found in [17]. We end this section by introducing a new functional space that will be used in Section 4.

Definition 2.10

Let T , M ∈ R + , and let s ⩾ 3 be an integer. We define the vector space ℰ T , M s by

ℰ T , M s = { u ∈ C ( [ 0 , T ] ; V ( Ω ) ) ∣ ‖ u ‖ L ∞ ( [ 0 , T ] ; X α s ) ⩽ M } .

It is easy to see that the vector space ℰ T , M s is a subspace of C ( [ 0 , T ] ; H 1 ( Ω ) ) , and we next show that ℰ T , M s is indeed a closed subspace.

Lemma 2.11

The vector space ℰ T , M s with the norm C ( [ 0 , T ] ; H 1 ( Ω ) ) is a Banach space.

Proof

Let { u n } n ∈ N + ⊂ ℰ T , M s be a sequence that converges to some u in C ( [ 0 , T ] ; H 1 ( Ω ) ) . We assert that u ∈ ℰ T , M s . Indeed, observing that u n is uniformly bounded by M in L ∞ ( [ 0 , T ] ; X α s ) , there must exists a subsequence { u n k } that converges weak-star to some u ˜ , and ‖ u ˜ ‖ L ∞ ( [ 0 , T ] ; X α s ) is also bounded by M . Observing that the subsequence { u n k } also converges to u strongly in C ( [ 0 , T ] ; H 1 ( Ω ) ) , it turns out that u ˜ ≡ u . Therefore, we have that u ∈ ℰ T , M s .□

3 Well-posedness of the linearized vorticity formulation

Let s ⩾ 3 be an integer, and let T > 0 be fixed arbitrarily. Suppose that u 0 ∈ X α s ( Ω ) ∩ V ( Ω ) and u ∈ L 1 ( [ 0 , T ] ; X α s ( Ω ) ) ∩ V ( Ω ) . We prove in this section that there exists a unique solution u ˜ ∈ C ( [ 0 , T ] ; X α s ( Ω ) ) to system (4). Furthermore, we establish some uniform estimates of u ˜ , which will be used in the next section.

We begin with the first subequation in system (4), that is

(26) ∂ t q + u ⋅ ∇ q = q ⋅ ∇ u , q ∣ t = 0 = q 0 ,

where q 0 = ∇ × ( u 0 − α 2 Δ u 0 ) . For this transport equation, we have the following result:

Proposition 3.1

Suppose that u 0 ∈ X α s ∩ V ( Ω ) and u ∈ L 1 ( [ 0 , T ] ; X α s ∩ V ( Ω ) ) . Then, the transport equation (26) has a unique solution q ∈ C ( [ 0 , T ] ; H s − 1 ( Ω ) ) . Furthermore, for t ∈ [ 0 , T ] , q ( t ) is divergence free and satisfies

(27) ‖ q ( t ) ‖ H s − 1 ⩽ ‖ u 0 ‖ X α s exp K s ∫ 0 t ‖ u ( t ′ ) ‖ X α s d t ′ ,

where K s is a constant that depends only on s.

Proof

Thanks to Lemma 2.5, we know that transport equations (26) have a unique solution q ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) .

We then show that q is divergence free for t ∈ [ 0 , T ] . Indeed, by applying the divergence operation to (26), we find that

(28) ∂ t ( ∇ ⋅ q ) + u ⋅ ∇ ( ∇ ⋅ q ) = 0 .

Now, we multiply (28) by ∇ ⋅ q , and integrate in space to obtain that

(29) 1 2 d d t ‖ ∇ ⋅ q ‖ L 2 2 = 0 .

Observing that q 0 is divergence free, the above inequality implies that q ( t ) is divergence free for t ∈ [ 0 , T ] .

Next, we prove that q satisfies (27). By applying the partial derivative ∂ β to (26), we obtain

(30) ∂ t ∂ β q + u ⋅ ∇ ∂ β q + ∑ ∣ γ + ω ∣ = ∣ β ∣ , ∣ γ ∣ ⩾ 1 C γ , ω ∂ γ u ⋅ ∇ ∂ ω q = ∑ ∣ γ + ω ∣ = ∣ β ∣ D γ , ω ∂ γ q ⋅ ∇ ∂ ω u ,

where C γ , ω , D γ , ω are constants that depend on the indexes γ , ω . We multiply (30) by ∂ β q , then sum over ∣ β ∣ ⩽ s − 1 , and integrate in space to obtain that

(31) d d t ‖ q ‖ H s − 1 2 ⩽ K s ‖ u ‖ H s ‖ q ‖ H s − 1 2 ⩽ K s ‖ u ‖ X α s ‖ q ‖ H s − 1 2 .

The Grönwall inequality then leads to (27) immediately.□

With the existence, uniqueness, and estimates of the solution q in hand, we can proceed considering the following problem:

(32) ∇ × ( u ˜ − α 2 Δ u ˜ ) = q , ∇ ⋅ u ˜ = 0 , u ˜ ∣ Γ = 0 .

In the 2D case, this problem is commonly handled by introducing a scalar potential called stream-function, and it leads to solve two boundary-value problems (i.e. the Poisson equations and the stationary Stokes equations). However, in the 3D case, the situation is much more complicated: (i) the potential is no longer a scalar function, but a vector function with three components; (ii) adequate boundary conditions must be enforced on the vector potential to ensure the existence and uniqueness. In this article, we decide to divide problem (32) into two parts by introducing a temporary unknown vector field v ˜ such that

(33) ∇ × v ˜ = q , ∇ ⋅ v ˜ = 0 , v ˜ ⋅ n ∣ Γ = 0 ,

and

(34) u ˜ − α 2 Δ u ˜ + ∇ p = v ˜ , u ˜ ∣ Γ = 0 .

We begin with equations (33), for which, we have the following result:

Proposition 3.2

Suppose that q ∈ H s − 1 ( Ω ) and is divergence free. Then, there exists a unique solution v ˜ ∈ H s ( Ω ) ∩ L σ 2 ( Ω ) to equation (33), and the following property holds:

(35) ‖ v ˜ ‖ H s ⩽ K ( ‖ v ˜ ‖ L 2 + ‖ q ‖ H s − 1 ) .

Remark 3.3

Since v ˜ is the vector potential of q , we have that ∇ × v ˜ ∣ t = 0 = q 0 . Then it is easy to deduce that v ˜ ∣ t = 0 ≡ u 0 .

Proof of Proposition 3.2

Since q ∈ H s − 1 ( Ω ) and is divergence free, it follows from Lemma 2.6 and Corollary 2.7 that there exists a unique v ˜ ∈ H s ( Ω ) to equation (33).

Next, by applying Lemma 2.8 to v ˜ , we obtain that

(36) ‖ v ˜ ‖ H s ( Ω ) ⩽ K ( ‖ γ v ˜ ‖ H s − 1 2 ( Γ ) + ‖ q ‖ H s − 1 ( Ω ) ) .

In view of Lemma 2.9, we deduce that (35) holds.□

In the 2D case, the zero-order term ‖ v ˜ ‖ L 2 ( Ω ) on the right-hand side of (35) can be absorbed by using the Poincaré inequality. This is mainly due to the fact that, in this case, the term ∇ ( ∇ ⋅ v ) does not appear on the right-hand side of (10). Indeed, for the 2D case, the vorticity q and the stream function Ψ satisfy the following Poisson equation:

(37) Δ Ψ = − q , Ψ ∣ Γ = 0 ,

and the velocity field v ˜ equals to ∇ ⊥ Ψ . The Poincare inequality implies that

(38) ‖ Ψ ‖ L 2 ≤ ‖ ∇ Ψ ‖ L 2 .

On the other hand, the Sobolev inequality tells that

(39) ‖ ∇ Ψ ‖ L 2 ≤ ‖ Ψ ‖ L 2 1 2 ‖ D 2 Ψ ‖ L 2 1 2 .

Collecting the above two inequalities gives that

(40) ‖ v ˜ ‖ L 2 ≲ ‖ ∇ Ψ ‖ L 2 ≲ ‖ D 2 Ψ ‖ L 2 ≲ ‖ q ‖ L 2 .

In 3D case, the “curl curl” operation on a vector field will produce a gradient part, which means that, in general, the stream-function and the vorticity do not satisfy the Poisson equations. Indeed, for a vector field Ψ ∈ Ω , let us set q = ∇ × ∇ × Ψ , then we have

(41) Δ Ψ = − q + ∇ ( ∇ ⋅ Ψ ) ,

owning to the presence of ∇ ( ∇ ⋅ Ψ ) , we cannot obtain (40) in general.

Therefore, for the 3D case, we should think of a different approach. Fortunately, we find that ( u , u ˜ ) satisfies a temporary equation such that the lower order derivatives of u ˜ are uniformly bounded. In fact, we have the following proposition.

Proposition 3.4

The vector fields u and v ˜ satisfy the following temporary equation:

(42) ∂ t v ˜ + P [ u ⋅ ∇ v ˜ + ∇ u i v ˜ i ] = 0 .

Furthermore, we have that, for t ∈ [ 0 , T ] ,

(43) ‖ v ˜ ( t , ⋅ ) ‖ L 2 ⩽ ‖ u 0 ‖ X α 0 exp ∫ 0 t ‖ ∇ u ( t ′ ) ‖ L ∞ d t ′ .

Proof

We define a vector field Q = Q ( t , x ) by

(44) Q = ∂ t v ˜ + u ⋅ ∇ v ˜ + ∇ u i v ˜ i ,

and it is easy to check that

(45) ∇ × Q = ∂ t q + u ⋅ ∇ q − q ⋅ ∇ u = 0 .

Observing that Ω is simply connected, it follows that (42) holds.

We then begin to prove inequality (43). We multiply (42) by v ˜ and integrate over Ω to obtain that

(46) 1 2 d d t ‖ v ˜ ‖ L 2 2 ⩽ ‖ ∇ u ‖ L ∞ ‖ v ˜ ‖ L 2 2 .

In view of the Grönwall inequality, we obtain (43) immediately.□

We then consider the stationary Stokes equations (34). From Lemma 2.4, we deduce that there exists a unique solution u ˜ ∈ X α s ( Ω ) ∩ V ( Ω ) , and u ˜ obeys

(47) ‖ u ˜ ‖ X α s ⩽ K α − s ‖ v ˜ ‖ H s .

Based on the above results, we obtain the following theorem immediately.

Theorem 3.5

Let s ⩾ 3 be an integer. Suppose that u 0 ∈ X α s ( Ω ) ∩ V ( Ω ) and u ∈ L ∞ ( [ 0 , T ] ; X α s ( Ω ) ∩ V ( Ω ) ) . Then, there exists a unique solution u ˜ ∈ C ( [ 0 , T ] ; X α s ( Ω ) ∩ V ( Ω ) ) to system (4) with the estimates

(48) sup t ∈ [ 0 , T ] ‖ u ˜ ( t ) ‖ X α s ⩽ K 1 α − s ‖ u 0 ‖ X α s exp K 2 ∫ 0 T ‖ u ( t ) ‖ X α s d t

where K 1 , K 2 are constants independent of α .

4 Proof of the main result

Proof of Theorem 1.1

The conclusion is trivial for the case where u 0 ≡ 0 , and therefore, we consider the case where ‖ u 0 ‖ X α s > 0 .

Assume that u ∈ ℰ T , M s and u ∣ t = 0 = u 0 , where the parameters T and M of the space ℰ T , M s will be determined later. From Proposition 3.5, we know that equation (4) have a unique solution u ˜ ∈ C ( [ 0 , T ] ; X α s ( Ω ) ∩ V ( Ω ) ) . We define a mapping G : ℰ T , M s → ℰ T , M s by

(49) G [ u ] = u ˜ .

Next, we divide into three steps to proceed with the proof. First, we choose the parameters T and M such that the mapping G is well defined. Second, we prove that the mapping G is a contraction. Finally, we prove that the fixed point of G is a unique solution of the Euler- α equations (1).

Step 1: The mapping G is well-defined. We observe that u ˜ satisfies (48), therefore, to ensure that u ˜ ∈ ℰ T , M s , it suffices to choose T , M such that

(50) K 1 α − s ‖ u 0 ‖ X α s exp K 2 M T ⩽ M ,

which is equivalent to

(51) T ⩽ ln M + s ln α − ln ( K 1 ‖ u 0 ‖ X α s ) K 2 M .

To maximize T , we choose that

(52) M = K 1 α − s ‖ u 0 ‖ X α s exp K 1 ‖ u 0 ‖ X α s , T = α s K 2 exp K 1 ‖ u 0 ‖ X α s .

It is ready to check that u ˜ ∈ ℰ T , M s , and therefore, the mapping G is well-defined.

Step 2: The mapping G is a contraction. Assume that u 1 , u 2 ∈ ℰ T , M s , we set

(53) u ˜ 1 = G [ u 1 ] , u ˜ 2 = G [ u 2 ] , v ˜ 1 = u ˜ 1 − α 2 Δ u ˜ 1 , v ˜ 2 = u ˜ 2 − α 2 Δ u ˜ 2 , U = u 1 − u 2 , U ˜ = u ˜ 1 − u ˜ 2 ,

then v ˜ 1 − v ˜ 2 = U ˜ − α 2 Δ U ˜ . Observing that ( u 1 , v ˜ 1 ) and ( u 2 , v ˜ 2 ) satisfy the temporary equation (42), we can subtract the equation for ( u 2 , v ˜ 2 ) from the one for ( u 1 , v ˜ 1 ) to obtain that

(54) ∂ t ( U ˜ + α 2 A U ˜ ) + P ( u 1 ⋅ ∇ v ˜ 1 − u 2 ⋅ ∇ v ˜ 2 + ∇ u i 1 v ˜ i 1 − ∇ u i 2 v ˜ i 2 ) = 0 .

Multiplying the above equation by U ˜ and integrating over Ω × [ 0 , t ) for any t ∈ ( 0 , T ] , we obtain, after integrating by parts,

(55) 1 2 ‖ U ˜ ( t ) ‖ H α 1 2 = ∫ 0 t ∫ Ω U ˜ ⋅ [ ( U ⋅ ∇ ) v ˜ 1 + ( u 2 ⋅ ∇ ) ( U ˜ − α 2 Δ U ˜ ) ] d x d s + ∫ 0 t ∫ Ω U ˜ ⋅ [ ∇ u i 1 ( U ˜ i − α 2 Δ U ˜ i ) − U i ∇ v ˜ i 2 ] d x d s ≕ K 1 + K 2 .

As usual, noting that ( U ˜ , u 2 ⋅ ∇ U ˜ ) = 0 and integrating by parts, we deduce that

(56) K 1 = ∫ 0 t ∫ Ω U ˜ ⋅ [ ( U ⋅ ∇ ) v ˜ 1 + ( u 2 ⋅ ∇ ) ( U ˜ − α 2 Δ U ˜ ) ] d x d s = ∫ 0 t ∫ Ω U ˜ ⋅ [ ( U ⋅ ∇ ) v ˜ 1 ] d x d s − α 2 ∫ 0 t ∫ Ω U ˜ ⋅ [ ( u 2 ⋅ ∇ ) ( Δ U ˜ ) ] d x d s = ∫ 0 t ∫ Ω U ˜ ⋅ [ ( U ⋅ ∇ ) v ˜ 1 ] d x d s + α 2 ∫ 0 t ∫ Ω Δ U ˜ ⋅ [ ( u 2 ⋅ ∇ ) U ˜ ] d x d s = ∫ 0 t ∫ Ω U ˜ ⋅ [ ( U ⋅ ∇ ) v ˜ 1 ] d x d s − α 2 ∫ 0 t ∫ Ω ∂ l u k 2 ∂ k U ˜ ⋅ ∂ l U ˜ d x d s ,

using the Hölder inequality, it follows that

∣ K 1 ∣ ⩽ ∫ 0 t [ ‖ U ˜ ‖ L 2 ‖ U ‖ L 2 ‖ ∇ v ˜ 1 ‖ L ∞ + α 2 ‖ ∇ u 2 ‖ L ∞ ‖ ∇ U ˜ ‖ L 2 2 ] d s .

Note that u 2 , u ˜ 1 ∈ ℰ T , M s , the above inequality implies

(57) ∣ K 1 ∣ ⩽ C M ∫ 0 t ‖ U ˜ ‖ H α 1 2 + ‖ U ‖ H α 1 2 d s .

We now estimate the second term K 2 , which is similar to K 1 . Indeed, by integration by parts, K 2 can be rewritten as:

(58) K 2 = ∫ 0 t ∫ Ω [ U ˜ i ∂ i u j 1 U ˜ j + α 2 ( ∂ k ∂ l u j 1 U ˜ l ∂ k U ˜ j + ∂ l u j 1 ∂ k U ˜ l ∂ k U ˜ j ) ] d x d s − ∫ 0 t ∫ Ω ∂ i v ˜ j 2 U ˜ i U j d x d s ,

using again the Hölder inequality, it follows that

∣ K 2 ∣ ⩽ ∫ 0 t [ ‖ U ˜ ‖ L 2 2 ‖ ∇ u 1 ‖ L ∞ + ‖ U ˜ ‖ L 2 ‖ U ‖ L 2 ‖ ∇ v ˜ 2 ‖ L ∞ ] d s + α 2 ∫ 0 t [ ‖ D 2 u 1 ‖ L 4 ‖ U ˜ ‖ L 4 ‖ ∇ U ˜ ‖ L 2 + ‖ ∇ u 1 ‖ L ∞ ‖ ∇ U ˜ ‖ L 2 2 ] d s .

Observing that u 1 , u ˜ 2 ∈ ℰ T , M s , we deduce that

(59) ∣ K 2 ∣ ⩽ C M ∫ 0 t ‖ U ˜ ‖ H α 1 2 + ‖ U ‖ H α 1 2 d s .

In view of (55), (57), and (59), it shows that, for all t ∈ [ 0 , T ] ,

(60) ‖ U ˜ ( t ) ‖ H α 1 2 ⩽ C M ∫ 0 t ‖ U ˜ ‖ H α 1 2 + ‖ U ‖ H α 1 2 d s .

Thanks to the Grönwall inequality, we have

(61) ‖ U ˜ ( t ) ‖ H α 1 2 ⩽ C M ∫ 0 t ‖ W ( s ) ‖ H α 1 2 exp { C M ( t − s ) } d s .

For arbitrary h ∈ ( 0 , T ] , the above inequality implies that

(62) sup t ∈ [ 0 , h ] ‖ U ˜ ( t ) ‖ H α 1 2 ⩽ ( e C M h − 1 ) sup t ∈ [ 0 , h ] ‖ U ( t ) ‖ H α 1 2 .

Consequently, if we choose h > 0 small enough such that e C M h − 1 < 1 , then we have that the mapping A is a contraction with respect to the H α 1 norm.

Step 3: Local-in-time existence. We choose h > 0 small enough such that the mapping G is a contraction. Then, by the Banach fixed point theorem, we conclude that there exists a unique fixed point u ∈ ℰ h , M s of the mapping G . We can see that this fixed point is also the limit of the fixed point iteration, with u 0 ≔ u 0 and u n ≔ G [ u n − 1 ] . In particular, we have that u = G [ u ] . As a result, ( u , u ) must satisfy the temporary equation (42), which turn out to be the Euler- α equations. In other words, u is the solution of the Euler- α equations (1), and the solution is unique.

Furthermore, from the definition of the parameter T and M , it is easy to check that u satisfies the inequality (7), and therefore, the solution can extend to [ 0 , T ] . Since u satisfies the Euler- α equations (1) and u ∈ ℰ T , M s , we deduce that u ∈ C ( [ 0 , T ] ; X α s ( Ω ) ∩ V ( Ω ) ) .□

5 Conclusions

We conclude from Theorem 1.1 that system (1) has a local unique solution when the initial data lie in H s ( Ω ) ( s ≥ 3 ). Moreover, equation (6) implies that the solution u α exists at least in [ 0 , T ] with T ≥ K α s , and the dependence on α originates from the Stokes equations (15) and its estimate (16). One particularly interesting question is that whether this dependence can be retrieved by using other techniques. This should be a difficult problem, due to the presence of the physical boundary.

Acknowledgements

The authors express their gratitude to the editors and the referees for valuable comments and suggestions.

Funding information: X. You was partially supported by the Doctoral Scientific Startup Fund of Jiangxi Science and Technology University (No. 2023BSQD24) and Jiangxi Provincial Natural Science Foundation (No. 20242BAB25008), and S. Yuan was partially supported by the Doctoral Scientific Startup Fund of Fujian Polytechnic Normal University (No. 404086).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. SY and XY proposed the main ideas and wrote the initial version. LH and LC brought out new ideas and valuable optimizations during the review process and pay a great role in correcting the mistakes and adopting reviewers’ suggestions.
Conflict of interest: The authors state no conflict of interests.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-11-06

Revised: 2024-08-16

Accepted: 2024-09-20

Published Online: 2024-11-06

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0077

Keywords for this article

non-Newtonian fluids; well-posedness for PDEs; Euler-α equations; Euler equations

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