Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension

1 Introduction

This article is concerned with the stability of equilibrium figure of uniformly rotating viscous incompressible fluid in R 3 with surface tension, where the equilibrium figure is rotationally symmetric about a certain axis. The fluid occupies a region Ω ( t ) at time t ≥ 0 , which is surrounded by a free interface Γ ( t ) . We denote the initial position of Γ ( t ) by Γ 0 . Besides, we denote the velocity and pressure of the fluid by v ( x , t ) and π ( x , t ) , respectively, and the unit outward normal on Γ ( t ) by ν Γ . The normal velocity and the doubled mean curvature of Γ ( t ) with respect to ν Γ are denoted by V Γ and H Γ , respectively. Then, the motion of the fluid is governed by the following system:

In this article, Ω ( 0 ) is assumed to be bounded. Here, S ( v , π ) is the stress tensor defined by

where μ is a positive constant and M ⊤ means the transpose of M . Without loss of generality, we may assume that external (atmospheric) pressure is zero in this article.

It is well known that the Navier-Stokes equations admits the following stationary solution that corresponds to a rigid rotation:

where ω ∈ R describes a constant angular velocity and p 0 is some constant, and we have set x ′ = ( x 1 , x 2 , 0 ) ∈ R 3 . Substituting ( v ∞ , π ∞ ) into the boundary condition (1.1) 3 , we obtain the equation for the doubled mean curvature H Γ ∞ of Γ ∞ :

This defines the equilibrium figure Ω ∞ of the rotating liquid. Notice that, if Ω ∞ is rotationally symmetric about the x 3 axis, the boundary condition (1.1) 4 is automatically satisfied. In the sequel, Ω ∞ is assumed to be rotationally symmetric with respect to the axis defined by e 3 . Obviously, in the case ω = 0 , we observe the classical Young-Laplace law σ H Γ ∞ + p 0 = 0 . However, our interest here is to investigate the case ω ≠ 0 and, especially, characterize the stability of the equilibrium figure by means of the second variation of the energy functional instead of a restriction on the value of ω .

We note that a solution to (1.1) satisfies the following equalities:

By passing a uniformly moving coordinate system x ^ = x − V ^ 0 t and v ^ = v − V ^ 0 , where V ^ 0 = ∣ Ω ( 0 ) ∣ − 1 ∫ Ω ( 0 ) v 0 d x , and by rotating coordinate axes, we suppose that

It follows form the Reynolds transport theorem that ( 1.4 ) 3 gives

which means that the barycenter point of the domain Ω ( t ) is always suited at the origin. Finally, to guarantee that ( 1.4 ) 2 holds with v = v ∞ and Ω ( t ) = Ω ∞ , the angular velocity ω and the value γ should satisfy the relation

Notice that the value ω should be determined from a given quantity γ . Namely, Γ ∞ is determined from γ , where Γ ∞ is a smooth solution to (1.2) subject to (1.6). If ∣ γ ∣ ≪ 1 , there exists a unique Γ ∞ satisfying (1.2) and (1.6), see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. Finally, the multiplication of (1.2) by x ⋅ ν Γ ∞ and integration over Γ ∞ leads to the expression for p 0 , where ν Γ ∞ is the unit outward normal on Γ ∞ . In fact, it follows from (1.2) that

The relation H Γ ∞ ν Γ ∞ = Δ Γ ∞ x , x ∈ Γ ∞ , and the divergence theorem imply

Hence, we deduce that

Notice that, by (1.6), we see that p 0 is strictly positive.

The free boundary problem of (1.1) is said to be finding a family of hypersurfaces { Γ ( t ) } t ≥ 0 and appropriately smooth solutions v and π . Notice that finding a family of hypersurfaces { Γ ( t ) } t ≥ 0 is equivalent to finding a family of { Ω ( t ) } t ≥ 0 . Since many authors have considered problems similar to (1.1) in various settings, we only mention the details of those papers that dealt with the effect of surface tension included on the free interface (the case σ > 0 ) and with assuming that Ω ( 0 ) is bounded. For the case of surface tension on the free boundary, we refer the reader to papers [1,3,4, 5,11,17, 18,35,36, 37] that deal with the case where the initial domain Ω ( 0 ) is an infinite layer of finite depth with a rigid bottom, see also a recent article by Saito and Shibata [23] that dealt with the case of the bottomless ocean. When Ω ( 0 ) is bounded, the first contribution to the solvability of (1.1) traces back to a long series of papers by Solonnikov [29,30,31]. Specifically, Solonnikov investigated the problem in L 2 regularity framework, i.e., he showed the local existence and uniqueness of solutions for (1.1) in Sobolev-Slobodetskiĭ spaces W 2 2 + α , 1 + α 2 with 1 / 2 < α < 1 . To obtain the local-in-time solutions in Hölder and anisotropic Sobolev regularity frameworks, we refer to the works by Moglilevskiĭ and Solonnikov [16] and Shibata [24], respectively.^[1] The unique global existence theorem in the L 2 regularity frameworks was proved by Solonnikov [30], where the solution converges to a uniform rigid rotation of the liquid about a certain axis, provided that the initial velocity and the initial angular momentum (i.e., ∣ γ ∣ ) are sufficiently small and Γ 0 is sufficiently close to a sphere. The similar result was also established in Hölder regularity framework, see Padula and Solonnikov [19]. More recently, the author [40] extends the result obtained in [19] the class of anisotropic Sobolev spaces W p , q 2 , 1 with 2 < p < ∞ and 3 < q < ∞ satisfying 2 / p + 3 / q < 1 , which can also be regarded as an extension of Shibata [26] dealing with the case ω = 0 . Notice that, in the previous studies [16,30,40], the equilibrium figure is uniquely determined by the constant γ . However, it was necessary to assume that ∣ γ ∣ is small since this assumption yields the smallness of ∣ ω ∣ , so that we could find a unique smooth solution to equation (1.2) subject to (1.6) based on a standard contraction mapping theorem; see Solonnikov [30, Thm. 5.1] and Watanabe [40, Prop. A.1]. On the other hand, Solonnikov [34] showed that the smallness condition on ∣ γ ∣ can be replaced by the condition of the positivity of the second variation of the functional E ω given by

provided that there exists a smooth surface Γ ∞ such that Γ ∞ satisfies (1.2) and (1.6) and is rotationally symmetric about the x 3 axis. In particular, he showed that the stability result can be obtained by the positivity of the second variation of the energy functional E ω ( h ) within the Hölder regularity framework.

The aim of this article is to extend the aforementioned Hölder regularity result obtained by Solonnikov [34, Thm. 2.1] in the L p -in-time and L q -in-space ( L p − L q ) setting with 2 < p < ∞ and 3 < q < ∞ satisfying 2 / p + 3 / q < 1 , which provides an optimal regularity on the initial data. In particular, in contrast to Shibata [26], we investigate the stability of nontrivial stationary solutions (i.e., ( v ∞ , π ∞ , Γ ∞ ) with ω ≠ 0 ) to the three-dimensional Navier-Stokes equations with surface tension. To prove our main result, we transform the system (1.1) into a problem on a domain F surrounded by a fixed surface G . To observe smoothing of the unknown interface, we rely on the direct mapping method via the Hanzawa transform, i.e., we approximate the free surface Γ ( t ) by a real analytic hypersurface G , in terms of the Hausdorff distance of the second-order normal bundles being as small as we wish. As the transformed problem becomes a quasilinear parabolic type PDE with inhomogeneous boundary data, it is widely known that the key idea to show the local-in-time existence of regular solution is to use the maximal L p − L q regularity result for an associated linearized problem. In this article, however, we will address the global existence issue, and hence, it is also required to derive some decay property of an associated linearized system – combining the local existence result and the decay property yields the global existence result due to a standard argument. To this end, we use the maximal L p − L q regularity result for the shifted linearized system to rewrite the linearized system as an abstract evolution equation with homogeneous boundary data. Since the shifted linearized system admits a unique solution in the maximal L p − L q regularity class that decays exponentially, it suffices to study the decay property for a solution to the abstract evolution equation. Although Solonnikov [34] established the decay estimate by energy estimates, his approach does not seem to be available in a general L p − L q setting. To overcome this difficulty, we study the decay property of an analytic C 0 -semigroup associated with the linearized system. On the basis of a spectral analysis of the corresponding linear operator A q defined in X 0 = J q ( F ) × B q , q 2 − 1 / q ( G ) , we will show that − A ˜ q ≔ − A q ∣ X ˜ 0 generates an analytic C 0 -semigroup { e − A ˜ q t } t ≥ 0 in X ˜ 0 , which is exponentially stable on some subspace X ˜ 0 of X 0 , see Section 5. Here, J q ( F ) and B q , q 2 − 1 / q ( G ) stand for the solenoidal space of ( L q ( F ) ) 3 and the inhomogeneous Besov spaces, respectively, and the space X ˜ 0 is defined as the set of all ( f , g ) ∈ X 0 that satisfies the following orthogonal conditions:

which are similar to but essentially different from the orthogonal condition considered in Shibata [26]. Thanks to these orthogonal conditions, we will observe that the resolvent set of − A ˜ q contains the right half-plane C + ≔ { λ ∈ C : Re λ ≥ 0 } including λ = 0 , which implies the exponential stability of the semigroup. Hence, our discussion generalizes the argument employed in Shibata [26] who deals with the case ω = 0 , and see also Shibata and Shimizu [28] for the case σ = ω = 0 . Notice that, if γ = 0 (i.e., ω = 0 ), then the equilibrium surface becomes a sphere so that a natural choice of G is a sphere as well. If we choose G as a sphere, we obtain a nice spectral property of the Laplace-Beltrami operator defined on G , which arises from the surface tension, see [26, Lemm. 4.5] (cf. [22, Prop. 10.2.1]). In our case, however, it follows from ω ≠ 0 and (1.2) that the equilibrium surface is not sphere. In addition, since we do not impose the smallness condition on ∣ γ ∣ (as well as ∣ ω ∣ ), the equilibrium surface cannot be understood as a normal perturbation of a sphere in general, which means that Shibata’s approach fails. If we assume that ∣ γ ∣ is sufficiently small, we can recover Shibata’s argument because the equilibrium surface is given by a normal perturbation of a sphere, and see a recent contribution by the author [40]. Thus, we have to introduce another technique to handle the term arising from the surface tension. To this end, we introduce the quadratic form that determines from the functional E ω given in (1.7).

It should be noted that our result even in the case γ = ω = 0 refines Shibata’s result [26]. In fact, in view of the trace theory (cf. Denk et al. [9]), if we study the linearized problem, the boundary data have to be in the intersection space:

where F p , q s denotes the vector-valued inhomogeneous Triebel-Lizorkin spaces. Hence, to find the strong solution to (1.1) via a contraction mapping principle, we have to estimate the nonlinear terms in this intersection space. However, in the argument in Shibata [26], the boundary data were not lying in (1.8), and hence, his result is not optimal in view of trace theory. Besides, Shibata [26] did not investigate that the height function h admits the higher regularity F p , q 2 − 1 / q ( J ; L q ( G ) ) , which implies that the free interface can be understood in the classical sense due to the embedding

To overcome these fallacious, we rely on the recent contributions established by the author [39,40], which were based on the studies of Lindemulder [13] and Meyries and Veraar [14,15]. Furthermore, it was not showed in the study by Shibata [25,26] that the solution can be understood in the classical sense, but the well-known parameter trick (cf. Prüss-Simonett [22, Ch. 9]) implies that the solution is indeed real analytic, jointly in time and space. This shows that the solutions to (1.1) are indeed classical.

To explain our main result, we shall introduce the quadratic form that characterize the stability of the stationary solution to (1.1). As we will explain in the next section, the free surface Γ ( t ) can be approximated by a real analytic surface G in the sense that the Hausdorff distance of the second-order bundles of Γ ( t ) and G is as small as we wish. In this case, we can write

where h is an unknown function but h 0 ∈ B q , p 2 + δ − 1 / p − 1 / q ( G ) is a given function. Here, we will prove that h is indeed smooth function defined on G for each t > 0 . Since we seek global solutions that converges to an equilibrium, in the following, we may set G = Γ ∞ . First, we observe that (1.2) is the Euler-Lagrange equation associated with E ω . In fact, the first variation of E ω at h = 0 is given by

In addition, the second variation of E ω at h = 0 is given by

where K G is the Gaussian curvature. We refer to [32, Sec. 2] for the derivations of (1.10) and (1.11), and see also [22, Ch. 2] for further geometric background. Recalling (1.2), we have

and hence, the second variation δ 0 2 E ω of E ω can be rewritten as follows:

where H G ′ ( 0 ) stands for the first variation of H G at h = 0 , since it holds

Since δ 0 E ω = 0 , we observe that G is a minimal surface. We now normalize ℬ G h by

and define the quadratic form Ψ G by

for g , h ∈ H 2 , 2 ( G ) . Then, we have δ 0 2 E ω = Ψ G ( h , h ) and ∫ G ℬ ^ G h d G = 0 .

The aim of this article is to show the stability of the stationary solution ( v ∞ , π ∞ , G ) to (1.1), provided that there exists a solution G to (1.2), and the quadratic form Ψ G satisfies the following assumption.

Our main result reads as follows.

In contrast to the previous article [40], Theorem 1.2 replaces the smallness condition on ∣ γ ∣ by the condition of the positivity of δ 0 2 E ω = Ψ G ( h , h ) , where it is assumed the existence of the equilibrium surface Γ ∞ satisfying (1.6). However, Theorem 1.2 does not conflict with the result obtained in [40]. In fact, if ∣ γ ∣ is suitably small, it follows from [40, Prop A.1] that there exists a unique Γ ∞ satisfying (1.2) and (1.6) with

where ∣ ( h ∞ , ∇ h ∞ ) ∣ L ∞ ( S R ) are small. Here, S R is a sphere centered at the origin with a radius R > 0 , and ∣ ( h ∞ , ∇ h ∞ ) ∣ L ∞ ( S R ) can be dominated by ∣ γ ∣ , see [40, Prop. A.1]. In this case, setting p 0 = 2 σ / R , the functional δ 0 2 E ω can be regarded as a perturbation from

which is positive definite on L ( 0 ) 2 ( S R ) ∩ H 2 , 2 ( S R ) (see [22, Prop. 10.2.1]). Hence, taking ∣ γ ∣ as small as possible, we observe that the smallness of ∣ γ ∣ implies the positivity of δ 0 2 E ω . Thus, we can conclude that Theorem 1.2 is an extension of the result obtained in the previous article. Accordingly, Theorem 1.2 includes all results obtained in [26,34,40].

The rest of the article is folded as follows. In Section 2, we recall the notation of functional spaces and preliminary propositions used throughout this article. In Section 3, we transform the system (1.1) to a problem on a domain F surrounded by a fixed interface G in terms of the Hanzawa transform. Section 4 is devoted to showing that the principal part of the linearization has the property of maximal L p − L q regularity. Then, some exponential decay property of the linearized system is proved in Section 5. Finally, the Section 6 presents the proof of the main result, Theorem 1.2.

2 Preliminaries