On the Viscous Camassa-Holm Equations with Fractional Diffusion

We study a class of the viscous Camassa-Holm equations (or the Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and the whole space in two and three dimensions. The order of the fractional diffusion is assumed to be $2s$ with $s\in [n/4,1)$, which seems to be sharp for the validity of the main results of the paper; here $n=2,3$ is the dimension of the space. If $s\in (n/4,1)$, global well-posedness in $C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2}))$ is established whenever the initial data $u_0\in D(A)$, where $A$ is the Stokes operator. When $s = n/4$, the global well-posedness is showed for $\|u_0\|_{D(A)}$ being suitably small. We also prove that such global solutions gain a regularity instantaneously after the initial time; a bound on a higher-order spatial norm is also obtained.


Introduction
The study of fluid dynamic equations with nonlocal effects or anomalous diffusion has attracted a great attention in recent years. While some of the problems are described by nonlocal equations to begin with, many others particularly those considering interface motion in fluids, are of nonlocal nature and often derived from local equations. See for examples [1,2,3,4,5,6,7] and references therein. Nonlocal evolution problems on a bounded domain are of particular interests from various analytic point of views.
In this paper we shall study the viscous Camassa-Holm equations with fractional diffusion in Ω ⊂ R n , n = 2, 3, where Ω is a smooth bounded domain with boundary ∂Ω, or Ω = R n . The equations can be written as follows: div u = 0, u| t=0 = u 0 .
Here α > 0 characterizes the scale at which the fluid motion is averaged, and ν > 0 is the viscosity. A = P(−∆) is the Stokes operator, with P being the Leray projection operator P : L 2 (Ω) → {v ∈ L 2 (Ω) : div v = 0, v · n = 0 on ∂Ω}; we always omit the Ω-dependence of A and P. A s with s ∈ [ n 4 , 1) is the spectral fractional Stokes operator defined below. The operator A s is obviously nonlocal. There are alternative ways (not necessarily equivalent) of defining nonlocal version of the fractional Stokes operator, but we find the spectral fractional Stokes operator is the easiest to work with for our purpose. The range of the power s can be seen to be sharp from the view of the energy method (see the proof of Theorem 3.1 below). When Ω is a smooth bounded domain, we also need boundary conditions u = A s u = 0 on ∂Ω.
Throughout the paper, unless otherwise stated, we shall always assume that Ω ⊂ R n is a smooth bounded domain, or Ω = R n , with n = 2, 3, and s ∈ [n/4, 1).
When s = 1, the equations (1)- (2) are often referred as the classic viscous Camassa-Holm equations, or equivalently the isotropic Lagrangian averaged Navier-Stokes equations (LANS-α) [8]. The inviscid version of the LANS-α equations, or the Lagrangian averaged Euler (LAE-α) equations, were first derived in [9,10] from a variational formulation, motivated by the fact that the Camassa-Holm equation in one dimension describes geodesic motion on certain diffeomorphism group. An alternative derivation can be found in [11]. Viscosities were later added to the LAE-α equations, giving rise to the LANS-α equations [12,13,14]. Its relation to the turbulence theory has been well investigated [15,16,17,18,19]. Both LAE-α and LANS-α equations can be viewed as closure models when the motion at the scales smaller than α is averaged out. Anisotropic generalizations of the LAE-α and LANS-α equations in bounded domains are presented in [20], which takes into account that the covariance tensor of the Lagrangian fluctuation field is not constantly identity matrix throughout the domain and it evolves with the flow. For a more comprehensive history of the LANS-α equation, we refer the readers to [8] and the references therein. From the analysis point of view, various global existence or well-posedness results of the LANS-α equation have been established on periodic boxes [16], bounded domains and whole space, [21,22,23], and Riemannian manifolds with boundaries [24]; decay of solutions in bounded domains and whole spaces was also investigated in [22,23].
Fractional diffusion arises naturally in many hydrodynamic problems, capturing nonlocal feature of certain dynamics, such as nonlocal diffusion [1,2,4,6] or thermal/electromagnetic effects [25,26]. Though it is not obvious how fractional diffusion can be incorporated into derivations of the Camassa-Holm equations, the very form of the fractional dissipation in (1) together with the boundary conditions (3) is quite natural from the analysis point of view; a similar choice is made in [8]. Also, for simplicity, we shall only focus on the isotropic version of the fractional LANS-α equations, that is the viscous Camassa-Holm equations, although it was suggested that the anisotropic LANS-α equation may be more relavent for bounded domains [20].
Our first result, Theorem 3.1, is the global well-posedness with sharp fractional power s. It may be viewed as a fractional version of the classical result by Kieslev-Ladyzenskaya and others for the Navier-Stokes equations [27,28]. It would also be interesting if one can build rather weak solutions as in [6] for suitable small positive powers s. Next, we show that the global solution admits an improved regularity, which is stated in Theorem 4.1 and Theorem 4.2. The latter, characterizing the critical case (n, s) = (2, 1/2), is in general not easy to establish, and it is a starting point for a further regularity theory. Here instead of dealing with technical issues with commutators associated with the nonlocal operators which could be rather tricky on a bounded domain, we make uses of the fractional semigroups to obtain desired estimates. One may need such nonlocal commutator estimates for higher regularity and the boundary regularity. These and related issues would be addressed elsewhere.
The rest of the paper is organized as follows. In Section 2, we introduce the spectral fractional Stokes operator and present an equivalent formulation of the equations (1)- (2). Section 3 will be devoted to proving our main result Theorem 3.1 on the global well-posedness of the Camassa-Holm equations with fractional diffusion in two and three dimensions. In Section 4, we show that when t > 0, the global solution actually admits higher spatial regularity than what it has been showed in Section 3; the main results in this section are summarized in Theorem 4.1 and Theorem 4.2, which contains also a bound on a higher-order norm of the solution in space.

Preliminaries
Before defining the operator A s , we introduce some notations. Let Σ = {φ ∈ C ∞ 0 (Ω) : ∇ · φ = 0}. As in much literature on mathematical hydrodynamics, the H 1 -completion of Σ is denoted to be V ; while the L 2 -completion of Σ is denoted by H.
When Ω ⊂ R n , n = 2, 3, is a smooth bounded domain, it is known that for the stationary Stokes equation in Ω with zero Dirichlet boundary condition, there exists a sequence of eigenvalues {µ j } j∈Z+ ⊂ R + and a sequence of eigenfunctions {w j } j∈Z+ ⊂ L 2 (Ω), both depending on Ω, solving such that {µ j } j∈Z+ is non-decreasing in R + and {w j } j∈Z+ forms an orthonormal basis of H. It is also known that w j ∈ C ∞ (Ω) ∩ V [28]. For all f ∈ H, we have its spectral decomposition The infinite sum is understood in the L 2 -sense. In fact, f L 2 (Ω) = {f j } j∈Z+ l 2 . For all r ∈ R, let For conciseness, we shall omit the Ω-dependence in D(A r )(Ω) whenever it is convenient. Then for all f ∈ D(A r ), Again the infinite sum is understood in the L 2 -sense. As a result, is the Fourier transform of f . Therefore, A s can be naturally defined asa Fourier multiplier .
In this case, the equation (1)-(2) can be rewritten as This simpler form coincides with the Camassa-Holm equation in R n [22] with fractional viscosity. We note that when Ω is a smooth bounded domain, the boundary condition (3) is well-defined and it is automatically satisfied in the space D(A 1+s/2 ) with s ∈ [n/4, 1), n = 2, 3. Indeed, we have the following lemma.
On one hand, this implies that {Af n } n∈Z+ forms a Cauchy sequence in L 2 (Ω) and thus {f n } n∈Z+ is a Cauchy sequence in V 2 due to ellipticity of A and zero boundary conditions of f n [28,29]. We assume To this end, for all f ∈ D(A), f, (−∆)f ∈ L 2 and f | ∂Ω = 0. This gives Remark 2.1. In fact, when Ω is a smooth bounded domain, D(A r ) = V 2r for all r ∈ [1/2, 5/4); in general, thanks to the interpolation theory [29], D(A r ) ֒→ V 2r for all r ≥ 1/2. When Ω = R n , D(A r ) = V 2r for all r ≥ 1/2.
Let (1 − α 2 ∆) −1 be the inverse of the elliptic operator (1 − α 2 ∆) on Ω (with zero Dirichlet boundary condition if Ω is a smooth bounded domain). In the view of u ∈ D(A 1+s/2 ) and Lemma 2.1, A s u = 0 on ∂Ω if Ω is a smooth bounded domain. Then it is valid to take (1 − α 2 ∆) −1 on both sides of (1), and we obtain the following equivalent formulation of the Camassa-Holm equation with fractional diffusion [8]: where with adaptation of notations in [8], and P α : is the Stokes projector [24] uniquely defined by It is also bounded for all r ≥ 1 [8].

Global Well-posedness
Our main result on the global well-posedness of the equations (6) and (2) (or equivalently, (1)-(2)), with boundary conditions (3) when Ω is a smooth bounded domain, is as follows.
We start from studying the nonlinear term in (6).
, and we have the following estimate for f (u 1 , u 2 ).
Proof. We first apply Lemma 3.1 with r = 2 − s and r ′ = 1 + 3s/4 > n/2 to find that Here we used interpolation in the last inequality. Taking integral in time, we obtain (10) by Hölder's inequality.
The following lemma states the local well-posedness result.

If
. For all u ∈ B, the estimate (11) holds due to energy estimate of the homogeneous solution e −tνA s u 0 and the definition of B.
Consider the map Q : Indeed, by Lemma 3.2, f (u, u) ∈ L 2 T (D(A 1−s/2 )). It is easy to establish (e.g. by Galerkin approximation) that there exists a unique w ∈ C [0,T ] (D(A)) ∩ L 2 T (D(A 1+s/2 )) solving (12); the continuity of w in D(A) is established through classic arguments [28]. In fact, where C = C(ν). Here we used the assumption T ≤ 1. By Lemma 3.2 and the fact that where C 1 = C 1 (α, ν, s, n, Ω). On the other hand, for all u 1 , By Lemma 3.2, To this end, we proceed in two different cases.
This completes the proof.
Now we can prove global well-posedness by combining Lemma 3.3 with a global H 1 -energy estimate, with a special consideration of the whole space case, where · D(A r ) and A r · L 2 are not equivalent.
Take inner product of (1 − α 2 ∆)u * and (16); it is valid to do so since ) for almost all time; the latter is showed in Lemma 3.2. By integration by parts, Here ·, · denotes the L 2 -inner product on Ω. Hence, On the other hand, it is easy to establish that ([28] Lemma 1.2 in Chapter III), in the scalar distribution sense on (0, T ), By a limiting argument and the continuity of Similarly, Therefore, We proceed in three different cases.
This completes the proof.
To this end, with abuse of notations, we state and prove an estimate used in the above proof.
Proof. For all r ≥ 0, By Young's inequality, Now it suffices to note that for all δ ∈ (0, n/2], by Cauchy-Schwarz inequality,

Improved Regularity of u *
In this section, we shall show that the global solution u * obtained in the previous section instantaneously gains regularity when t > 0. We proceed in two different cases.

Non-critical case: s > 1/2
We remark that Lemma 3.1 roughly shows that the regularity of f (u * , u * ) is one order lower than that of u * if we can take r = r ′ ∈ (n/2, 2] although the estimate is nonlinear; on the other hand, the backbone equation ∂ t u * + νA s u * = f (i.e. (6)) implies that u * admits regularity 2s-order higher than f . When s > 1/2, we immediately gain regularity of u * when t > 0 by bootstrapping. This is why we call the case s > 1/2 non-critical. We have the following theorem.
Proof. For given k ∈ N sufficiently small such that r k − 1 ∈ [1, 2], we apply Lemma 3.1 with r = r k = 2 + s + (2s − 1)k and r ′ = 1 + s to find that where K = 1−s 2s−1 is the largest k-value that can be achieved. The reason why we need r k −1 ∈ [1, 2] is that we wish the first factor on the right hand side of (25) to be u * D(A) instead of any other higher norms of u * .
To this end, we shall prove that for all j = 0, · · · , K, We are going to use induction. For the case j = 0, (29) is trivial; (30) can be proved by putting t = t 1 and t = T in (27). Now suppose (29) and (30) hold for j ≤ k − 1, with some k ≥ 1. Since which proves (29) for j = k. We let t = t k+1 and t = T in (28), and obtain that This proves (30) for the case j = k. By induction, (29) and (30) are established for j = 0, · · · , K. In a similar spirit of (31), we can also show that (29) is true for j = K + 1, i.e.

Critical case: s = 1/2
In this section, we consider the case (n, s) = (2, 1/2). It is called critical since no easy bootstrapping argument as above can be applied to show higher regularity of u * . More sophisticated analysis is involved. In what follows, we shall prove, in the fashion of constructing a solution, that u * has local Hölder continuity in time away from t = 0 as a function valued in D(A 1+s/2 ); while the Hölder norm admits a singularity at t = 0 with certain growth rate as t → 0 + . This idea comes from the earlier studies of regularity of L p -solution of the Navier-Stokes equation and semilinear parabolic equations [30,31,32]. To be more precise, we introduce the following definition. 3. For all 0 < t ≤ t + h ≤ T , It is nice to have homogeneous solutions given by the semigroup {e −tνA s } t≥0 to be in this type of sets.
Proof. It is trivial that w(t) D(A) ≤ w 0 D(A) and In the last inequality, we used interpolation and the fact that t ≤ 1.
To prove (34), we derive that Similarly, In the last inequality, we used the fact that This completes the proof.
The following lemma is the key to show existence of the solution in the type of sets B β R,T . It plays a similar role of (13) in constructing a solution, but the characterization is much more refined.  Then v[w 1 , w 2 ](t) ∈ B β CR1R2,T , where C = C(α, ν, Ω, β).
Proof. Before we check the definition of B β CR1R2,T , it is useful to state the following estimates involving f (w 1 , w 2 ). We apply Lemma 3.1 with r = 2 − s and r ′ = r to find that where C = C(α, Ω). It is also useful to derive that where C = C(α, Ω). For simplicity, we shall write v[w 1 , w 2 ](t) as v(t) in the sequel.
Combining the above estimates with (38), we find that which is (34).
We focus on the second term as the first term can be easily handled using Step 3. We calculate that I 1 + I 2 + I 3 + I 4 + I 5 + I 6 .
This completes the proof.
With Lemma 4.2 in hand, we have the following result in the critical case as a refined version of Lemma 3.3. It is already known that Q is well-defined from B ′ to B as long as M is sufficiently small. We claim that it is also well-defined from B ′ to itself if M is set to be sufficiently small, depending on α, ν, Ω and β. Indeed, thanks to Lemma 4.1 and Lemma 4.2, for all u ∈ B ′ , Qu − e −tνA s u 0 = v[u, u] ∈ B β CM 2 ,T , where C = C(α, ν, Ω, β).
To this end, we define u (0) = e −tνA s u 0 ∈ B ′ , and u (j) = Qu (j−1) ∈ B ′ for all j ∈ N + . It is not difficult to show by induction that for all j ∈ N + , where C 3 = C 3 (α, ν, Ω, β) while C 1 and C 2 arise in (14) and (15) respectively. Indeed, (44) follows immediately from (14) and (15). To show (45), we note that u )) and C β loc ((0, T ]; D(A 1+s/2 )) to u * * . It is easy to show that u * * is a fixed point of Q, and thus a local solution of (6). By uniqueness result in Lemma 3.3, such u * * is unique and u * * = u * . It satisfies the following estimates u * − e −tνA s u 0 L ∞ T (D(A))∩L 2 Assuming M ≤ 1, we obtain the desired estimates. This completes the proof.