Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl

. In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i


Introduction
Let N ≥ 3 be an integer and γ ∈ R be a real number. In what follows, D γ (R N ) N denotes the set of all smooth vector fields u : R N → R N , u(x) = u 1 (x), u 2 (x), · · · , u N (x) for x = (x 1 , x 2 , · · · , x N ) with compact support such that u(0) = 0 if γ ≤ 1 − N 2 . Then, the Hardy-Leray inequality with weight γ ∈ R is given by for all u ∈ D γ (R N ) N , where the constant γ + N 2 − 1 2 is known to be sharp. This was proved for N = 3, γ = 0 by J. Leray [10] along his study on the Navier-Stokes equations, as an N -dimensional generalization of the one-dimensional inequality by H. Hardy [6]. In the context of hydrodynamics, it is an interesting problem whether the value of the optimal constant increases by imposing u to be solenoidal, i.e., div u = 0. Costin and Maz'ya [3] obtained a positive answer in every dimension N ≥ 3, under the additional assumption of axisymmetry.
It is clear that C γ > (γ+ 1 2 ) 2 for γ = − 1 2 , so we see the solenoidal and axisymmetric constraint improves the best constant of the Hardy-Leray inequality. However, since the sole assumption of axisymmetry does not change the optimality of the constant (γ + 1 2 ) 2 in (1) N =3 , it is expected that the condition of axisymmetry in Theorem 1 can be relaxed. Indeed, we shall show in this paper that Theorem 1 does hold by imposing only one component of u to be axisymmetric. To be more precise, let us introduce the spherical polar coordinates (ρ, θ, ϕ) For each (θ, ϕ), define the orthonormal frame (σ, e θ , e ϕ ) ∈ SO(3) by e θ = (− sin θ, cos θ cos ϕ, cos θ sin ϕ) , Then a vector field u : R 3 → R 3 at every point x = ρσ is expanded in that frame as u = σu ρ + e θ u θ + e ϕ u ϕ , where the last term e ϕ u ϕ is called the swirl part of u, which we abbreviate as u ϕ = e ϕ u ϕ . Also, the scalar function u ϕ is called the swirl component of u. 1 A vector field u is called axisymmetric if its three components u ρ , u θ and u ϕ are independent of ϕ.
Now, let us assume that u ∈ D γ (R 3 ) 3 , and that its swirl part u ϕ is axisymmetric, i.e., u ϕ is independent of ϕ. Then, as we shall show later, u ϕ becomes a solenoidal field and satisfies the inequality with the optimal constant (γ + 1 2 ) 2 + 2 . Since it is just the same as C γ in Theorem 1 if γ > 1 , we observe that the effect of the swirl part is dominant in this case. Accordingly, it is also interesting to evaluate the constant if we fix the swirl part of u. Now our main theorem is the following: Theorem 2. Let u ∈ D γ (R 3 ) 3 be a solenoidal field. If u is swirl-free, i.e., u ϕ ≡ 0, then the inequality holds with the optimal constant C = C γ,0 given by 1 In many papers, swirl component is defined in the cylindrical coordinates in R 3 . However, there are no differences between the two definitions.
More generally, for a given non-zero scalar function g : R 3 → R 3 which is independent of ϕ such that g = ge ϕ ∈ D γ (R 3 ) 3 , let us define Then (3) holds for any u ∈ G with the sharp constant C = C γ,g , where (2) , we have C γ,g ≥ C γ for all γ ∈ R. Then, it directly follows from Theorem 2 that: be a solenoidal vector field. We assume that u ϕ is axisymmetric, i.e., the swirl component u ϕ is independent of ϕ. Then the inequality holds with the same constant C γ in Theorem 1.
This corollary shows that the axisymmetry assumption of u in Theorem 1 can be weakened to that of the swirl part u ϕ . In other words, the non-swirl part u − u ϕ = e ρ u ρ + e θ u θ need not be axisymmetric to obtain the optimality of the constant in Theorem 1.
In the context of fluid mechanics, the effect of the swirl component of a velocity vector field is well-studied from various view points; see for example, [9], [13], [1], [7]. [8], [11], [14], to name a few. By Corollary 3, we see that the effect of the swirl component is significant also from the view point of general optimal inequalities, such as Hardy-Leray inequality with the improved best constant.
Lemma 4. Let u = σu ρ +e θ u θ +e ϕ u ϕ be a smooth vector field in R 3 \{0}. Assume that the swirl part u ϕ = e ϕ u ϕ is axisymmetric. Then we have where the two terms in the right-hand side are expressed in terms of components as proof. By use of (6), (4) and (5), we directly have the following calculations: We now assume that u ϕ is axisymmetric. Then ∂ ϕ u ϕ = 0 and integration by parts yield where the last equality follows from (8) and the commutation relation Then, adding to the both sides of the above integral equation by This, together with letting u ϕ = 0 or u ρ = u θ = 0, gives the desired formula.

Proof of Theorem 2
As in [3], let u ≡ 0 and let the right-hand side of (3) be finite: Since β > 0, this ensures the finiteness of the left-hand side of (3): R 3 |u| 2 |x| 2γ−2 dx < ∞. We now introduce the vector field v : R 3 → R 3 as the left-hand side of (10): which is called the Brezis-Vázquez-Maz'ya transformation [2], [12]. Then the righthand side of (3) is written in terms of v as where the last equality follows from |v(0)| = 0 and the support compactness of v. Dividing the both sides by Therefore, the minimization problem of the left-hand side, the Hardy-Leray quotient for u with weight γ, is reduced to that for v with weight −1/2 .
for (λ, σ) ∈ R × S 2 . Then the radial and spherical components of ρ∇v are given by By use of these relations and Lemma 4, the Hardy-Leray quotient of v with weight −1/2 in (11) is calculated as follows : Here the last equality follows from the isometric relation On the other hand, we now represent the solenoidal condition div u = 0 in terms of v : which is equivalent to (7) and the assumption v ϕ = 0. That is, since ρ∂ ρ = ∂ t . Integrating both sides of (13) with the complex measure e −iλt dt over R , we find the equivalent solenoidal condition written in terms of v = σh+e θ f as Now let us substitute (14) into (12). Then, integration by parts in each of the numerator and the denominator in (12) yields where we have introduced the second-order differential operator and where q(λ, −T θ ) and Q(λ, −T θ ) are operators defined by the polynomials in α, by putting α = −T θ . To evaluate (15), we expand f by using eigenfunctions (See Lemma 5 in Appendix.) Discarding the non-negative term | ¡ ∂ ϕ h| 2 + | ¡ ∂ ϕ f | 2 in the right-hand side of (15), we then see where F γ is defined by Here we note that F γ (x, α ν ) is just the same as the equation (2.36) n=3 in [3]. As in [3] again, by observing that Combining (17) to (16), we arrive at let λ γ ∈ R denote a value of λ that attains the minimum of F γ (λ 2 , α 1 ). Define the sequence {v n : R 3 → R 3 } n∈N of smooth vector fields by v n (x) = v n (e t σ) = − σD θ + e θ (∂ t − γ + 3 2 ) ξ( t n ) cos(λ γ t) sin θ for every n ∈ N , where ξ : R → R is an even smooth function ≡ 0 with compact support on R. Then, it is clear that v n satisfies (13), and so u n = ρ −γ− 1 2 v n is certainly solenoidal with compact support on R 3 \{0}. Also note that ψ 1 (θ) = sin θ is the first eigenfunction of −T θ = −∂ θ D θ associated with α 1 = 2, see Lemma 5 in Appendix. Now let us denote the radial and angular components of v n respectively as h n and f n . Then we see Therefore, {v n } n∈N is certainly a minimizing sequence for the v-part of the Hardy-Leray quotient, which completes the proof of equation (19). Returning to (18), we compute the minimum value of F γ (x, α 1 ) = 2 + x + . It is easy to check that the quadratic function G γ has the two roots By numerical calculation, they satisfy Thus it turns out that otherwise.
Now the computation of (18) is done. Finally, combining this to (19)=(18) and returning to (11), we arrive at: which completes the proof of Theorem 2 for u ϕ ≡ 0.

3.2.
The case u − u ϕ ≡ 0. In this case, u = u ϕ ≡ 0 is an axisymmetric swirl field by assumption. This also implies that v − v ϕ ≡ 0, and that v = v ϕ ≡ 0 is also axisymmetric swirl. By Lemma 4, the Hardy-Leray quotient for v = v ϕ with weight −1/2 is estimated from below as To see the infimum of the left-hand side among such v is equal to the right-hand side, we choose a sequence of axisymmetric swirl fields {v n } n∈N as v n (e t σ) = e ϕ ξ(t/n) sin θ , where ξ : R → R is a smooth function ≡ 0 with compact support. Then it is easy to check that Returning to (11), we have the inequality (2) for any axisymmetric u ϕ with the optimal constant (γ + 1 2 ) 2 + 2.
3.3. The case u ∈ G. In this case, the swirl part u ϕ = g = ge ϕ ∈ D γ (R 3 ) 3 is non-zero and axisymmetric. We may assume u − u ϕ ≡ 0 by the results in the former subsections. By Lemma 4, we can split the Hardy-Leray quotient for u into the swirl and the non-swirl parts: where the last inequality follows from the result in subsection 3.1, since u − u ϕ is swirl-free and solenoidal by (9). To see that the infimum of the left-hand side of (20) among G is equal to the right-hand side, we choose a sequence {ũ n } n∈N of solenoidal and swirl-free fields such that R 3 |ũ n | 2 |x| 2γ−2 dx = 1 and On the other hand, since ∂ ϕ g ≡ 0, the vector field g = ge ϕ is also solenoidal by (7). Then it follows that the sequence {u n = nũ n + g} n∈N belongs to G and that R 3 |∇un| 2 |x| 2γ dx R 3 |un| 2 |x| 2γ−2 dx → C γ,0 as n → ∞. Consequently, we reach to the desired result.
This holds also with respect to the measure sin θdθ, which concludes the lemma.