On the constants in a Kato inequality for the Euler and Navier-Stokes equations

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) ->v . D w, where v, w : T^d ->R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality |_n |<= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.


Introduction
(the subscripts 0, Σ recall the vanishing of the mean and of the divergence, respectively). For each n, we equip H n 0 with the standard inner product and the norm v|w n := √ −∆ n v| √ −∆ n w L 2 , v n := v|v n , (1. 4) which can be restricted to the (closed) subspace H n Σ0 . Our aim is to analyze quantitatively, in terms of the Sobolev inner products, the quadratic map appearing in (1.1). Some aspects of this map have been already examined in the companion paper [10]; here we have considered the bilinear maps sending two vector fields v, w on T d into v•∂w or L(v•∂w), and we have discussed some inequalities about them, the basic one being . (1.5) Our attention has been focused on the sharp constants K n ≡ K nd appearing therein, for which we have given fully quantitative upper and lower bounds.
In the present work we discuss other inequalities related to the quadratic Euler/NS nonlinearity, discovered by Kato in [6], and establish upper and lower bounds for the unknown sharp constants appearing therein. First of all we consider the inequality | v•∂w|w n | G ′ n v n w 2 n for n ∈ ( d 2 + 1, +∞), v ∈ H n Σ0 , w ∈ H n+1 0 , (1. 6) writing G ′ n ≡ G ′ nd for the sharp constants therein. With the additional assumption that w be divergence free, we can write (1. 7) with the sharp constant G n ≡ G nd fulfilling the obvious relation G n G ′ n . Let us observe that (1.7) can be rephrased in terms of the Leray projection L; indeed, with the assumptions therein we have w = Lw and this fact, combined with the symmetry of L in the Sobolev inner product, gives v•∂w|w n = v•∂w|Lw n = L(v•∂w)|w n for v ∈ H n Σ0 , w ∈ H n+1 Σ0 . (1.8) Due to (1.8), Eq. (1.7) is more directly related to the incompressible Euler/NS equations (1.1); in the sequel, (1.7) is referred to as the Kato inequality, and we call (1.6) the auxiliary Kato inequality. These inequalities (and similar ones) are well known, but little has been done previously to evaluate with some accuracy the constants which appear therein. On the other hand, quantitative bounds on such constants are useful to estimate the time of existence of the solution of (1.1) for a given initial datum, or its distance from any approximate solution.
In the present paper we derive fully computable upper and lower bounds G ± n ≡ G ± nd such that G − n G n G ′ n G + n (1. 9) for all n > d/2 + 1. As examples, the bounds G ± n are computed in dimension d = 3, for some values of n. In these cases the upper and lower bounds are not too far, at least for the purpose to apply them to the Euler/NS equations.
To be more precise about such applications, let us exemplify a framework already mentioned in [10]; the starting point of this setting is a result of Chernyshenko, Constantin, Robinson and Titi [4], that can be stated as follows. Consider the Euler/NS equation (1.1) with a specified initial condition u(x, 0) = u 0 (x); let u ap : (G n u ap (t) n + K n u ap (t) n+1 )dt (1.11) (u ap (t) := u ap (·, t), ǫ(t) := ǫ(·, t)). For a given datum u 0 , one can try a practical implementation of the above criterion after choosing a suitable u ap (say, a Galerkin approximate solution). Of course, T can be evaluated via (1.11) only in the presence of quantitative information on K n and G n , which are missing in [4]. In a forthcoming paper [11], our estimates on K n and G n will be employed together with the existence condition (1.11) (or with some refinement of it, suited as well to get bounds on u(t) − u ap (t) n ). For completeness we wish to mention that a program similar to the one described above, but based on technically different inequalities, has been developed in [8] [9] for the incompressible NS equations in Sobolev spaces of lower order. For example, in [9] we have considered the NS equations in H 1 Σ0 (T 3 ); here we have derived a fully quantitative upper bound on the vorticity curl u 0 L 2 of the initial datum, which ensures global existence of the solution. Again for completeness, we remark that the fully quantitative attitude proposed here for the Euler/NS equations is more or less close to the viewpoints of other authors about these equations, or about different nonlinear evolutionary PDEs [1] [3] [7] [12] [13] [14].
Organization of the paper. Section 2 summarizes our standards about Sobolev spaces on T d and the Euler/NS quadratic nonlinearity. Section 3 states the main results of the paper; here we present our upper and lower bounds G ± n on the constants in the inequalities (1.6) (1.7), which are treated by Propositions 3.5 and 3.7. The upper bounds are determined by the sup of a positive function G n , defined on the space Z d \ {0} of nonzero Fourier wave vectors; at each point k ∈ Z d \ {0}, G n (k) is a sum (of convolutional type) over Z d \ {0, k}. The lower bounds are determined by suitable trial functions. As examples, in Eq. (3.21) we report the numerical values of G ± n , for d = 3 and n = 3, 4, 5, 10. Section 4 contains the proofs of the previously mentioned Propositions 3.5, 3.7. Several appendices are devoted to the practical evaluation of the function G n mentioned before, and of the bounds G ± n . Appendix A presents some preliminary notations and results. Appendix B contains the main theorem (Proposition B.1) about the evaluation of G n and of its sup. Appendix C gives details on the computation of G n , and on the corresponding upper bounds G + n , for the previously mentioned cases d = 3, n = 3, 4, 5, 10. Appendix D describes the computation of the bounds G − n , for the same values of d and n. For all the numerical computations required by this paper, as well as for some lengthy symbolic manipulations, we have used systematically the software MATH-EMATICA. Throughout the paper, an expression like r = a.bcde... means the following: computation of the real number r via MATHEMATICA produces as an output a.bcde, followed by other digits not reported for brevity.

Some preliminaries
We use for Sobolev spaces and the Euler/NS bilinear map the same notations proposed in [10]; for the reader's convenience, these are summarized hereafter. Throughout the paper, we work in any space dimension d 2 ; (2.1) we use r, s as indices running from 1 to d. For a, b ∈ C d we put where a := (a r ) is the complex conjugate of a. We often refer to the d-dimensional torus 3) whose elements are typically written x = (x r ) r=1,...d .
Distributions on T d , Fourier series and Sobolev spaces.
The relevant Fourier coefficients of zero mean distributions are labeled by the set The distributional derivatives ∂/∂x s ≡ ∂ s and the Laplacian ∆ : The space of real distributions is +∞] we often consider the real space especially for p = 2. L 2 is a Hilbert space with the inner product v|w L 2 := The zero mean parts of D ′ and L p are this is a real Hilbert space with the inner product v|w n : (2.12) In this way we can define, e.g., the spaces Any v = (v r ) ∈ D ′ is referred to as a (distributional) vector field on T d . We note that v has a unique Fourier series expansion (2.4) with coefficients as in the scalar case, the reality of v ensures v k = v −k . L 2 is a real Hilbert space, with the inner product and the norm (2.14) We define componentwise the mean v ∈ R d of any v ∈ D ′ (see Eq. (2.5)); D ′ 0 is the space of zero mean vector fields, and L p 0 = L p ∩ D ′ 0 . We similarly define componentwise the operators ∂ s , ∆, √ −∆ n : D ′ → D ′ 0 . For any real n, the n-th Sobolev space of zero mean vector fields H n 0 (T d ) ≡ H n 0 is made of all d-uples v with components v r ∈ H n 0 ; an equivalent definition can be given via Eq.(2.11), replacing therein L 2 with L 2 . H n 0 is a real Hilbert space with the inner product and the induced norm Divergence free vector fields. Let div : Hereafter we introduce the space D ′ Σ of divergence free (or solenoidal) vector fields and some subspaces of it, putting H n Σ0 is a closed subspace of the Hilbert space H n 0 , that we equip with the restrictions of | n , n . The Leray projection is the (surjective) map where, for each k, L k is the orthogonal projection of C d onto the orthogonal complement of k; more explicitly, if c ∈ C d , (2.20) From the Fourier representations of L, , etc., one easily infers (2.21) Furthermore, L is an orthogonal projection in each one of the Hilbert spaces L 2 , H n 0 ; in particular, Making contact with the Euler/NS equations. The quadratic nonlinearity in the Euler/NS equations is related to the bilinear map sending two (sufficiently regular) vector fields v, w on T d into v•∂w; we are now ready to discuss this map.
Hereafter we often refer to the case v ∈ L 2 , ∂ s w ∈ L 2 (s = 1, ..., d) ; (2.23) the above condition on the derivatives of w implies w ∈ L 2 . The results mentioned in the sequel are known: the proofs of Lemmas 2.1, 2.2 are found, e.g., in [10], and the proof of Lemma 2.3 is reported only for completeness.

Lemma.
For v, w as in (2.23), consider the vector field v•∂w on T d , of components this is well defined and belongs to L 1 . With the additional assumption div v = 0, one has v•∂w = 0 (which also implies L(v•∂w) = 0, see (2.21)).
2.2 Lemma. Assuming (2.23), v•∂w has Fourier coefficients In particular, (2.26) holds if v, w are C 1 and div v = 0. By a density argument, one extends (2.26) to all v, w as in the statement of the Lemma.
The following result, essential for the sequel, is also well known (see, e.g., [10]).

The Kato inequality
Throughout this section we assume The following Proposition 3.1 is known, dating back to [6] (see [5] for a more general formulation, similar to the one proposed hereafter). As a matter of fact, the quantitative analysis presented later in this paper also gives, as a byproduct, an alternative proof of this Proposition. With the language of the Introduction, G ′ n and G n are, respectively, the sharp constants in the "auxiliary Kato inequality" (1.6) and in the Kato inequality (1.7); we recall that L could be inserted into (3.4), due to the relation (1.8) v•∂w|w n = L(v•∂w)|w n . It is obvious that G n G ′ n ; (3.5) in the rest of the section (which is its original part) we present computable upper and lower bounds on G ′ n and G n , respectively. The upper bound requires a more lengthy analysis; the final result relies on a function G nd ≡ G n , appearing in the forthcoming Definition 3.3. To build this function, as in [10] we refer to the exterior power 2 R d , identified with the space of real, skew-symmetric d × d matrices A = (A rs ) r,s,=1,...,d . We consider the (bilinear, skewsymmetric) operation ∧ and the norm | | defined by In the sequel, for p, q ∈ R d , we often use the relations where ϑ ≡ ϑ(p, q) ∈ [0, π] is the convex angle between p and q (defined arbitrarily, if p = 0 or q = 0); we use as well the inequality Keeping in mind these facts, let us stipulate the following.

Definition. We put
and 2n > d.
(iii) In Appendix B we will prove that and give tools for the practical evaluation of G nd and of its sup.
The main result of the present section is the following.
3.5 Proposition. The constant G ′ n defined by (3.4) has the upper bound
The practical calculation of the above upper bound is made possible by a general method, illustrated in Appendix B; the results of such calculations, for d = 3 and some illustrative choices of n, are reported at the end of this section. Let us pass to the problem of finding a lower bound for the constant G n ; this can be obtained directly from the tautological inequality choosing for v and w two suitable non zero "trial functions"; hereafter we consider a choice where v k = 0 for k ∈ Z d 0 \ V and w k = 0 for k ∈ Z d 0 \ W with V, W two finite sets. For the sake of brevity in the exposition of the final result, let us stipulate the following.
3.6 Definition. We put (the set U can depend on the family (u k ), and −U : (or any lower approximant for this) , (3.20) Proof. See Section 4. Here, we anticipate the main idea: the vector fields v := k∈V v k e k , w := k∈W w k e k belong to H m Σ0 for each real m, and v n = N n ((v k )), w n = N n ((w k )), v•∂w|w n = (2π) −d/2 P n ((v k ), (w k )); so, (3.19) is just the relation (3.17) for this choice of v, w.
Putting together Eqs. (3.5) (3.15) (3.19) we obtain a chain of inequalities, anticipated in the Introduction, here, the bounds G ± n can be computed explicitly from their definitions (3.16) (3.20).  For the reader's convenience, we report a Lemma from [10].
and ϑ(p, q) ≡ ϑ ∈ [0, π] be the convex angle between q and p. Then From now on, n ∈ ( d 2 + 1, +∞). Hereafter we present an argument proving (Proposition 3.1 and, simultaneously) Proposition 3.5. This is divided in several steps; in particular, Step 1 relies on an idea of Constantin and Foias [5]. These authors use their idea to obtain a proof of the Kato inequalities, but are not interested in the quantitative evaluation of the sharp constants therein; our forthcoming argument can be regarded as a refined, fully quantitative version of their approach, developed for the specific purpose to estimate G ′ n . Proof of Propositions 3.1, 3.5. We choose v ∈ H n Σ0 , w ∈ H n+1 0 and proceed in some steps. Step To prove all this, we first recall the Sobolev imbedding H n 0 ⊂ L ∞ , holding because n > d/2 (see, e.g., [2]); this obviously implies H n In the above: the first equality corresponds to the definition of | n , the sec- Step 2. The vector field z in (4.3) has Fourier coefficients To prove this, let us start from the Fourier coefficients of v•∂w; this has zero mean, so (v•∂w) 0 = 0. The other coefficients are this follows from (2.25) taking into account that, in the sum therein, the term with h = 0 vanishes due to v 0 = 0, and the term with h = k is zero for evident reasons.
Step 3. Estimating the Fourier coefficients of z. Let k ∈ Z d 0 ; Eq. (4.6) implies (4.10) To go on, we note that h•v h = 0 due to the assumption div v = 0; so, we can apply Eq.
Inserting the inequality (4.11) into (4.10), we get (in the definition of Q n (k) one can write as well h∈Z d 0 , since the general term of the sum vanishes for h = k).
Step 4. Estimates on z L 2 . Eq. (4.13) implies The sup of G n is finite, as we will show (by an independent argument) in Proposition B.1; making reference to the definition of G + n in terms of this sup (see Eq. (3.16)), we can write the last result as On the other hand, Inserting this result into (4.14), we obtain Step 5. Concluding the proofs of Propositions 3.1, 3.5. Eqs. (4.5) (4.16) imply These statements are self-evident; of course, the conditions u k = u −k and k•u k = 0 in (3.18) ensure u to be real, and divergence free.
Step 2. Consider two families (v k ) k∈V , (w k ) k∈W ∈ H, and define v : where, as in (3.20), P n (v, w) := −i h∈V,ℓ∈W,h+ℓ∈W |h + ℓ| 2n (v h •ℓ)(w ℓ •w h+ℓ ). In fact, the Fourier coefficients of v•∂w have the expression (2.25) which proves the thesis (4.20). In the above chain of equalities, the third passage relies on a change of variable k = h + ℓ, and the fourth passage depends on the Step 3. Conclusion of the proof. We consider two nonzero families (v k ) k∈V , (w k ) k∈W ∈ H, and define v := k∈V v k e k , w := k∈W w k e k . According to Steps 1 and 2, we have v n = N n ((v k )), w n = N n ((w k )), v•∂w|w n = (2π) −d/2 P n ((v k ), (w k )); so, the inequality G n | v•∂w|w n |/ v n w 2 n takes the form (3.19-3.20).
A Some tools preparing the analysis of the function G n In the sequel d ∈ {2, 3, ...}. Let us fix some notations, to be used throughout the Appendices. (ii) Γ is the Euler Gamma function, · · are the binomial coefficients.
But c n (z, u) C n , so we obtain the thesis (A.4).
(ii) In general, D nℓ and E nℓ are polynomials in c of degrees ℓ+4 and ℓ+2, respectively; as functions of c, these have the same parity as ℓ. (Some of the subsequent computations require as well the values of m n6 , M n6 for n = 4, 5, 10; these are reported in [10].) In the sequel we present a lemma on a function of two vector variables h, k, to be used later (see Eq.(B.4)); as indicated below, this is related to the functions D n , E n in (A.13) and to their Taylor expansions.
A.9 Lemma. Let h, k ∈ R d \ {0}, h = k, and let ϑ(h, k) ≡ ϑ be the convex angle between them. Furthermore, let n ∈ R; then the following holds.
To conclude, let us introduce some variantsD nℓ andÊ nℓ of the polynomials defined before (Ê nℓ was already considered in [10]).
(i) The function G n can be evaluated using the inequalities this function can be reexpressed as with C n as in (A.3).
(ii) As in Remark 3.4, consider the reflection operators R r (r = 1, ..., d) and the permutation operators P σ (σ a permutation of {1, ..., d}). Then (so, the computation of G n (k) can be reduced to the case k 1 k 2 ...
Step 1. One has The above decomposition follows noting that Z d 0k is the disjoint union of the domains of the sums defining G n (k) and ∆G n (k). G n (k) is finite, involving finitely many summands; ∆G n (k) is finite as well, since we know that G n (k) < +∞.
Step 2. For each k ∈ Z d 0 , one has the representation (B.4) If |k| 2ρ, in the above one can replace Z d 0k with Z d 0 and θ(|k − h| − ρ) with 1. To prove (B.4) we reexpress the sum in Eq. (B.3), using Eq.
Step 3. For each k ∈ Z d 0 one has with δG n as in Eq. (B.5). The obvious relation 0 < ∆G n (k) was already noted; in the sequel we prove that ∆G n (k) δG n . The definition (B.17) of ∆G n (k) contains the term |h ∧ k| 2 (|k| n − |k − h| n ) 2 , for which we have: The domain of the above two sums is contained in each one of the sets {h ∈ Z d | |h| ρ} and {h ∈ Z d | |k − h| ρ}; so, Step 4. One has the inequalities (B.2) G n (k) < G n (k) G n (k)+δG n . These relations follow immediately from the decomposition (B.16) G n (k) = G n (k)+∆G n (k) and from the bounds (B.18) on ∆G n (k).
Step 5. One has the equalities (B.6) G n (R r k) = G n (k), G n (P σ k) = G n (k), involving the reflection and permutation operators R r , P σ . Again, we can invoke the argument employed for the analogous properties of the function K n in [10].
The polynomial nature of the functions P nℓ , P ′ nℓ and P ′′ nℓ follows from their definition (B.8) in terms of the polynomialsÊ nℓ ,D nℓ . The inequalities (B.12) are obvious.