SCUOLA DI DOTTORATO VITO VOLTERRA DOTTORATO DI RICERCA IN MATEMATICA XXV CICLO. Inequalities with angular integrability and applications

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1 SCUOLA DI DOTTORATO VITO VOLTERRA DOTTORATO DI RICERCA IN MATEMATICA XXV CICLO Ineualities with angular integrability and alications Dottorando Renato Lucà Relatore Prof. Piero D Ancona

2 ANNO ACCADEMICO

3 Contents Introduction 3 Chater 1. Classical ineualities with angular integrability 6 1. The Stein-Weiss ineuality 7. Weighted Sobolev embeddings Caffarelli-Kohn-Nirenberg weighted interolation ineualities Strichartz estimates for the wave euation 3 Chater. Introduction to the regularity roblem for the Navier-Stokes euation 5 1. Euivalence between the differential and integral formulation 5. The Leray-Hof solutions 8 3. Regularity criteria 3 4. Well osedness with small data 35 Chater 3. Results in weighted setting with angular integrability Decay estimates for convolutions with heat and Oseen kernels 4. Regularity criteria in weighted saces with angular integrability Well osedness with small data in weighted L saces 53 Outlooks and remarks 59 Acknowledgments 6 Bibliograhy 61

4 Introduction We study the imrovements due to the angular regularity in the context of Sobolev embeddings and PDEs. It is well known that many fundamental ineualities in mathematical analysis get imrovements under some additional symmetry assumtions. Such imrovements are related to the geometric nature of the sace and in articular to the action of a certain grou of symmetry. This is not surrising because a symmetric function on a differentiable manifold can be considered as a function defined on lower dimensional manifold on wich stronger estimates are often available. Then such imroved estimates can be extended on the whole manifold by the action of a certain grou. In articular we work in a very simle setting by considering radially symmetric functions defined on R n. Such functions are indeed defined on R + and SO(n) acts to go back to R n. For instance the Hardy-Littelwood ([4], [13], [9], [7]), Caffarelli-Kohn-Nirenberg ([6]) and Strichartz ([3], [16], []) ineuality on R n get imrovements if one restricts to consider just radially symmetric functions. A natural uestion is if and how this henomenon occurs if we relace the symmetry hyotesis with merely higher integrability in the angular variables. We show that imrovements can be obtained by working with the norms: ( ) 1 + f L x L f L x L = f(ρ ) L (S n 1 ) ρn 1 dρ = su ρ> f(ρ ) L (S ); n 1 we observe that such results interolate beetween the imroved versions for radially symmetric functions and the classical ones. We develo widely this technology in the first chater. The main results roved are extensions of Hardy-Littelwood-Sobolev (theorem 1.3, corollary 1.4) and Caffarelli- Kohn-Nirenberg ineuality (theorem 1.11). Another interesting asect is that the ineualities on which we focus are fundamental tools in the study of PDEs. For instance the Caffarelli-Kohn-Nirenberg ineuality is really imortant in the study of regularity of the Navier-Stokes euation s solutions, because it rovides a riori estimates through interolation of uantities related to the energy dissiation. So it is exectable that the technology develoed can lead to succesfully results also in this areas. We focus basically on the small data theory and on regularity criteria. We actually extended the criteria in [33] and we made a conjecture on a ossible extension of the small data result in [7]. As well known the well osedness of the Cauchy Problem for the Navier-Stokes euation t u + (u )u + = u in R + R n u = in R + R n u = u in {} R n, is a big mathematical challenge, and only artial results have been obtained. Leray has roved global existence of weak solutions for L initial data in his ioneering work [6]. On the other hand the unieness of Leray s solutions is still oen, as it is the roagation of regularity of the initial datum u. The well osedness theory 3,

5 is well develoed for short times, or if one restricts to small u. In this scenario is useful to estabilish at least regularity or unieness criteria. We mean to find some a riori assumtions on the solutions under which the regularity (or the unieness) is guaranteed. We focus on the regularity in sace variables, being the time regularity a different and more difficult roblem. This is actually exectable by a hysical viewoint, because of the assumtion of incomressibility of the fluid; see [9]. It turns out that the a riori assumtions reuested in order to get regularity are basically boundedness assumtions on u, u, or u. As mentioned we refer basically to the criteria in [33], in which the main novelty is to consider boundedness in weighted L saces with weights x α. More recisely the author works with solutions u : [, T ] R n R n, euied with the norms: x α u L s T L, x s + n = 1 α, (.1) where, of course, the indexes relations follow by scaling considerations. Under boundedness assumtions the regularity in the segment (, T ) {} is achieved, more recisely u is C in the sace variables. So the introduction of weights allows to get local regularity criteria, where we mean only in a neighborhood of the origin. At first we show how for some choices of the indexes the criteria in [33] are indeed global (the regularity is achieved in (, T ) R n ). Then we get imrovements by working with saces with different integrability in radial and angular directions; so instead of the norms (.1) we use: x α u L s T L x L = ( T ( + ) s u(t, ρ ) L (S n 1 ) ρα+n 1 dρ dt ) 1 s, with s + n = 1 α. In this setting we get global regularity if sufficiently high values of are considered. We observe, as exectable, two different behaviour in the ranges α < and α > ; we show regularity in the case G where and of course: G = (n 1) α + n 1 ; < G, if α <, G <, if α >. A similar analysis has been erformed about the well osedness with small data in mixed angular-radial weighted saces euied with the norms: ( + ) 1 x α u L x L = u (ρ ) L (S n 1 ) ρα+n 1 dρ, (.) with again the critical scaling relationshi α = 1 + n. Here we find out the critical value (n 1) G = 1, and the well osedness is achieved for small data, with G. Actually we have closely looked at the following heuristic: the weights x α, α < localize, in some sense, the norms of the data (or of the solutions) near to the origin. In such a way local results are still available, but a loss of informations far from the origin occurs. These informations can be recovered every times by a suitbale amount of angular integrability. 4

6 On the other hand the local (in the sense of localized near to the origin) results have an intrinsic interest, and we also look at this roblem. In theorem 3.6 we rove local regularity for bounded solutions in x α u L s T L x L, with. This imroves in a different direction the results in.15. We get local regularity under the assumtion of a sufficiently high angular regularity, i.e for L where (n 1) (α+1)+(n 1) if 1 α < L = (n 1) +(n 1) if < α < 1. The main technical tools we use consist in decay estimates for convolutions with the heat and Oseen kernels in the context of weighted L L x saces. Estimates in weighted saces have been considered in literature, but the information rovided by the angular integrability leads to a really satisfactory admissibility range for the weights. The recise relation between the weights and the angular integrability is basically contained in the relation (1.7) in the corollary 1.4. The small data theory in the context of weighted L L x saces with < is more delicate and we just make a conjecture about a ossible imrovement of theorem.4, in which the authors show regularity of the Leray s solutions in the interior of the sace-time arabola: Π = } {(t, x) s.t. t > x, ε ε for a sufficiently small ε > and data with x 1/ L x = ε < ε. The conjecture in section 3 is made in order to cover the ga between this localized result and the classical well osedness results (in articular we refer to theorem 3.7 that s a articular case of the Koch-Tataru theorem.3). Remark.1. Of course by translations all the results are still valid if the norms and the weights are centered in a oint x. So if we consider ( ) + f L x x L = f( x + ρ) 1 L (S n 1 ) ρn 1 dρ, ( ( x x α T ) + u L s T L x x L = u( x + ρ) s ) 1 s L (S n 1 ) ρα+n 1 dρ dt, ( ) x x α + u L x x L = u ( x + ρ) 1 L (S n 1 ) ρα+n 1 dρ, and so on. The content of the Chater 1 is taken from [4]. 5

7 CHAPTER 1 Classical ineualities with angular integrability The goal of this section is to extend some classical estimates in the context of L saces to a setting in which the role of the angular and radial integrability is well distinguished. In order of exlaining our urose we start by a well known estimate of Walter Strauss [3] that roves x n 1 u(x) C u L, x 1, (1.1) for radial functions u Ḣ1 (R n ), n, This is an examle of a well known general henomenon: under suitable assumtions of symmetry, notably radial symmetry, classical estimates and embeddings of saces admit substantial imrovements. In the case of (1.1), a control on the H 1 norm of u gives a ointwise bound and decay of u. Both are false in the general case. Radial and more general symmetric estimates have been extensively investigated, in view of their relevance for alications, esecially to differential euations. This henomenon is uite natural; indeed, symmetric functions can be regarded as functions defined on lower dimensional manifolds, on which stronger estimates are available, then extended by the action of some grou of symmetries. Radial functions are essentially functions on R +, while the norms on R n are recovered by the action of SO(n) that introduces suitable dimensional weights connected to the volume form. In view of the ga between the symmetric and the non symmetric case, an interesting uestion arises: is it ossible to uantify the defect of symmetry of functions and rove more general estimates which encomass all cases, and in articular reduce to radial estimates when alied to radial functions? Heuristically, one should be able to imrove on the general case by introducing some measure of the distance from the maximizers of the ineuality, which tyically have the greatest symmetry. The aim of this aer is to give a artial ositive answer to this uestion, through the use of the following tye of mixed radial-angular norms: f L x L f L x L ( + = f(ρ ) L (S n 1 ) ρn 1 dρ = su ρ> f(ρ ) L (S ). n 1 ) 1, When the context is clear we shall write simly L L. For = the norms reduce to the usual L norms u L x L u L (R n ), while for radial functions the value of is irrelevant: u radial = u L L u L (R n ), [1, ]. Notice also that the norms are increasing in. The idea of distinguishing radial and angular directions is not new and has roved successful in the context of Strichartz estimates and disersive euations (see [16], [], [3]; see also []). To give a flavour of the results which can be obtained, Strauss estimate (1.1) can be extended as 6

8 follows: x n σ u(x) D σ u L L, n < σ < n for arbitrary non radial functions u and all 1 < <, 1. Remark 1.1. Of course by translations all the results we will rove hold with the norm ( ) + f L x x L = f( x + ρ) 1 L (S n 1 ) ρn 1 dρ, f L x x L = su ρ> f( x + ρ) L (S ); n 1 1. The Stein-Weiss ineuality A central role in our aroach will be layed by the fractional integrals φ(y) (T γ φ)(x) = dy, < γ < n. x y γ R n Weighted L estimates for T γ are a fundamental roblem of harmonic analysis, with a wide range of alications. Starting from the classical one dimensional case studied by Hardy and Littlewood, an exhaustive analysis has been made of the admissible classes of weights and ranges of indices (see [] and the references therein). In the secial case of ower weights the otimal result is due to Stein and Weiss: Theorem 1.1 ([1]). Let n 1 and 1 < <. Assume α, β, γ satisfy the set of conditions (1 = 1/ + 1/ ) β < n, α < n, < γ < n α + β + γ = n + n n (1.) α + β. Then the following ineuality holds x β T γ φ L C(α, β,, ) x α φ L. (1.3) Conditions in the first line of (1.) are necessary to ensure integrability, while the necessity of the condition on the second line is due to scaling. On the other hand, the sharness of α + β is less obvious and follows from the results of [18]. In the radial case the last condition can be relaxed and α + β is allowed to assume negative values. Radial imrovements were noticed in [4], [13], and the shar result was obtained by Rubin [9] and more recently by De Naoli, Dreichman and Durán: Theorem 1. ([9],[5]). Let n,,, α, β, γ be as in the statement of Theorem (1.1) but with the condition α + β relaxed to ( 1 α + β (n 1) 1 ). (1.4) Then estimate (1.3) is valid for all radial functions φ = φ( x ). Using the L L x norms we are able rove the following general result which extends both theorems: 7

9 Theorem 1.3. Let n and 1 < <, 1. Assume α, β, γ satisfy the set of conditions Then the following estimate holds: β < n, α < n, < γ < n α + β + γ = n + n n ( 1 α + β (n 1) 1 ) (1.5) x β T γ φ L x L C x α φ L x L. (1.6) The range of admissible, indices can be relaxed to 1 in two cases: (i) when the third ineuality in (1.5) is strict, or (ii) when the Fourier transform φ has suort contained in an annulus c 1 R ξ c R (c c 1 >, R > ); in this case (1.6) holds with a constant indeendent of R. Remark 1.. Notice that: (a) with the choices = and = (i.e. in the usual L norms) Theorem 1.3 reduces to Theorem 1.1; (b) if φ is radially symmetric, with the choice =, Theorem 1.3 reduces to Theorem 1.. Indeed, if φ is radially symmetric then T γ φ is radially symmetric too, so that all choices for, are euivalent; (c) obviously, the same estimate is true for general oerators T F with nonradial kernels F (x) satisfying T F φ(x) = F (x y)φ(y)dy, F C x γ. The roof of Theorem 1.3 is based on two successive alications of Young s ineuality for convolutions on suitable Lie grous: first we use the strong ineuality on the rotation grou SO(n); then we use a Young ineuality in the radial variable, which in some cases must be relaced by the weak Young-Marcinkewicz ineuality on the multilicative grou (R +, ) with the Haar measure dρ/ρ. The convenient idea of using convolution in the measure dρ/ρ was introduced in [5]. Remark 1.3. The oerator T γ is a convolution with the homogenous kernel x γ. Consider instead the convolution with a nonhomogeneous kernel φ(y) S γ φ(x) = x y γ dy. By the obvious ointwise bound S γ φ(x) T γ φ (x) it is clear that S γ satisfies the same estimates as T γ. However the scaling invariance of the estimate is broken, and indeed something more can be roved, thanks to the smoothness of the kernel (see Lemma 1.7): Corollary 1.4. Let n and 1, 1. Assume α, β, γ satisfy the set of conditions β < n, α < n ( 1, α + β (n 1) 1 ) +, (1.7) 1 1 ( α + β + γ > n ). (1.8) 8

10 Then the following estimate holds: x β S γ φ L x L C x α φ L x L. (1.9) The first result we need is an exlicit estimate of the angular art of the fractional integral T γ φ. Notice that a similar analysis in the radial case was done in [5] (see Lemma 4. there). The following estimates are shar: Lemma 1.5. Let n, ν >, and write x = (1 + x ) 1/. Then the integral I ν (x) = x y ν ds(y) x R n S n 1 satisfies I ν (x) x ν for x, (1.1) while for x we have 1 if ν < n 1 I ν (x) log x if ν = n 1 (1.11) x 1 n 1 ν if ν > n 1. Proof. We consider four different regimes according to the size of x. We write for brevity I instead of I ν. First case: x. For x large and y = 1 we have x y x, hence I(x) x ν x ν. This roves (1.1). Second case: x 1. Clearly we have x y 1 when y = 1, and this imlies I(x) 1 x ν. This is euivalent to (1.11) when x 1/. Third case: 1 x. This is the bulk of the comutation since it contains the singular art of the integral, as x 1. We write the integral in olar coordinates using the sherical angles ( 1,,..., n 1 ) on S n 1, oriented in such a way that 1 is the angle between x and y. Using the notation σ = x y, by the symmetry of I(x) in (,..., n 1 ) we have I(x) π σ ν (sin 1 ) n d 1. In order to rewrite the integral using σ as a new variable, we comute so we have σdσ = d( x y ) = d( x + 1 x cos 1 ) = x sin 1 d 1 (sin 1 ) n d 1 = σ(sin 1) n 3 dσ x and, noticing that x 1 x y = σ x + 1, I(x) x +1 x 1 σ 1 ν (sin 1) n 3 dσ. x Now let A be the area of the triangle with vertices, x andd y: we have A = x sin 1 so that x +1 I(x) x n σ 1 ν A n 3 dσ. x 1 Recalling Heron s formula for the area of a triangle as a function of the length of its sides we obtain x +1 I(x) x n σ 1 ν[ ] n 3 ( x +σ +1)( x +σ 1)( x +1 σ)(σ +1 x ) dσ. x 1 Notice that this formula is correct for all dimensions n. 9

11 Now we slit the integral as I I 1 + I with I 1 (x) = x n x and x 1 σ 1 ν[ ( x + σ + 1)( x + σ 1)( x + 1 σ)(σ + 1 x ) x +1 I (x) = x n σ 1 ν[ ( x + σ + 1)( x + σ 1)( x + 1 σ)(σ + 1 x ) x In the second integral I, recalling that 1 x, we have so that x σ x + σ + 1 x + σ 1 σ + 1 x 1 I x +1 x 1 ( x + 1 σ) n 3 dσ = (1 σ) n 3 dσ 1. ] n 3 ] n 3 In the first integral I 1, using that 1 x and x 1 σ x, we see that moreover, and we have x ( x + σ + 1) ( x + 1 σ) 1; 1 x + σ 1 σ I 1 (x) x x 1 or, after the change of variable σ σ( x 1), so that x + σ 1 σ n 3 n 1 ν+ σ (σ + 1 x ) dσ 1+ 1 I 1 (x) ( x 1) n 1 ν x 1 (σ 1) n 3 n 1 σ ν dσ. Now slit the last integral as A + B where and we have immediately A = ( x 1) n 1 ν (σ 1) n n 1 σ ν dσ 1+ 1 B = ( x 1) n 1 ν x 1 (σ 1) n 3 n 1 σ ν dσ; A ( x 1) n 1 ν while, keeing into account that σ σ 1 for σ in (, x 1 ), dσ dσ. which gives B = ( x 1) n 1 ν 1+ 1 x 1 σ n ν dσ 1 if ν < n 1 B log x if ν = n 1 x 1 n 1 ν if ν > n 1 (1.1) 1

12 1 Fourth case: x 1. Using the change of variable x = 1/ x, we see that I(x) I(1/ x ), thus the fourth case follows immediately from the third one, and this concludes the roof of the Lemma. We shall also need the following estimate which is roved in a similar way: Lemma 1.6. Let n, ν >. Then the integral J ν (x, ρ) = x ρ ν ds() S n 1 x R n, ρ satisfies: J ν (x, ρ) x ν for ρ 1 or x ρ, (1.13) J ν (x, ρ) ρ ν for x 1 or ρ x, (1.14) while in the remaining case, i.e. when x 1 and ρ 1 and 1 x ρ x, ρ ( ν ) if ν < n 1 J ν (x, ρ) ρ ν log ρ x ρ if ν = n 1 (1.15) ρ 1 n x ρ n 1 ν if ν > n 1. As a conseuence, one has J ν ρ + x ν when ν < n 1 and J ν ρ + x ν log( ρ + x ) when ν = n 1. Proof. The roof is similar to the roof of Lemma 1.5; we sketch the main stes. Estimates (1.13) and (1.14) are obvious, thus we focus on (1.15). Write r = x, so that we are in the region 1/ r/ρ ; we shall consider in detail the case 1 r ρ, the remaining region being similar. integral is reduced to where A is given by Heron s formula Using the same coordinates as before, the x +1 J ν ( x, ρ) = x n ρσ ν A n 3 σ dσ x 1 A( x, σ) = ( x + σ + 1)( x + σ 1)( x + 1 σ)(σ + 1 x ). We slit the integral on the intervals x σ x + 1 and x 1 σ x. The first iece gives x +1 I 1 ρ ν ( x + 1 σ) n 3 dσ and by the change of variable σ σ( x + 1) we obtain x I 1 ( x, ρ) ρ ν. For the second integral on x 1 σ x, noticing that we have 1 x + σ 1 σ x I ρσ ν σ n 1 n 3 (σ + 1 x ) dσ x 1 x =( x 1) n 1 x 1 (r ρ)σ ν σ n 1 n 3 (σ 1) dσ 1 via the change of variables σ σ( x 1) which gives ρσ (r ρ)σ. The art of the integral bewteen 1 and roduces ( x 1) n 1 r ρ ν = ρ 1 n (r ρ) n 1 r ρ ν 11

13 while the remaining art between and x /( x 1) gives r ( x 1) n 1 r ρ (r ρ)σ ν σ n dσ =ρ 1 n r ρ 1 n r (r ρ) (r ρ) σ ν σ n dσ σ n 1 + σ ν dσ which can be comuted exlicitly. Summing u we obtain (1.15). We are ready for the main art of the roof. By the isomorhism S n 1 SO(n)/SO(n 1) we can reresent integrals on S n 1 in the form g(y)ds(y) = c n g(ae)da, n S n 1 SO(n) where da is the left Haar measure on SO(n), and e S n 1 is a fixed arbitrary unit vector. Thus, via olar coordinates, a convolution integral can be written as follows (aart from inessential constants deending only on the sace dimension n): F φ(x) = F (x y)φ(y)dy = F (x ρω)φ(ρω)ds ω ρ n 1 dρ R n S n 1 F (x ρbe)φ(ρbe)dbρ n 1 dρ SO(n) Hence the L norm of the convolution on the shere can be written as F φ( x ) L (Sn 1 ) F φ( x Ae) L A (SO(n)) F ( x Ae ρbe)φ(ρbe)db SO(n) L A (SO(n)) ρ n 1 dρ where e is any fixed unit vector. By the change of variables B AB 1 in the inner integral (and the invariance of the measure) this is euivalent to = F (AB 1 ( x Be ρe))φ(ρab 1 e)db If F satisfies SO(n) L A (SO(n)) ρ n 1 dρ F (x) Cf( x ) (1.16) for a radial function f, we can write F (AB 1 ( x Be ρe)) Cf ( x Be ρe ) and we notice that the integral f ( x Be ρe ) φ(ρab 1 e) db = g h(a) SO(n) is a convolution on SO(n) of the functions g(a) = f ( x Ae ρe ), h(a) = φ(ρae). We can thus aly the Young s ineuality on SO(n) (see e.g. Theorem 1..1 in [1]) and we obtain, for any, r, [1, + ] with 1 + = +, 1 1 r 1 1

14 the estimate F φ( x ) L (Sn 1 ) f( x e ρ ) L r (S n 1 ) φ(ρ) L (Sn 1 ) ρn 1 dρ (1.17) where we switched back to the coordinates of S n 1. Notice that the conditions on the indices imly in articular. Secializing f to the choice f( x ) = x γ we get F φ( x ) L ρ γ ρ 1 x e γ L r φ(ρ) L ρ n 1 dρ which can be written in the form ( ) α+ n = x n α n x γ n+γ ρ 1 x e γ n dρ ρ L r ρα+ φ(ρ) L ρ or euivalently, recalling (1.), ( ) β+ n = x β n x ρ 1 x e γ n dρ ρ L r ρα+ φ(ρ) L ρ Following [5], we recognize that the last integral is a convolution in the multilicative grou (R, ) with the Haar measure dρ/ρ, which imlies x β+ n F φ( x ) L g 1 h 1 ( x ), with g 1 (ρ) = ρ β+ n ρe γ L r, h 1 (ρ) = ρ α+ n φ(ρ) L. By the weak Young s ineuality in the measure dρ/ρ (Theorem in [1]) we obtain x β x F φ L L β+ n F φ( x ) L that is to say rovided x β F φ L L h 1 L (ρ 1 dρ) g 1 L r, (ρ 1 dρ) L (ρ 1 dρ) ρ φ β+ n L L ρe γ L L r. (1.18) r, (ρ 1 dρ), r, (1, + ) = 1 r + 1. In articular this imlies >. (1.19) In order to achieve the roof, it remains to check that the last norm in (1.18) is finite. Notice that, when r <, ρe γ L r = I γ r (ρe) 1 r where I ν was defined and estimated in Lemma 1.5. On the other hand, when r = one has directly r = = ρe γ L r ρ 1 γ. (1.) Using cutoffs, we slit the L r, norm in three regions ρ 1/, ρ and 1/ ρ. In the region ρ 1/, recalling (1.1)-(1.11) or (1.), we have I γ r (ρe) 1 r 1 = ρ β+ n Iγ r (ρe) 1 r L 1 (, 1/; dρ/ρ) 13

15 since by assumtion β < n/; thus the contribution of this art to the L r, (dρ/ρ) norm is finite. In the region ρ we have I γ r (ρe) 1 r ρ γ since the condition = ρ β+ n Iγ r (ρe) 1 r ρ β γ+ n L 1 (, ; dρ/ρ) β γ + n < α < n is satisfied by (1.7), and again the contribution to the L r, norm is finite. For the third region 1/ ρ, by estimate (1.11), we see that in the case γ r n 1 one has again, for some σ, I γ r (ρe) 1 r log ρ 1 σ = ρ β+ n Iγ r (ρe) 1 r L 1 (1/, ; dρ/ρ) On the other hand, in the case γ r > n 1 (which includes the choice r = ), we see that ρ β+ n Iγ r (ρe) 1 r n 1 ρ 1 r γ L r, (1/, ; dρ/ρ) n 1 γ 1 r r. Recalling the relation between, r, (res., r, ) the last condition is euivalent to ( 1 γ (n 1) 1 ) n + n n which is recisely the third of conditions (1.7). The weak Young ineuality can be used in (1.18) only in the range, r, (1, + ), which forces 1 < < <. To cover the cases 1 < we use instead the strong Young ineuality: we can write x β ρ F φ L L φ β+ n L L ρe γ Lr L r (ρ 1 dρ) (1.1) for the full range, r, [1, + ]. The revious arguments are still valid aart from the last ste which must be relaced by ρ β+ n Iγ r (ρe) 1 r n 1 ρ 1 r γ L r (1/, ; dρ/ρ) n 1 γ > 1 r r and this imlies that the ineuality in the last condition (1.7) must be strict. The case 1 < = < has already been covered. Indeed, in this case the scaling condition (1.7) imlies α + β + γ = n = α + β > since γ < n. Thus when = the last ineuality in (1.7) is strict and we can aly the second art of the roof; the cases follow from the case =. To comlete the roof, it remains to consider the case (ii) where we assume that the suort of the Fourier transform φ is contained in an annular region of size R. By scaling invariance of the ineuality, it is sufficient to consider the case R = 1. Now let ψ(x) be such that ψ Cc and recisely ψ(ξ) = 1 for c 1 ξ c, ψ(ξ) = for ξ > c 1 and ξ < 1 c, for some constants c > c c 1 > c 1 >. This imlies φ = F 1 ( ψ φ) = ψ φ 14

16 and we can write T γ φ = x γ ψ φ = (T γ ψ) φ. Since T γ ψ = cf 1 ( ξ γ n ψ(ξ)) is a Schwartz class function, we arrive at the estimates T γ φ(x) C µ,γ x µ φ µ 1. (1.) Here we can take µ arbitrarily large. Thus the roof of case (ii) is concluded by alying the following Lemma: Lemma 1.7. Let n. Assume 1, 1 and α, β, µ satisfy β < n, α < n ( 1, α + β (n 1) 1 ) +, (1.3) 1 1 ( µ > α β + n ). (1.4) Then the following estimate holds: x β x µ φ L x L φ L x L. (1.5) Proof. Notice that, by (1.3), the right hand side in (1.4) is always strictly ositive and never larger than n 1, thus it is sufficient to rove the lemma for µ in the range < µ n. By (1.17) we have, for all,, r [1, + ] with 1 + 1/ = 1/ r + 1/, µ φ ( x ) L (Sn 1 ) Notice that when r = we have We write for brevity Q( x ) x Thus (1.6) becomes n 1 β+ J µ r ( x, ρ) 1 r φ(ρ) L (Sn 1 ) ρn 1 dρ. (1.6) x e ρ µ L x ρ µ. µ φ ( x ) L, P (ρ) = ρ J( x, ρ) = J 1 r µ r (x, ρ) n 1 β+ Q(σ) σ n 1 α+ (res. x ρ µ if r = ). and the estimate to be roved (1.5) can be written as φ(ρ) L J(σ, ρ)ρ n 1 α P (ρ)dρ (1.7) Q L (,+ ) P L (,+ ) (1.8) Recall that the integrals of the form J(σ, ρ) have been estimated in Lemma 1.6. We slit Q into the sum of several terms corresonding to different regions of ρ, σ. In the region σ 1 we have J(σ, ρ) ρ µ so that n 1 β+ Q 1 (σ) σ ρ µ ρ n 1 α P (ρ)dρ (1.9) Thus we see that in this region (1.8) follows simly from Hölder s ineuality and the fact that α < n/ and β < n/. Similarly, it is easy to handle the art of the integral with ρ 1 since we have then J(σ, ρ) σ µ. Thus in the following we can restrict to σ 1, ρ 1. When 1 σ ρ/ we have again J(σ, ρ) ρ µ and (1.7) becomes n 1 β+ Q (σ) σ σ 15 ρ µ ρ n 1 α P (ρ)dρ (1.3)

17 If we assume µ > n α (1.31) we can aly Hölder s ineuality and we get n 1 β+ Q 3 (σ) σ σ n µ α P L. Now the right hand side is in L (σ 1) rovided µ > n α + n ( β α β + n ) (1.3) and we see that (1.3) imlies (1.31) since β < n/ by assumtion. When 1 ρ σ/ we have J(σ, ρ) σ µ and (1.7) becomes Q 4 (σ) σ n 1 β+ σ µ σ and by Hölder s ineuality we have as before σ n 1 β+ σ n µ α P L ρ n 1 α P (ρ)dρ (1.33) so that (1.3) is again sufficient to obtain (1.8). Finally, let σ 1, ρ 1 and 1 σ ρ σ. In this region we must treat differently the values of µ r larger or smaller than n 1, and the case r = is considered at the end. Assume that n 1 < µ r n; then J(σ, ρ) ρ 1 n σ ρ n 1 r µ, and using the relations we see that (1.7) reduces to σ ρ, Q 5 (σ) σ α β+(n 1)( ) σ 1 = 1 + r 1 1 σ/ σ ρ n 1 r µ P (ρ)dρ. (1.34) The last integral is (bounded by) a convolution of P (ρ) with the function ρ n 1 r µ. In order to estimate the L (σ 1) norm of Q 5, we use first Hölder s then Young s ineuality: Q 5 L σ ɛ L ρ n 1 r µ L 1 P L where ( 1 ɛ = α β + (n 1) 1 ) +, = By assumtion we have ɛ. When ɛ >, in order for the norms to be finite we need n 1 ɛ > 1, µ < 1 r 1 which can be rewritten ( ) (n 1) 1 + µ < 1 < ɛ and we see that we can find a suitable rovided the first side is strictly smaller than the last side; this condition is recisely euivalent to (1.3) again (recall also that n 1 < µ n). The argument works also in the case ɛ = by choosing =. If on the other hand < µ < n 1, we have J 1 r µ r ρ µ also in this region, so that Q 5 (σ) σ σ n 1 β+ σ n 1 α µ 16 σ/ P (ρ)dρ

18 by σ ρ. Hölder s ineuality gives Q 5 (σ) σ n 1 β+ σ n 1 α µ σ 1 P L which leads to exactly the same comutations as above and in the end to (1.3). The case µ = n 1 introduces a logarithmic term which does not change the integrability roerties used here. It remains the last region when r = so that J(σ, ρ) = σ ρ µ and 1/ 1/ = 1. Then n 1 β+ Q 5 (σ) σ σ σ/ σ ρ µ ρ n 1 α P (ρ)dρ which is identical with (1.34) with r =, thus the same comutations aly and the roof is concluded.. Weighted Sobolev embeddings In this section we write estimate (1.6) in the form of a Sobolev embedding. In this way we get also critical estimates in Besov Saces. Recalling the ointwise bound u(x) CT λ ( D n λ u ), < λ < n (1.35) where D σ = ( ) s, we see that an immediate conseuence of (1.9) is the weighted Sobolev ineuality x β u L L x α D σ u L L (1.36) rovided 1 < <, 1 and β < n, α < n, < σ < n α + β = σ + n n ( 1 α + β (n 1) 1 ) (1.37) As usual, if the last condition is strict we can take, in the full range 1. For instance, this imlies the ineuality rovided 1 and x β u(x) x α D σ u L L (1.38) β <, α < n, < σ < n α + β = σ n α + β > (n 1) ( 1 1 ). If we choose α = we have in articular for (1, ), [1, ] x n σ u(x) D σ u L L, n < σ < n. (1.39) This extends to the non radial case the radial ineualities in [3], [17], [] (see also [8]) and many others; notice that in the radial case we can choose = to obtain the largest ossible range. When σ is an integer we can relace the fractional oerator D σ with usual derivatives; see Corollary 1.1 below for a similar argument. By similar techniues it is ossible to derive nonhomogeneous estimates in terms of norms of tye D σ u L ; we omit the details. 17

19 Critical estimates in Besov saces. Case (ii) in Theorem 1.3 is suitable for alications to saces defined via Fourier decomositions, in articular Besov saces. We recall the standard machinery: fix a Cc radial function ψ (ξ) eual to 1 for ξ < 1 and vanishing for ξ >, define a Littlewood-Paley artition of unity via φ (ξ) = ψ(ξ) ψ(ξ/), φ j (ξ) = φ ( j ξ), and decomose u as u = j Z u j where u j = φ j (D)u = F 1 φ j (ξ)fu. Then the homogeneous Besov norm Ḃs,1 is defined as u Ḃs = js u j L. (1.4),1 j Z We can aly Theorem 1.3-(ii) to each comonent u j in the full range of indices 1, with a constant indeendent of j. By the standard trick ũ j = u j 1 + u j + u j+1, u j = φ j (D)ũ j we obtain the estimate x β T γ u L x L C x α ũ j L x L j Z (1.41) for the full range 1, 1, with α, β, γ satisfying (1.37). The right hand side can be interreted as a weighted norm of Besov tye with different radial and angular integrability; this kind of saces were already considered in [8]. In the secial case α =, = > 1 we obtain a standard Besov norm (1.4) and hence the estimate (with the otimal choice = = ) reduces to x β T γ u L C u x L Ḃ. (1.4),1 This estimate is weaker than (1.9) when the third condition in (1.7) is strict, but in the case of euality it gives a new estimate: recalling (1.35), we have roved the following Corollary 1.8. For all 1 < we have x n 1 n 1 u L x L C u 1 Ḃ 1,1. (1.43) If we restrict (1.43) to radial functions and =, we obtain the well known radial ointwise estimate (see [], [19]). x n 1 u C u Ḃ1/,1 1 < < (1.44) 3. Caffarelli-Kohn-Nirenberg weighted interolation ineualities In this section we use the technology outlined before toghether with interolation. In this way we can extend to L L setting also the Caffarelli-Kohn-Nirenberg ineualities. Such ineualities are fundamental tools in mathematical analysis and in PDEs theory. In articular are really useful in the context of Navier-Stokes euation because rovide a riori estimate for weak solutions by interolation of uantities related to the energy dissiation. Let s start by the family of ineualities on R n, n 1 x γ u L r C x α u a L x β u 1 a L. (1.45) for the range of arameters n 1, 1 <, 1 <, < r <, < a 1. (1.46) Some conditions are immediately seen to be necessary for the validity of (1.45): to ensure local integrability we need γ < n r α < n β < n (1.47) 18

20 and by scaling invariance we need to assume γ n ( r = a α + 1 n ) + (1 a) ( β n ). (1.48) In [1] the following remarkable result was roved, which imroves and extends a number of earlier estimates including weighted Sobolev and Hardy ineualities: Theorem 1.9 ([1]). Consider the ineualities (1.45) in the range of arameters given by (1.47), (1.46), (1.48). Denote with the uantity ( 1 = γ aα (1 a)β a + n r 1 a a ) (1.49) (the identity in (1.49) is a reformulation of the scaling relation (1.48)). Then the ineualities (1.45) are true if and only if both the following conditions are satisfied: (i) (ii) a when γ n/r = α + 1 n/. Remark 1.4. Notice that in the original formulation of [1] also the case a = was considered, but with the introduction of an additional arameter forcing β = γ when a =. Thus the case a = becomes trivial in the original formulation; however, at least for r > 1, a much larger range γ β < n can be obtained by a direct alication of the Hardy-Littlewood-Sobolev ineuality, so strictly seaking the additional reuirement β = γ is not necessary. We think the formulation adoted here is cleaner. On the other hand, the necessity of (i) follows from the uniformity of the estimate w.r.to translations, while the necessity of (ii) is roved by testing the ineuality on the sikes x γ n/r log x 1 truncated near x =. In [5] the authors rove the following radial imrovement of Theorem 1.9: Theorem 1.1 ([5]). Let n, let α, β, γ, r,,, a be in the range determined by (1.47), (1.46), (1.48), define as in (1.49), and assume that ( a 1 n ) a, α < n 1, (1.5) the first ineuality being strict when = 1. Then estimate (1.45) is true for all radial functions u C c (R n ). We somewhat simlified the statement of Theorem 1.1 in [5], and in articular conditions (1.8)-(1.1) in that aer are euivalent to (1.5) here, as it is readily seen. Notice that the condition a forces r to be larger than 1. Using the L L norms we can extend both Theorems 1.9 and 1.1. For greater generality we rove an estimate with fractional derivatives Our result is the following: D σ = ( ) σ, σ >. Theorem Let n, r, r,,,, [1, + ), < a 1, < σ < n with γ < n r, β < n, n n < α < n σ (1.51) satisfying the scaling condition γ n ( r = a α + σ n ) ( + (1 a) β n ). (1.5) r Define the uantities ( 1 = aσ + n r 1 a a ) (, = aσ + n 1 r 1 a ã ). (1.53) 19

21 and assume further that + (n 1), (1.54) ( 1 <, a σ n ) ( < aσ, a σ ñ ) aσ. (1.55) Then the following interolation ineuality holds: x γ u L r L r C x α D σ u a x β u 1 a. (1.56) x L x L L x L If one assumes strict ineuality in (1.54), then the ineualities in (1.55) can be relaxed to non strict ineualities. When σ is an integer, the condition on α from below can be droed, and a slightly stronger estimate can be roved. We introduce the notation x α D σ u L L = x α D ν u L L, ν = (ν 1,..., ν n ) N n. Then we have: ν =σ Corollary 1.1. Assume σ = 1,..., n 1 is an integer. Then the following estimate holds x γ u L r L r C x α D σ u a x β u 1 a. (1.57) x L x L L x L rovided the arameters satisfy the same conditions as in the revious theorem, with the excetion of the condition α > n + n/ which is not necessary. Remark 1.5. If σ = 1, Corollary 1.1 contains both the original result of [1] (for a) and the radial imrovement of [5]. Indeed, if we choose =, =, r = r in Corollary 1.1 we get of course =, and selecting σ = 1 we reobtain the original ineuality (1.45) in the range a. On the other hand, if u is a radial function, estimate (1.57) does not deend on the choice of,, r and we can let assume an arbitrary value in the range (1.55). Thus if > a(σ n/) we can choose =, while if = a(σ n/) we can choose = ɛ > arbitrarily small, recovering the results of Theorem 1.1. Again we can get theorem 1.11 as a conseuence of 1.3. Proof. We begin by taking < a 1, and indices r, r, s, s,, [1, + ] such that 1 r = a s + 1 a 1, r = ã s + 1 a. (1.58) Then by two alications of Hölder s ineuality we obtain the interolation ineuality x γ u L rl r = ( x δ u) a ( x β u) 1 a L rl r ( x δ u) a L s/a L s/a ( x β u) 1 a L /(1 a) L /(1 a) = x δ u a L x β u 1 a sl s L L rovided the exonents γ, δ, β are related by (1.59) γ = aδ + (1 a)β. (1.6) Now the main ste of the roof. By Theorem (1.3) we know that x δ T λ u Ls L s x α u L L under suitable conditions on the indices. Now using the well known estimate u(x) C λ,n T λ ( D n λ u ) (1.61)

22 the revious ineuality can be euivalently written which together with (1.59) gives x δ u L s L s x α D σ u L L, σ = n λ x γ u L rl r x α D σ u a L L x β u 1 a. (1.6) L L The conditions on the indices are those given by (1.58), (1.6), lus those listed in the statement of Theorem (1.3) (notice that we are using α instead of α). The comlete list is the following: r, s,, r, s, [1, + ], a < 1, < σ < n, (1.63) 1 r = a s + 1 a 1, r = ã s + 1 a. (1.64) 1 < s <, 1 s, (1.65) γ < n r, β < n, α < n, δ < n s, (1.66) γ = aδ + (1 a)β, (1.67) α + δ + n σ = n + n s n, (1.68) ( 1 α + δ (n 1) s 1 ) +. (1.69) 1 1 s Recall also that, when the last ineuality (1.69) is strict, we can allow the full range 1 s. Our final task is to rewrite this set of conditions in a comact form, eliminating the redundant arameters δ, s, s. Define the two uantities ( 1 = aσ + n r 1 a a ) (, = aσ + n 1 r 1 a ã ). Then (1.64) are euivalent to ( = a σ + n s n ) (, = a σ + ñ s ñ ) while (1.68) is euivalent to (1.7) δ = α + (1.71) a and we can use (1.7), (1.71) to relace δ, s, s in the remaining relations. Condition (1.67) becomes = γ aα (1 a)β, (1.7) which is recisely the scaling condition, while (1.69) becomes The last ineuality in (1.66), δ < n/s, can be written so that (1.66) is relaced by + (n 1). (1.73) α < n σ γ < n r, β < n, n n < α < n σ. (1.74) Finally, conditions (1.65) translate to ( 1 <, a σ n ) ( < aσ, a σ ñ ) aσ. (1.75) 1

23 When the ineuality in (1.73) is strict, the last condition can be relaxed to ( 1, a σ n ) ( aσ, a σ ñ ) aσ. (1.76) We ass now to the roof of Corollary 1.1. Assume now σ is integer, and the ineuality x γ u L rl r C x α D σ u a L L x β u 1 a L L is true for a certain choice of the arameters as in the theorem, so that in articular α < n σ < n. Then we shall rove that also the following ineualities are true x k γ u Lr L r C x k α D σ u a L L x k β u 1 a L L (1.77) for all integers k, where we are using the shorthand notation x k α D σ u L L = x k α D ν u L L, (ν = (ν 1,..., ν n ) N n ). ν =σ This in articular imlies that the condition on α from below can be droed when σ is integer. When k =, (1.77) is obtained just by relacing D σ with D σ in the original ineuality. The roof of this estimate is identical to the revious one; the only modification is to use, instead of (1.61), the stronger ointwise bound u(x) C λ,n T λ ( D n λ u ) (1.78) which is valid for all λ = 1,..., n 1. Now if we aly (1.77) (with k = ) to a function of the form x k u for some k 1, we obtain x k γ u L r L r C x α D σ ( x k u) a L L x k β u 1 a L L and to conclude the roof we see that it is sufficient to rove the ineuality x α D σ ( x k u) L L x k α D σ u L L (1.79) for all α < n/, 1, <, and integers σ = 1,..., n 1, k 1. Notice indeed that all the conditions on the arameters (aart from α > n+n/) are unchanged if we decrease γ, α, β by the same uantity. By induction on k (and writing δ = α), we are reduced to rove that for all, [1, ) and 1 σ n 1 x δ D σ ( x u) L L x 1+δ D σ u L L, δ > σ n. (1.8) Using Leibnitz rule we reduce further to x 1+δ l u L L x 1+δ D l u L L, δ > l n (1.81) for l = 1,..., n 1, and by induction on l this is imlied by x δ u L L x 1+δ u L L, δ > 1 n. (1.8) In order to rove (1.8), consider first the radial case. When u = φ( x ) is a radial (smooth comactly suorted) function, we have x δ u L L ρ δ+n 1 φ(ρ) dρ.

24 Integrating by arts we get = δ + n (ρ δ+n 1 φ ) 1 1 ρ δ+n φ 1 φ(ρ) dρ x δ u x 1+δ u L L L L (ρ δ++n 1 φ ) 1 dρ which imlies (1.8) in the radial case. If u is not radial, define ( φ(ρ) = u(ρ) L (Sn 1 ) = so that ( x δ u L L The roof in the radial case imlies moreover we have S n 1 u(ρ) ds ) 1 ρ δ+n 1 φ(ρ) dρ. x δ u L L x δ+1 φ ( x ) L ; φ (ρ) φ 1 S n 1 u(ρ) 1 u ds and in conclusion we obtain ( ) 1 ( ) 1 φ 1 u u S S ) 1 = u(ρ) L (Sn 1 ) as claimed. x δ u L L x δ+1 u L L 4. Strichartz estimates for the wave euation As a last examle, we mention an alication of our result to Strichartz estimates for the wave euation; a more detailed analysis will be conducted elsewhere. The wave flow e it D on R n, n, satisfies the estimates, which are usually called Strichartz estimates: rovided the indices, r satisfy Here the L t L r x norms are defined as D n r + 1 n e it D f L t Lr x f L (1.83) [, ], < 1 r 1 (n 1). (1.84) u(t, x) L = u(t, ) t Lr x L r L x. t In their most general version, the estimates were roved in [11], [15]. Notice that in (1.83) we included the extension of the estimates which can be obtained via Sobolev embedding on R n. If the initial value f is a radial function, the estimates admit an imrovement in the sense that conditions (1.84) can be relaxed to [, ], < 1 r < 1 1 (n 1). (1.85) 3

25 This henomenon is connected with the finite seed of roagation for the wave euation and is usually deduced using the sace-time decay roerties of the euation. For a thourough discussion and a comrehensive history of such estimates see e.g. [14] and the references therein. A different set of estimates are the smoothing estimates, also known as Morawetztye or weak disersion estimates. These aear in a large number of versions; a articularly shar one is the following, from [1]: x ζ D 1 ζ e it D f L t L Λ 1 ζ 1 f x L, < ζ < n. (1.86) Here the oerator Λ = (1 S n 1) 1/ is a function of the Lalace-Beltrami oerator on the shere and acts only on angular variables, thus we see that the flow imroves the angular regularity. Morawetztye estimates are concetually simler than (1.83), being related to more basic roerties of the oerators; indeed L estimates of this tye can be roved for uite large classes of euations via multilier methods. Corresonding estimates are known for the Schrödinger flow e it, and M.C. Vilela [4] noticed that in the radial case they can be used to deduce Strichartz estimates via the radial Sobolev embedding. Following a similar idea for the wave flow, in combination with our recised estimates (1.36), gives an even better result, which strengthens the standard Strichartz estimates (1.83)-(1.84) in terms of the mixed L L x norms. Indeed, a secial case of (1.36) gives, for arbitrary functions g(x), ( g L x L x α D α+ n n n 1 g L,, [, ), > α (n 1) 1 ) (1.87) with the exclusion of the case α =, = =. Then by (1.87) and (1.86) we obtain the recised Strichartz estimates rovided x δ D n + 1 n δ e it D f L t L x L Λ ɛ f L (1.88), [, + ), δ < n, < ɛ < n 1, < 1 < 1 1 (n 1) and ( 1 ɛ δ + (n 1) 1 (1.89) ). (1.9) (n 1) 1 We will not exloit the conseuence of this articular section in the thesis. The results contained can be consider just as an alication of our oint of view to a different class of roblems. We hoe to came back on Strichartz estimates in L L saces in future works. 4

26 CHAPTER Introduction to the regularity roblem for the Navier-Stokes euation In this chater we introduce the Cauchy roblem for the Navier-Stokes aroximation of the fluid motion in the whole sace, that s t u + (u )u + u = in [, T ) R n u = in [, T ) R n u = u in {} R n. (.1) Here u = (u 1, u, u 3 ) is the velocity field, is the ressure and the viscosity have been set to one. No external forces is working. The first euation is the Newton law while the second guarantees the incomressibility of the fluid. To reuire incomressibility also at time t = have to be considered just initial data u such that u =. So is useful to define the sace { } L σ(r n ) = u C (Rn ) s.t. u dx < +, u =. (.) R n We will use the same notation for the norm of scalar, vector or tensor uantities, the meaning will be clear by the situation; for instance we will use L (R n ) = dx, u R n L (R n ) = 3 R n i=1 u i dx, u L (R n ) = 3 R n i,j=1 ( iu j ) dx. We will use also the notation u L (R n ) instead of u (L (R n )) 3, and so on. The well osedness of (.1) is a well-known mathematical challenge and just artial results have been obtained. The main uestion is: if we consider an initial datum u in the Schwartz class, there exists a uniue global solution of the roblem (.1)? In this chater we will give an excursus on classical theorems starting by the ioneering work of Leray, Hof, Serrin and Kato [6, 8, 9]. This classical results have been imroved in many different directions and in the thesis we will focus basically on the weighted norm aroach, which also has a wide reference literature, see [33], [3], et al. We will also briefly focus on [35, 7]. The first is articulary relevant because seems to be a shar version for the existence with small data. The second is a celebrated landmark for the regularity theory. 1. Euivalence between the differential and integral formulation In this secion we give the integral formulation of roblem (.1). This formulation is very useful in order to study both local (in time) well osedness and global well osedness with small data. In such a case, starting by the integral formulation is immediate to look at the euation (.1) as a erturbed heat euation, and fixed oint techniues are available. Of course this is irrelevant when we look for global solutions with large initial data. We follow basically [45], and we omit, almost comletely, the roofs to get a comact resentation. Anyway all the roofs are classical and can be easily found in literature. Let come back on the system: t u + (u )u + = u u = u = u. 5

27 or in comonents (i = 1,..., n): { t u i + n j=1 u j j u i + i = n j=1 jju i n i=1 iu i =. By taking the divergence of the first euation and using u = we get: n n = u j j u i + (.3) = i i=1 j=1 n i j (u i u j ) +, (.4) i,j=1 so can, at least formally, be recovered by u througt: n i,j=1 = i j (u i u j ). (.5) By using his relation the system can be written as: { u = e t u + t e(t s) P (u u)ds in [, T ) R n u = in [, T ) R n (.6). where (u u) i,j = u i u j and P is formally: or in comonents Pf = f 1 f, (.7) (Pf) i = f i i j f j. The oerator P, that s a really useful tool in the study of the Navier-Stokes roblem, has been introduced by Leray in [6]. It is a rojection oerator on the subsace of the divergence free vector fields. It is infact easy to show that Pf = f f =. In order to give recise definition of the formal comutation above at first we have to make sense to the oerator P. This is easy if we restrict to f L (R n ), in this case Pf is defind by: Pf = f (R R)f or in comonents: (Pf) i = f i R i R j f j, j where R j is the Riesz transform in the direction j defined by the symbol i ξj ξ. This is a simle way to define P, even if it can be defined on larger Banach saces as a Calderon-Zygmund oerator; details can be found in [45]. Anyway we are interested basically in the oerator P( f), that aears in the integral formulation (.6); it is defined through comonents by: (P (u u)) i = j (u u) i,j 1 i j k (u u) j,k. In such a case the differentiation allows to extend the definition to a larger class of Banach saces. To give a recise definition (again in [45]) we need the auxiliary saces: Definition.1. Let define the dual saces W L (R n ), L 1 uloc (Rn ): the sace W L (R n ) is the Banach sace of the Lebesgue measurable functions φ on R n such that su φ(x) < +, (.8) k Z n x {k+[,1] n } euied with the norm (.8); 6

28 the sace L 1 uloc (Rn ) is the sace of locally intgrable functions euied with the norm: It holds the following: L 1 uloc(r n ) = su [,1] n 1 [,1] nf L 1 (R n ). Lemma. ([45]). The oerator 1 i j k is a convolution oerator with a kernel T i,j,k such that the following decomosition holds: T i,j,k = α i,j,k + i j β k where α i,j,k W L (R n ) and β k L 1 loc (Rn ). By lemma (.) and inclusions L 1 loc L 1 uloc L 1 uloc, W L L, it turns out that P( ) can be defined on the sace (L 1 uloc (Rn )) n n. Now we focus on some roerties of convolutions with the Oseen Kernel, so we consider: It holds the following: 1 i j e t. Lemma.3 ([45]). Let 1 i, j n. The oerator 1 i j e t is a convolution oerator O i,j (t) f j, with O i,j (t) (C (R n )) n n, the homogeneity: O i,j (t) = 1 ( ) x t O t n i,j, and the decay: for each multi-index η. (1 + x ) n+ η η O i,j (L (R n )) n n, This is the main technical tool we need in order to study the roerties of e t P( ) that acts on the tensor u u through (e t P( (u u))) i = e t j (u u) i,j e t 1 i j k (u u) j,k. It holds the following: Proosition.4 ([45]). Let 1 i, j, k n. The oerator e t P( ) is a convolution oerator K i,j,k (t) f j,k, with K i,j,k (t) (C (R n )) n n, the homogeneity: K i,j,k (t) = 1 ( ) x t K t n+1 i,j,k, and the decay: (1 + x ) n+1+ η η K i,j,k (L (R n )) n n, for each multi-index η. We finish by stating an useful euivalence result: Theorem.5 ([45]). Let u s<t ( L t L uloc,x (, s) Rn ). Then the following are euivalent: (1) u is a weak solution of: t u + P (u u) = u in [, T ) R n u = in [, T ) R n u = u in {} R n. (.9) 7

29 () u solves the integral roblem: { u = e t u + t e(t s) P (u u)ds in [, T ) R n u = in [, T ) R n. (.1). The Leray-Hof solutions The modern theory of Navier-Stokes euation starts with the Leray s work [6] in which global existence of weak solutions of (.1) for L initial data is roved. Such weak solutions have also hysical meaning because they resond to the energy dissiation. On the other hand the existence theorem is by comactness and neither regularity nor unieness have been roved in the general case. In this section we will briefly sketch the ideas of the Leay s theory. We start by definition of weak solution in the context of [6] Definition.6 (Leray s solutions). The air (u, ) is a weak Leray solution of the Navier-Stokes system (.1) in [, T ) R n if the following holds: (1) Exist some constants E, E 1 such that: R n u(t, ) dx E, (.11) T for almost every t (, T ), and T u dxdt E 1 ; (.1) R n () (u,) satisfy (.1) in the sense of distributions in [, T ) R n, that s T ( t φ+(u )φ)u dxdt+ u φ(x, ) dx = ( φ )u dxdt, (.13) R n R n R n for each φ Cc ([, T ) R n ) with φ = and T T n u φ dxdt =, φ + u i u j i j φ =, (.14) R n R n i,j=1 for each φ Cc ([, T ) R n ). (3) u satisfy the energy ineuality: t u(t, ) + R n u dxdt R n u R n (.15) for each t (, T ). Condition (.15) is exression of the dissiation of kinetic energy (the first term of the sum) caused by the frictions (the second term). It can be justified by multilication of (.1) with φu and integration by arts. It is well know that Leray weak solutions are weakly continuous (see [41]), i.e. lim u(t, x)w(x) dx = u(s, x)w(x) dx (.16) t s R n R n for all w L (R n ), and so, if u L (R n ) then lim u(t, x)w(x) dx = u (x)w(x) dx, (.17) t R n R n for all w L (R n ). This is how u attend its initial datum. In [6] is roved the existence of a weak solution u R + R n of (.1) for every u L σ(r n ). If we set the roblem in [, T ] Ω, Ω R n oen and bounded, and we reuire zero Dirichlet condition on [, T ] Ω, an analogous result is due to Hof [8]. Let then state the recise Leray s result: 8