Kinetic energy represented in terms of moments of vorticity and applications
Tomasz Cieślak, Krzysztof Oleszkiewicz, Marcin Preisner, Marta Szumańska
aa r X i v : . [ m a t h - ph ] A ug Kinetic energy represented in terms of moments of vorticityand applications.
Tomasz Cie´slak
Institute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, Warsaw, Polande-mail: [email protected]
Krzysztof Oleszkiewicz
Institute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, Warsaw, PolandInstitute of Mathematics, University of Warsaw, ul. Banacha 2, Warsaw, Polande-mail: [email protected]
Marcin Preisner
Instytut Matematyczny, Uniwersytet Wroc lawski, Pl. Grunwaldzki 2/4, Wroc law, Polande-mail: [email protected]
Marta Szuma´nska
Institute of Mathematics, University of Warsaw, ul. Banacha 2, Warsaw, PolandInstitute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, Warsaw, Polande-mail: [email protected]
August 15, 2018
Abstract
We study 2d vortex sheets with unbounded support. First we show a version of the Biot-Savart law related to a class of objects including such vortex sheets. Next, we give a formulaassociating the kinetic energy of a very general class of flows with certain moments of theirvorticities. It allows us to identify a class of vortex sheets of unbounded support being only σ -finite measures (in particluar including measures ω such that ω ( R ) = ∞ ), but with locallyfinite kinetic energy. One of such examples are celebrated Kaden approximations. We studythem in details. In particular our estimates allow us to show that the kinetic energy of Kadenapproximations in the neighbourhood of an origin is dissipated, actually we show that theenergy is pushed out of any ball centered in the origin of the Kaden spiral. The latter resultcan be interpreted as an artificial viscosity in the center of a spiral. Key words: vortex sheet, spherical averages, Biot-Savart law, Kaden spirals.
MSC 2010: Introduction
In the present paper we study vortex sheets which are not compactly supported. In engineeringor physics literature vortex sheets are usually two-dimensional divergence-free (in the sense ofdistributions) vector fields such that their vorticities are zero except on a curve c , along whichtangential components of velocity are discontinuous. For us vortex sheets are a wider class ofobjects, namely 2d divergence-free velocity fields whose vorticity ω are σ − finite measures only. Inparticular unbounded measures ω , i.e. such that ω ( R ) = ∞ , are included. The usual definition of vortex sheets assumes that such objects have vorticities beingcompactly supported finite Radon measures, see [6]. However such restriction eliminates from theconsiderations well-known spirals of vorticity, self-similar objects well-established in engineeringand physics literature like Kaden spirals (see [9, 7]), Prandtl spirals (we refer the reader to [13,10, 17, 3]) or recent hyperbolic spirals introduced in [19]. Extension of the theory to such self-similar vortex spirals seems important and required. Moreover we would like to restrict ourselvesto vector fields with locally finite kinetic energy. The importance of such objects is emphasizedin the introduction of [3]. It was noticed in [3] that a crucial property of a compactly supportedvorticity measure ω ω ( B (0 , r )) = cr α , (1.1)where c and α are positive constants, yields that local H − -norm of ω is finite. By the lemma ofSchochet [18] it means that the kinetic energy generated by a compactly supported part of such avortex sheet is locally finite. Property (1.1) is satisfied at least by well-known examples of Kadenand Prandtl, see [3].The main concern of the present paper is to extend the previous study in [3] to the case ofvortex sheets which are not necessarily compactly supported, moreover such that their vorticity ω is an unbounded measure, i.e. ω ( R ) = ∞ . One of the main questions we address is whether the kinetic energy generated by such objectsis locally finite or not. We shall give precise conditions yielding sharp estimates of local kineticenergy from above and below for a class of objects satisfying (1.1), see Theorem 1.1 below.When speaking about kinetic energy carried by a vorticity we need to know the divergence-free velocity associated to the vorticity. In case of vorticity being a compactly supported regularfunction, velocity is given by the usual Biot-Savart operator. One of the tasks of the presentpaper is to identify the velocity given by a vorticity being σ -finite measure satisfying (1.1). Thisis discussed in Section 2. We shall say more on it also at the end of Introduction.The question concerning kinetic energy is very important for several reasons. On the one handit is a natural expectation for an object which is supposed to have a physical meaning. Next, whenlooking for vortex sheets weak solutions of the 2d Euler equations one has to make sure that thefollowing integral Z T Z R v ( x, t ) ⊗ v ( x, t ) : ∇ φ ( x, t ) dxdt is finite for a divergence-free velocity field v : R → R and any smooth compactly supporteddivergence-free test function φ : R → R . To this end it suffices that the local kinetic energy of v ,2hich is defined by E r ( ω ) = Z B (0 ,r ) | v ( x ) | dx, (1.2)is finite.Our main theorem states that the local kinetic energy E r ( ω ) of a nonnegative σ -finite measureof vorticity ω satisfying (1.1) undergoes a precise estimate from below and above. Theorem 1.1
Let ω be a nonnegative σ -finite measure satisfying (1.1) with α ∈ (0 , . Then, for c being a constant appearing in (1.1) , we have c πα r α ≤ E r ( ω ) ≤ c απ ( πα ) r α , r > . (1.3)In the proof of Theorem 1.1 we study the spherical averages A r ( ω ) = (2 π ) − Z π (cid:12)(cid:12) v ( re iθ ) (cid:12)(cid:12) dθ, r > E r ( ω ) by E r ( ω ) = 2 π Z r A s ( ω ) s ds. (1.5)Our main tool to estimate A r ( ω ) is the following formula expressing A r ( ω ) in terms of inner andouter moments of vorticity. Since 2d plane can be viewed as a set of complex numbers z = x + iy ,for r > n ≥ m r, ( ω ) = ω ( B (0 , r )) , (1.6) m r,n ( ω ) = Z B (0 ,r ) u n dω ( u ) , (1.7) M r,n ( ω ) = Z C \ B (0 ,r ) u − n dω ( u ) , (1.8)where the powers are taken in the sense of complex numbers.Actually, our formula for A r ( ω ) is proven under less restrictive assumptions on the measure ω than (1.1). Assume that a nonnegative σ -finite measure ω on R is given such that Z R (1 + | x | ) − dω ( x ) < ∞ . (1.9)For such measures the following result holds. Theorem 1.2
Assume that a nonnegative σ -finite measure ω satisfies (1.9) . Then π A r ( ω ) = ∞ X n =0 r − n − | m r,n ( ω ) | + ∞ X k =1 r k − | M r,k ( ω ) | (1.10) for a.e. r > . Remark 1.3
Let us emphasize that Theorems 1.1 and 1.2 state results holding for measures whichare only σ -finite, in particular it applies also to measures ω such that ω ( R ) = ∞ (provided theysatisfy the assumptions). Moreover, we study Kaden’s approximations, see [9], and examine their properties. In particulartime evolution of the energy in any ball surrounding the origin of the Kaden spiral is computed.It is shown that such an energy is dissipated. When time approaches infinity, the kinetic energycontained in any ball surrounding the origin of the Kaden approximation tends to the minimalpossible value given by the left-hand side of (1.3), while for small times Kaden’s spiral’s kineticenergy approaches the upper bound in (1.3). Actually, we even compute the limiting objectsreached by the divergence-free velocities related to the Kaden spiral when t approaches 0 as wellas when t tends to infinity. The results concerning time evolution are obtained using our momentformula (1.10) applied to the difference of two unbounded measures. Such a difference does nothave to be a signed measure. Thus, extension of (1.10) requires a precise definition of some newobjects.First of all, let us notice that when we consider a difference of two nonnegative measures ω and ω , such that ω ( R ) = ω ( R ) = ∞ , then ˜ ω := ω − ω cannot be defined as a signed measure.Indeed, one has a problem to decide what is the value ˜ ω ( A ), where A is such that ω i ( A ) = ∞ for i = 1 ,
2. Such technical difficulties are the main reason for which we work with the followingobjects, which we shall call vorticities . By definition, a vorticity is a distribution ˜ ω which can berepresented as a difference of ω and ω , such that for i = 1 , σ -finite measures ω i satisfy (1.9).In Section 3 we extend (1.3) to vorticities ˜ ω = ω − ω , for which ω i satisfy (1.9). Complexmoments m r,n and M r,n , appearing in (1.3), as linear in ω are extended to ˜ ω in a natural way.We also show an extension of the Biot-Savart law to vorticities from a wide subclass of σ -finite measures, in particular our results are applicable also to measures ω such that ω ( R ) = ∞ .Actually, again our theorem works for quite a wide class of objects. Moreover, for σ -finite measuressatisfying (1.1) with α ∈ (0 ,
1) it is shown that velocities obtained via the Biot-Savart law are in L loc , so that the kinetic energy is finite.Let us explain ourselves from the slightly non-orthodox structure of the paper. Namely, thetechnical core with the proof of Theorem 1.2 is a content of Subsection 3.2. It is self-contained.Moreover, some of main results which we prove with the help of Theorem 1.2 are presented in earliersections. We use there results proven later in Subsection 3.2. This way technical computations arepostponed.At the end of Introduction let us state an easy fact concerning the assumptions we providedfor measures ω . We show that measures on C satisfying (1.1) with α ∈ (0 ,
1) satisfy also theassumption (1.9). Indeed, we have the following proposition.
Proposition 1.4
Let us fix α ∈ (0 , . Let a nonnegative σ -finite measure ω satisfy (1.1) withsuch α . Then (1.9) is also satisfied. roof. Notice that Z C dω ( u )1 + | u | = Z ∞ ω (cid:18)(cid:26)
11 + | u | > t (cid:27)(cid:19) dt = Z ω ( B (0 , /t − dt = c Z (cid:18) t − (cid:19) α dt ≤ c Z t − α dt < ∞ , and so (1.9) holds true. (cid:3) However, the class of measures satisfying (1.9) is much wider than those fulfiling (1.1), forinstance measures considered in [8] are also fine.
Notation.
We need to fix a convention which we use to speak about 2d vorticity. By definition,( x , x ) ⊥ = ( − x , x ). The partial derivatives in R are denoted by ∂ and ∂ , while a vorticity ofa vector field v = ( v , v ) is defined as curl( v ) = ∂ v − ∂ v . When it is convenient, we shall usecomplex notation for R . Main objects of our studies are two-dimensional velocity fields associated with vortex sheets. In thecase when a vorticity ω related to the divergence-free velocity v = ( v , v ) is a regular compactlysupported function, div( v ) = 0 and so there exists a potential ψ such that ∇ ψ = v ⊥ . In otherwords v = − ⊥ ∇ ψ . Taking the curl of both sides we arrive at − △ ψ = ω. (2.1)Then the velocity field v is recovered from vorticity ω by the Biot-Savart formula, i.e., v ( x ) = 12 π Z R ( x − y ) ⊥ | x − y | dω ( y ) . (2.2)The same procedure works for more general vorticities, even for compactly supported measures.Then ψ , a solution to (2.1), still exists, and if ω ∈ H − loc then ψ is regular enough to make surethat v is given by (2.2). However it is not known whether the Biot-Savart law is still valid in thecase of a vorticity which is not a bounded measure. Yet, a very important class of vortex sheetsare the so-called self-similar spirals of vorticity (one of them, the Kaden spiral is studied later onin the present paper). For instance measures satisfying (1.1) are immediately unbounded, indeed ω ( R ) = lim r →∞ ω ( B (0 , r )) = lim r →∞ cr α = ∞ . The question which we address is a validity of Biot-Savart’s law for nonnegative measuressatisfying (1.9). We show that for such objects the Biot-Savart formula is still well-defined andrecovers velocity field v related to vortex sheet ω , i.e. curl ( v ) = ω , provided ω satisfies (1.9). Con-sequently, in view of Proposition 1.4, the Biot-Savart law holds in particular for σ -finite measuressatisfying (1.1) with α ∈ (0 , v ( x ) is well-defineda.e., divergence-free in the sense of distributions as well as curl ( v ) = ω holds in the sense ofdistributions. 5otice that our result is not trivial since the procedure described at the beginning of the presentsection to derive the Biot-Savart law seems to require strong regularity assumptions. Indeed,solving (2.1) with ω being only σ -finite measure, in particular possibly ω ( R ) = ∞ , seems nottrivial. Notice that due to the contribution from infinity of both ω and a fundamental solutionof Laplace operator, ψ might not exist, and so one cannot tell that v = − ⊥ ∇ ψ . Our result holdsin a more general situation, when vorticity is a nonnegative measure which might not possess astream function. A reader might check that actually stream function does not exist for Kaden’sspirals (that are introduced in Section 4). Nevertheless, we show that the velocity field can still beexpressed by the Biot-Savart formula (2.2) if a measure ω satisfies (1.1) with α ∈ (0 , σ -finite measures satisfying a conditionstated already in [3] and being a consequence of Prandtl’s similitude laws (see [3] and the referencestherein).As to the proof of Theorem 2.1, the only non-standard part is a justification of the use ofFubini’s theorem in (2.6). This requires a sort of potential theory type estimate. The requiredresult is a claim of Corollary 3.6, a technical lemma yielding integrability of the integrand in (2.6).Consequently, see Proposition 1.4, the Biot-Savart law is also well-defined for vorticities for which(1.1) holds with α ∈ (0 , Theorem 2.1
Let ω be a nonnegative σ -finite measure satisfying (1.9) . Assume that v ( x ) is givenby v ( x ) = 12 π Z R ( x − y ) ⊥ | x − y | dω ( y ) . (2.3) Then v ∈ L loc ( R ) (in particular is well-defined a.e.) and for all ϕ ∈ C ∞ ( R ) it holds Z R ∇ ϕ ( x ) · v ( x ) dx = 0 , (2.4) Z R ⊥ ∇ ϕ ( x ) · v ( x ) dx = − Z R ϕ ( y ) dω ( y ) . (2.5) Proof.
Let us prove (2.5). We have12 π Z R ⊥ ∇ ϕ ( x ) · v ( x ) dx = 12 π Z R ⊥ ∇ ϕ ( x ) Z R ( x − y ) ⊥ | x − y | dω ( y ) dx. First, we notice that measure ω ( x ), satisfies assumptions of Corollary 3.6, so the latter can beapplied to show that v given by (2.3) satisfies v ∈ L loc , in particular, v is finite a.e. On the otherhand, the same Corollary 3.6, again applied to the measure ω ( x ), allows us to use Fubini’s theoremin the integral Z R Z R ∂ i ϕ ( x ) y j − x j | y − x | dω ( y ) dx, (2.6)where i, j ∈ { , } . Indeed, there exist R , M such that supp ϕ ⊂ B (0 , R ) and k ∂ i ϕ k ∞ ≤ M , it isenough to make sure that R B (0 ,R ) R C dω i ( y ) | y − x | dx < ∞ . But this is exactly (3.11), the main claim ofCorollary 3.6. 6nowing that the Fubini theorem can be applied below, the rest of the reasoning is fullystandard. We have Z R ⊥ ∇ ϕ · v dx = 12 π Z R Z R ⊥ ∇ ϕ ( x ) · ⊥ ∇ ln | x − y | dω ( y ) dx F ubini = 12 π Z R Z R ∇ ϕ ( x ) · ∇ ln | x − y | dx dω ( y ) = 12 π Z R F ( y ) dω ( y ) , where F ( y ) = R R ∇ ϕ ( x ) · ∇ ln | x − y | dx . Let ε > F ( y ) = Z B ( y,ε ) ∇ ϕ ( x ) · ∇ ln | x − y | dx + Z R \ B ( y,ε ) ∇ ϕ ( x ) · ∇ ln | x − y | dx := F ( y ) + F ( y ) . Obviously, F ( y ) ≤ Cε k∇ ϕ k ∞ . Denote by ν the inward normal unit vector on ∂B ( y, ε ). Byintegrating by parts, F ( y ) = − Z R \ B ( y,ε ) ϕ ( x )∆(ln | x − y | ) dx + Z ∂B ( y,ε ) ϕ ( x ) ∂∂ν ln | x − y | dl ( x )= − ε − Z ∂B ( y,ε ) ϕ ( x ) dl ( x ) → − πϕ ( y ) . The proof of (2.4) is analogous. In the last step there we use ∂∂ν (ln | x − y | ) ⊥ = 0. (cid:3) Let us conclude this section with a remark concerning higher integrability of velocity v givenby (2.2) if ω satisfies (1.1) with α ∈ (0 , α ∈ (0 ,
1) implies (1.9). Next, a consequence of Theorem 1.1, which we prove in Section 3, isa higher integrability of v associated with ω via (2.2). Remark 2.2
Let ω be a nonnegative σ -finite measure which satisfies (1.1) with α ∈ (0 , . Then v associated with ω via (2.2) satisfies (2.4) and (2.5) and belongs to L loc ( R ) . This section is devoted to the proof of the main result. We prove Theorem 1.2, which yields aformula representing radial averages of the square of the divergence-free velocity associated withthe vorticity being nonnegative σ -finite measure satisfying (1.9). Moreover, we show how to inferTheorem 1.1 from our moment representation formula. As a consequence we obtain a preciseestimate (from below and above) of the kinetic energy contained in a ball B (0 , r ) carried by avorticity satisfying (1.1).Finally, we also state a variational problem related to the local kinetic energy estimates andfind its lower and upper bounds. Moreover, we identify the measures at which the maximal andminimal values are taken.The next theorem provides estimates required to obtain Theorem 1.1 as a consequence ofTheorem 1.2. It also gives very precise constants in the bounds of local kinetic energy which willbe used further to identify minimizers as well as maximizers of a variational problem leading tothe local kinetic energy estimates. 7 heorem 3.1 Let ω be a nonnegative σ -finite measure on C that satisfies (1.1) with α ∈ (0 , and c > . Then we obtain c π r α − ≤ A r ( ω ) ≤ c α ( πα ) r α − . (3.1) Proof.
We shall use Theorem 1.2. For the lower estimate, observe that4 π A r ( ω ) ≥ r − m r, ( ω ) = r − ω ( B (0 , r )) = c r α − . For the upper estimate, we notice that (1.1) allows us to estimate the moments, | m r,n ( ω ) | ≤ Z B (0 ,r ) | u | n dω ( u ) = c Z r s n αs α − ds = c αn + α r n + α ( n ≥ , (3.2) | M r,k ( ω ) | ≤ Z C \ B (0 ,r ) | u | − k dω ( u ) = c Z ∞ r s − k αs α − ds = c αk − α r − k + α ( k ≥ . (3.3)This leads to4 π A r ( ω ) ≤ c ∞ X n =0 r − n − (cid:18) αn + α (cid:19) r n +2 α + c ∞ X k =1 r k − (cid:18) αk − α (cid:19) r − k +2 α = c α ∞ X n = −∞ n + α ) r α − = c π α sin ( πα ) r α − . (cid:3) We finish this section with the proof of Theorem 1.1 provided that Theorem 1.2 holds. Thelatter is proven in Section 3.2.
Proof of Theorem 1.1.
We recall (1.5) and see that the kinetic energy E r ( ω ) is given as2 π R r sA s ( ω ) ds . Hence, integrating the bounds in (3.1) in r , we obtain the claim of Theorem 1.1. (cid:3) In what follows we introduce a functional over a certain subset of σ -finite measures which givesthe value of local kinetic energy associated with these measures. We identify the possible extremevalues and provide the extremizers. Let us fix α ∈ (0 , A as A := { nonnegative σ -finite measures ω satisfying (1.1) with α ∈ (0 , } . Fix r > E r ( ω ) for any ω ∈ A . We look for its minimumand maximum. The lower and upper bounds are given in Theorem 1.1. We show that those areactually achieved and provide the examples of extremals.Before proceeding with the argument let us notice that in the case of slightly more regularvelocities such a functional was used in literature to construct steady states of the incompressible2d Euler system, see [1] in the case of regular solutions and [21] in the case of vortex patches. It8s related to the hamiltonian structure of the Euler system. It is not clear to us whether the sameapproach could work in the case of steady vortex sheets (like those introduced in [5]).Denote by ω ∞ and ω respectively dω ∞ ( x + ix ) = cα π | x + ix | α − dx dx , (3.4) dω ( x + ix ) = cαx α − χ (0 , ∞ ) ( x ) δ ( x ) dx dx . (3.5)The first one is a radially symmetric measure and the latter one is a vortex sheet supported on thehalf-line. In both cases they are chosen so that (1.1) holds.First we notice that all the moments for ω ∞ (except m r, ( ω ∞ )) vanish. Thus A r ( ω ∞ ) = c π r α − and E r ( ω ∞ ) = c πα r α . From the proof of Theorem 3.1 one observes that for ω all the upper estimates become equalities,thus A r ( ω ) = c α ( πα ) r α − and E r ( ω ) = c απ ( πα ) r α . This proves that the estimates given in (1.3) and (3.1) are optimal, meaning that w ∞ and ω areminimizer and maximizer of E r over A , respectively. This subsection is essentially self-contained and can be read independently of the rest of the paper.We shall use the complex notation, that is z = x + iy ∈ C = R . The goal of the section is to proveTheorem 1.2. Assume that a nonnegative measure ω on C is given such that (1.9) is satisfied.In the complex notation the velocity is given by v ( z ) = 12 π Z C i ( z − u ) | z − u | dω ( u ) = 12 π Z C iz − u dω ( u ) . (3.6)We assume only (1.9). It turns out that this is enough to guarantee that (3.6) is well-defined fora.e. z ∈ C . As we shall see in Corollary 3.6, v ∈ L loc and so v is finite a.e. Moreover, we have seenin Section 2 that Corollary 3.6 is an important factor of the proof of the Biot-Savart formula forvortex sheets satisfying (1.9).The proof of Theorem 1.2 splits into several steps. Let us start with the following definition,set G = ( r > Z C dω ( u ) p | r − | u | | < ∞ ) . In particular, ω ( rS ) = 0 for r ∈ G , where S = { z ∈ C : | z | = 1 } . We show below that thisset is of full measure in (0 , ∞ ). It will be essential in showing that the divergence-free velocityassociated with ω is well-defined a.e.. Lemma 3.2
Let ω be a nonnegative σ -finite measure satisfying (1.9) . Then the Lebesgue measureof G c = (0 , ∞ ) \ G is zero. roof. For R ≥ W = Z R r Z C dω ( u ) p | r − | u | | dr < ∞ , (3.7)which obviously implies the lemma. Let W = W + W = Z | u | Lemma 3.3 Let u, v ∈ C , r > , | u | , | v | 6 = r . Then (2 π ) − Z π dθ ( u − re iθ )( v − re − iθ ) = ( uv − r ) − | u | , | v | > r, ( r − uv ) − | u | , | v | < r, | u | − r )( | v | − r ) < . (3.8) Proof. In the region | z | < r we have1 z − re iθ = − re iθ − zre iθ = − ∞ X n =1 z n − r n e − inθ , (3.9)whereas for | z | > r we have 1 z − re iθ = ∞ X n =0 r n z n +1 e inθ . (3.10)10onsider first the case | u | , | v | < r . Then in light of (3.9) one obtains(2 π ) − Z π dθ ( u − re iθ )( v − re − iθ ) = (2 π ) − ∞ X n,m =1 u n − v m − r n + m Z π e i ( m − n ) θ dθ = ∞ X n =1 ( uv ) n − r n = 1 r − uv . In the case when | u | , | v | > r the proof follows similarly using (3.10).Finally, let us consider the case | u | < r, | v | > r . Here we observe the crucial cancellations.(2 π ) − Z π dθ ( u − re iθ )( v − re − iθ ) = − ∞ X n =1 ,m =0 u n − r n r m v m +1 Z π e − i ( n + m ) θ dθ. Since for n ≥ , m ≥ n + m > 0, the integral on the right-hand side of the above identityis zero. Hence the claim follows. (cid:3) Let us notice that Lemma 3.3 can also be proved by the residue theorem. In a special case v = u we get the following. Corollary 3.4 For | u | 6 = r we have (2 π ) − Z π dθ | u − re iθ | = (cid:12)(cid:12) r − | u | (cid:12)(cid:12) − . Moreover, using the Cauchy-Schwarz inequality, Corollary 3.4, and the definition of the set G ,we arrive at the following. Corollary 3.5 For a nonnegative σ -finite measure ω , which satisfies (1.9) , and r ∈ G we have Z C Z C Z π dθ | u − re iθ | | w − re − iθ | dω ( u ) dω ( w ) < ∞ . Corollary 3.5 is essential in the proof of Theorem 1.2. It justifies an application of the Fubinitheorem in a crucial moment.Before proceeding with the proof of Theorem 1.2, let us state the next corollary, which on theone hand, guarantees that v , as defined in (3.6), is finite a.e., on the other hand gives a strongestimate which is used in Section 2 to state a general version of the Biot-Savart law. Corollary 3.6 Let R > . For a nonnegative σ -finite measure ω , which satisfies (1.9) , there holds Z B (0 ,R ) Z C dω ( u ) | u − z | dz < ∞ . (3.11) In particular, v defined by (3.6) , satisfies v ∈ L loc and so v < ∞ a.e.. roof. By H¨older’s inequality we have Z π dθ | u − re iθ | ≤ (2 π ) / (cid:18)Z π dθ | u − re iθ | (cid:19) / , hence by Fubini’s theorem Z B (0 ,R ) Z C dω ( u ) | u − z | dz ≤ Z R r Z C (2 π ) / (cid:18)Z π dθ | u − re iθ | (cid:19) / dω ( u ) dr. According to Corollary 3.4, the latter equals Z R r Z C π dω ( u ) p | r − | u | | dr, which is finite by (3.7). Hence, (3.11) is proven, which immediately guarantees that v ∈ L loc . (cid:3) Proof of Theorem 1.2. Assume that r ∈ G . By (3.6), A r ( ω ) = (2 π ) − Z π (cid:12)(cid:12) v ( re iθ ) (cid:12)(cid:12) dθ = (2 π ) − Z π (cid:12)(cid:12)(cid:12)(cid:12)Z C idω ( u ) re − iθ − u (cid:12)(cid:12)(cid:12)(cid:12) dθ = (2 π ) − Z C Z C Z π u − re iθ )( w − re − iθ ) dθ dω ( u ) dω ( w ) . In the last equality we have used Fubini’s theorem, see Corollary 3.5. Applying Lemma 3.3 weobtain4 π A r ( ω ) = Z B (0 ,r ) Z B (0 ,r ) r − uw dω ( u ) dω ( w ) + Z C \ B (0 ,r ) Z C \ B (0 ,r ) uw − r dω ( u ) dω ( w )= Z B (0 ,r ) Z B (0 ,r ) ∞ X n =0 u n w n r n +2 dω ( u ) dω ( w )+ Z C \ B (0 ,r ) Z C \ B (0 ,r ) ∞ X k =0 r k u k +1 w k +1 dω ( u ) dω ( w )= ∞ X n =0 r − n − m r,n ( ω ) m r,n ( ω ) + ∞ X k =1 r k − M r,k ( ω ) M r,k ( ω ) , This ends the proof of Theorem 1.2, provided that we justify the last equality. It suffices to have I = Z B (0 ,r ) Z B (0 ,r ) ∞ X n =0 | u | n | w | n r n +2 dω ( u ) dω ( w ) < ∞ and I = Z C \ B (0 ,r ) Z C \ B (0 ,r ) ∞ X k =0 r k | u | k +1 | w | k +1 dω ( u ) dω ( w ) < ∞ . 12o this end, notice that I = Z B (0 ,r ) Z B (0 ,r ) dω ( u ) dω ( w ) r − | u || w | ≤ Z B (0 ,r ) Z B (0 ,r ) dω ( u ) dω ( w ) p r − | u | p r − | w | = Z B (0 ,r ) dω ( u ) p r − | u | ! , which is finite since r ∈ G . We have used the fact that ( r − | u || w | ) ≥ ( r − | u | )( r − | w | ). Ina similar way, I = Z C \ B (0 ,r ) Z C \ B (0 ,r ) dω ( u ) dω ( w ) | u || w | − r ≤ Z C \ B (0 ,r ) Z C \ B (0 ,r ) dω ( u ) dω ( w ) p | u | − r p | w | − r = Z C \ B (0 ,r ) dω ( u ) p | u | − r ! < ∞ . (cid:3) For future issues let us at this point extend the moment formula to vorticities . It will be usefulwhen studying limits of Kaden’s spiral when time approaches 0 or ∞ . It seems to us that it can beapplicable in many other situations concerning convergence or simply computation of the differenceof velocities based on the difference of their vorticities.Consider ˜ ω = ω − ω , where ω i satisfy (1.9). Notice that all the moments (1.6)–(1.8) for either ω or ω are finite, thus we can define same moments for ˜ ω as a difference of the moments for ω and ω . Completely the same argument as in Lemma 3.2 leads us to the claim that the set G sign = G ∩ G ⊂ (0 , ∞ ), G i = ( r > Z C dω i ( u ) p | r − | u | | < ∞ ) , i = 1 , , is of full measure.Next, proceeding in the same way as in the proof of Corollary 3.5, we arrive at Corollary 3.7 Let ˜ ω = ω − ω be such that (1.9) is satisfied. Assume moreover that r ∈ G sign .Then for k, l ∈ { , } we have Z C Z C Z π dθ | u − re iθ | | w − re − iθ | dω k ( u ) dω l ( w ) < ∞ . Corollary 3.7 allows us to use Fubini’s theorem in the present context and this way extend Theorem1.2 to its version for vorticities . First, notice that due to linearity in ω of the Biot-Savart operatoras well as m r,n and M r,k , one immediately sees how to understand quantities occurring in (1.10).Moreover, by linearity, the proof of Theorem 1.2 goes in the same way also for ˜ ω . The followingholds. Corollary 3.8 Let ˜ ω = ω − ω , where nonnegative σ -finite measures ω and ω satisfy (1.9) .Then ω satisfies (1.10) . Applications - Kaden spirals In the present section our aim is to introduce and study some properties of Kaden’s spirals. Itturns out that the framework of moment formula we introduced in Theorem 1.2 is very helpfulin this respect. In order to define the Kaden spiral we need to first consider the Birkhoff-Rottequation. It was introduced in [2, 15] independently and it describes the time evolution of thecurve c -an interface between the zero vorticity regions of the vortex sheet whose velocity fieldattains the tangential velocity discontinuity along the curve c . Let us denote the position of thecurve c at time t and cumulative vorticity Γ by Z (Γ , t ). Then the following equation is satisfied(we refer an interested reader to the handbook [12] for details). ∂∂t Z (Γ , t ) = 12 πi p.v. Z R d Γ ′ Z (Γ , t ) − Z (Γ ′ , t ) , where Z : R × (0 , ∞ ) → C . For µ > Z (Γ , t ) = t µ z ( γ ) , γ = t − µ Γ . (4.1)Such solution in the new variables γ ∈ R and t > − µ ) γ z ′ ( γ ) + µz ( γ ) = i π p.v. Z R dγ ′ z ( γ ) − z ( γ ′ ) . (4.2)The construction of the Kaden spirals, introduced in [9] and reviewed recently in [7], is basedon the following ansatz. Arcs of spirals appearing in nature, when packed densely, are similar toarcs of a circle. Thus each arc (each 2 π turn around the origin) can be approximated by a circle.Now assume that the whole vorticity is concentrated on a circle, then if P is the point inside thecircle, then the velocity in P generated by the vorticity is zero, while for any point lying outsidethe circle the velocity is the same as the velocity generated by the point vortex placed at the centreof the circle with the mass/strength/vorticity equal to the total vorticity of the circle. Using theabove approach we assume that the part of the spiral further away to the origin than consideredpoint z ( γ ) does not influence the velocity in the point, while the part closer to the origin than z ( γ )extorts the velocity equal to iγ π · z ( γ ) . Inserting conclusion of the above heuristics into (4.2) we get(1 − µ ) γz ′ ( γ ) + µz ( γ ) = iγ πz ( γ ) . Switching to polar coordinates, z ( γ ) = r ( γ ) e iθ ( γ ) , we get(1 − µ ) γ (cid:0) r ′ ( γ ) e iθ ( γ ) + iθ ′ ( γ ) r ( γ ) e iθ ( γ ) (cid:1) + µr ( γ ) e iθ ( γ ) = iγ πr ( γ ) e − iθ ( γ ) . By dividing by e iθ ( γ ) and separating the real and imaginary part we obtain a system of ordinarydifferential equations for r and θ , namely (cid:26) (1 − µ ) γr ′ ( γ ) + µr ( γ ) = 0 , (1 − µ ) γr ( γ ) θ ′ ( γ ) = γ πr ( γ ) . (4.3)14or µ = 1 / ( r ( γ ) = C | γ | µ µ − ,θ ( γ ) = πC | γ | − µ + C . Further on we take C = 1 , C = 0 and consider only γ > θ ( r ) = 12 π r − /µ , r > . We are interested in an evolution in time of the vorticity and the velocity field. Thus we goback to the original variable Γ (see (4.1)) getting Z (Γ , t ) = t µ z ( t − µ Γ) = R (Γ , t ) e i Θ(Γ ,t ) , with ( R (Γ , t ) = Γ µ µ − , Θ(Γ , t ) = t π Γ − µ . (4.4)We will denote the spiral curve that is the support of the vorticity for a given time t > 0, by c t . In polar coordinates c t is given by the equationΘ( R ) = t π R − µ . (4.5)The measure corresponding to the vorticity for the Kaden spiral c t at time t > ω t . The support of ω t is c t , moreover since Γ is a cumulative vorticity in a ball of radius | Z | ω t ( B (0 , | Z (Γ , t ) | )) = Γ , or, equivalently, ω t ( B (0 , r )) = r − µ . (4.6)Notice that the spiral concentrates at the origin and for µ ∈ (1 / , 1) the length of c t ∩ B (0 , r )is infinite. Indeed, the following fact holds. Proposition 4.1 For each t > and µ ∈ (1 / , the Kaden spiral c t restricted to a ball B (0 , r ) , r > , has infinite length. Proof. Let us parametrize the Kaden spiral (4.5), by (0 , ∞ ) ∋ s s exp (cid:0) it π s − /µ (cid:1) . Thenthe length of c t on B (0 , r ) is given by l ( c t ) = Z r s s t (2 πµ ) s − − /µ ds ≥ t πµ Z r s − /µ ds = ∞ for µ ∈ (1 / , (cid:3) emma 4.2 Let c t be the Kaden spiral with µ ∈ (1 / , . Then for f ∈ L ( R , ω t ) we have Z R f ( x ) dω t ( x ) = (cid:18) − µ (cid:19) Z ∞ f (cid:18) s exp (cid:18) it π s − /µ (cid:19)(cid:19) s − /µ ds. Proof. We shall use the same parametrization of c t as in the proof of Proposition 4.1. Aswe know that ω t is supported on c t and (4.6) holds, we can find a non-negative density g t ( s ), suchthat (see [16]) Z R f ( x ) dω t ( x ) = Z ∞ f (cid:18) s exp (cid:18) it π s − /µ (cid:19)(cid:19) g t ( s ) ds. By taking f = χ B (0 ,r ) and using (4.6) again, we get r − /µ = Z r g t ( s ) ds. Differentiating both sides we obtain g t ( s ) = (2 − /µ ) s − /µ . (cid:3) We end this section with a simple observation concerning kinetic energy of the consideredvelocity field. Proposition 4.3 Consider the velocity field generated by the Kaden spiral c t via (2.2) . Then wehave the following scaling of the kinetic energy in a ball B (0 , r ) , E r ( ω t ) = t µ − E t − µ r ( ω ) . Proof. Using Lemma 4.2 and changing variables twice ( R Rt µ and x xt µ ),4 π E r ( ω t ) = (cid:18) − µ (cid:19) Z B (0 ,r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:0) x − R exp (cid:0) it π R − /µ (cid:1)(cid:1) ⊥ (cid:12)(cid:12) x − R exp (cid:0) it π R − /µ (cid:1)(cid:12)(cid:12) R − µ dR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = (cid:18) − µ (cid:19) t µ − Z B (0 ,r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:0) x − Rt µ exp (cid:0) i π R − /µ (cid:1)(cid:1) ⊥ (cid:12)(cid:12) x − Rt µ exp (cid:0) i π R − /µ (cid:1)(cid:12)(cid:12) R − µ dR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = (cid:18) − µ (cid:19) t µ − Z B (0 ,t − µ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:0) x − R exp (cid:0) i π R − /µ (cid:1)(cid:1) ⊥ (cid:12)(cid:12) x − R exp (cid:0) i π R − /µ (cid:1)(cid:12)(cid:12) R − µ dR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = 4 π t µ − E t − µ r ( ω ) . (cid:3) The present subsection is devoted to study time evolution of the velocity field carried by Kaden’sspiral, in particular we can estimate the evolution of the kinetic energy E r ( ω t ) of Kaden’s spiral,when r > t goes either to zero or to infinity. The constants α and µ are alwaysrelated by α = 2 − /µ, (4.7)16ee (1.1), (4.6) for the definitions of α and µ . We assume that µ ∈ (1 / , ω and ω ∞ , themeasures defined in Subsection 3.1.The results presented below show the applicability of our main Theorem 1.2 in the examinationof Kaden’s spiral. On the one hand we show that the velocity field related to Kaden’s spiral, bythe results of Theorem 2.1 of Section 2 such exists and is an element of L loc , dissipates the energyin any ball surrounding the origin of a spiral. Indeed, we prove below that when time approacheszero, kinetic energy contained in a ball centered in an origin of Kaden’s spiral tends to the maximalpossible value of local kinetic energy carried by the vorticity satisfying (1.1) with α ∈ (0 , ∞ , then the kinetic energy in a ball centered in an origin approaches the minimalpossible value. This means that in the meantime the energy is pushed out from any ball surroundingthe origin of the Kaden approximation, the latter indicates a sort of viscosity in the center of thespiral.On the other hand, we show that velocity field associated with Kaden’s spiral converges in L loc to (3.5) with c = 1 when time tends to 0. This convergence is interesting in view of the problemof uniqueness of Delort’s solutions of 2d Euler equation constructed in [4]. Such solutions havevorticities being compactly supported nonnegative Radon measures. Whether they are unique isstill an open question. If the requirement that vorticities are measures is relaxed, it is known thatthere are infinitely many vortex sheet solutions satisfying the 2d Euler equations, see [20]. However,velocity fields constructed in [20] are extremely oscillating, so that their vorticities are not evenmeasures. Numerical simulations suggest that spirals of vorticity could be the counterexamplesto the uniqueness problem in the Delort’s class of solutions with vorticities being measures. It isobserved in the computations that such spirals detach from the steady solution of the form similarto that given by (3.5), see for instance [11, 14]. Hence our result concerning the convergence ofKaden’s spiral with time approaching 0 is of interest in this respect, in particular since Kaden’sspiral is a nonnegative measure.Finally, let us notice that the evolution of the Kaden spiral is a path in the class of nonnegative σ -finite measures linking the object with maximal value of the energy functional with the one withminimal value (see Subsection 3.1). Proposition 4.4 Assume that r > is fixed. Let ω t be the vorticity of the Kaden spiral defined in (4.4) with µ ∈ (1 / , and ω ∞ be given by (3.4) with c = 1 , α related to µ via (4.7) , in particular α ∈ (0 , . Next, consider u ( t ) and u ∞ as divergence-free velocity fields related to ω t and ω ∞ ,respectively, via the Biot-Savart law. Then lim t →∞ Z B (0 ,r ) | u ( t ) − u ∞ | dx = 0 . (4.8) In particular, lim t →∞ E r ( ω t ) = E r ( ω ∞ ) . (4.9) Proof. First, we notice that m r, ( ω t ) = m r, ( ω ∞ ). Indeed, ω ∞ satisfies (1.1) with c = 1, α = 2 − /µ and ω t satisfies (4.6). Next, as we have already observed, all the other moments for ω ∞ vanish.17ence, using Corollary 3.8, we arrive at4 π | A r ( ω t − ω ∞ ) | = ∞ X n =0 r − n − | m r,n ( ω t − ω ∞ ) | + ∞ X k =1 r k − | M r,k ( ω t − ω ∞ ) | = ∞ X n =0 r − n − | m r,n ( ω t ) − m r,n ( ω ∞ ) | + ∞ X k =1 r k − | M r,k ( ω t ) − M r,k ( ω ∞ ) | = ∞ X n =1 r − n − | m r,n ( ω t ) | + ∞ X k =1 r k − | M r,k ( ω t ) | . (4.10)Using Lemma 4.2 and integration by parts, for n ≥ m r,n ( ω t ) = α Z r s n +1 − /µ exp (cid:18) itn π s − /µ (cid:19) ds = − πµαitn Z r s n +2 (cid:18) exp (cid:18) itn π s − /µ (cid:19)(cid:19) ′ ds = − πµαitn (cid:20) r n +2 exp (cid:18) itn π r − /µ (cid:19) − ( n + 2) Z r s n +1 exp (cid:18) itn π s − /µ (cid:19) ds (cid:21) and, bounding the absolute value of both exponential terms by 1, | m r,n ( ω t ) | ≤ C r n +2 tn . (4.11)Now we proceed to estimate the outer moments. We shall consider M r, and M r, separately.By changing variables ts − /µ = a we get M r, ( ω t ) = α Z ∞ r s − /µ exp (cid:18) − it π s − /µ (cid:19) ds = αµt µ − Z t r − /µ a − µ exp (cid:18) − ia π (cid:19) da. Since the integral R ∞ a − µ exp( − ia π ) da is convergent, we deduce that | M r, ( ω t ) | ≤ Ct µ − . (4.12)Next M r, ( ω t ) = α Z ∞ r s − − /µ exp (cid:18) − itπ s − /µ (cid:19) ds = απµit Z ∞ r (cid:18) exp (cid:18) − itπ s − /µ (cid:19)(cid:19) ′ ds = απµit (cid:18) − exp (cid:18) − itπ r − /µ (cid:19)(cid:19) and | M r, ( ω t ) | ≤ Ct − . (4.13)18or n ≥ M r,n ( ω t ) = α Z ∞ r s − n +1 − /µ exp (cid:18) − itn π s − /µ (cid:19) ds = 2 πµαitn Z ∞ r s − n +2 (cid:18) exp (cid:18) − itn π s − /µ (cid:19)(cid:19) ′ ds = 2 πµαitn (cid:20) − r − n +2 exp (cid:18) − itn π r − /µ (cid:19) + ( n − Z ∞ r s − n +1 exp (cid:18) − itn π s − /µ (cid:19) ds (cid:21) . Estimating similarly as in (4.11), | M r,n ( ω t ) | ≤ C r − n tn . (4.14)Summarizing, plugging the estimates (4.11)–(4.14) into (4.10), for t > | A r ( ω t − ω ∞ ) | ≤ Ct µ − (1 + r ) , (4.15)which, together with (1.5), implies (4.8). (cid:3) Proposition 4.5 Assume that r > is fixed. Let ω t be the vorticity of the Kaden spiral defined in (4.4) with µ ∈ (1 / , and ω be given by (3.5) with c = 1 , α related to µ via (4.7) . Next, let u ( t ) and u be divergence-free velocity fields associated with ω t and ω , respectively, by the Biot-Savartoperator (2.2) . Then lim t → + Z B (0 ,r ) | u ( t ) − u | dx = 0 . (4.16) In particular, lim t → + E r ( ω t ) = E r ( ω ) . (4.17) Proof. Fix ε > 0. By (1.5) and Corollary 3.8 (a signed measures version of Theorem 1.2)it suffices to show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =0 A n + ∞ X k =1 B k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =0 r − n − | m r,n ( ω t − ω ) | + ∞ X k =1 r k − | M r,k ( ω t − ω ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε (4.18)for t > A = m r, ( ω t − ω ) = 0.Since | m r,n ( ω t − ω ) | = | m r,n ( ω t ) − m r,n ( ω ) | ≤ | m r,n ( ω t ) | + | m r,n ( ω ) | as well as | M r,k ( ω t − ω ) | = | M r,k ( ω t ) − M r,k ( ω ) | ≤ | M r,k ( ω t ) | + | M r,k ( ω ) | , and due to the fact that both ω t and ω satisfy assumptions of Theorem 3.1, in view of (3.2) and(3.3), we find N ∈ N such that ∞ X n = N | A n | + ∞ X k = N | B k | < ε/ . (4.19)19ssume that k = 1 , ..., N − 1. Using Lemma 4.2 and the estimate (cid:12)(cid:12) e iτ − (cid:12)(cid:12) ≤ C | τ | for τ ∈ R weget | M r,k ( ω t − ω ) | ≤ α Z ∞ r s − k +1 − /µ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − itk π s − /µ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ Ctk Z ∞ r s − k +1 − /µ ds ≤ Ctr − k +2 − /µ . Consequently, N − X k =1 | B k | ≤ Ct N − X k =1 r k − r − k +4 − /µ ≤ Ct N r − /µ < ε/ , (4.20)where the last inequality holds for t > A n for n = 2 , ..., N − n = 1 is a bit more delicate andwill be treated separately). By using Lemma 4.2 one more time, | m r,n ( ω t − ω ) | ≤ α R r s n +1 − /µ (cid:12)(cid:12) exp (cid:0) itn π s − /µ (cid:1) − (cid:12)(cid:12) ds ≤ Ctn R r s n +1 − /µ ds ≤ Ctr n +2 − /µ . (4.21)We have used n + 2 − /µ > 0. The latter is a consequence of n ≥ µ ∈ (1 / , n = 1 and µ ∈ (2 / , − /µ > − | m r, ( ω t − ω ) | ≤ α Z r s − /µ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) it π s − /µ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ Ct Z r s − /µ ds ≤ Ctr − /µ . Consequently, see also (4.21), for µ ∈ (2 / , 1) and t > N − X n =1 | A n | ≤ Ct r − /µ < ε/ , which together with (4.19) and (4.20) gives the claim of Proposition 4.5 in the range of parameters µ ∈ (2 / , µ ∈ (1 / , / 3] we still have to deal with m r, . Again we split ourconsiderations into two cases, µ < / µ = 2 / 3. In the first one we have | m r, ( ω t − ω ) | ≤ α Z r s − /µ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) it π s − /µ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ds = α Z t µ ... + Z rt µ ... ! ≤ C Z t µ s − /µ ds + Ct Z rt µ s − /µ ds ≤ Ct µ − + Ct h s − /µ i s = t µ s = ∞ ≤ Ct µ − . Therefore, in the case µ ∈ (1 / , / N − X n =1 | A n | ≤ Ct µ − r − + Ct N − X n =2 r − n − r n +4 − /µ ≤ Ct µ − N max( r − , r − /µ ) < ε/ , (4.22)20f t > µ ∈ (1 / , / µ = 2 / | m r, ( ω t − ω ) | ≤ Ct · / − + Ct Z rt / s − ≤ Ct + Ct (ln r − / t ) . Hence for small enough t > N − X n =1 | A n | ≤ C ( t − t ln t ) r − + Ct r − ln r + Ct r − < ε/ . (4.23)Estimates (4.22) and (4.23) finish the proof of (4.18) for µ ∈ (1 / , / (cid:3) At the end let us remark that Kaden’s spiral is continuous in a certain sense. The proof is verysimilar to the proof of Proposition 4.5, so we only provide a sketch. Proposition 4.6 Assume that r > is fixed. Moreover take < t, t < ∞ . Let ω t , ω t bevorticities of Kaden spirals at times t, t defined in (4.4) with µ ∈ (1 / , . Next, let u ( t ) and u ( t ) be the divergence-free velocity fields associated to ω t and ω t , respectively, by the Biot-Savartoperator (2.2) . Then lim t → t Z B (0 ,r ) | u ( t ) − u ( t ) | dx = 0 . (4.24) Proof. The beginning of the proof goes the same way as in Proposition 4.5. Let E n and D k be defined as E n := r − n − | m r,n ( ω t − ω t ) | , D k := r k − | M r,k ( ω t − ω t ) | , then, due to the fact that ω t and ω t (in view of (4.6)) satisfy assumptions of Theorem 3.1, estimates(3.2),(3.3) guarantee that for any ε > N > ∞ X n = N | E n | + ∞ X k = N | D k | < ε/ . (4.25)To show (4.24) and thus finish the proof of Proposition 4.6 we only need to estimate E n for n = 0 , , ..., N − D k for k = 1 , ..., N − 1. On the one hand, in view of (4.6), E = 0. Next,utilizing Lemma 4.2, we notice that | m r,n ( ω t − ω t ) | ≤ α Z r s n +1 − /µ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) i ( t − t ) n π s − /µ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ds, | M r,k ( ω t − ω t ) | ≤ α Z ∞ r s − k +1 − /µ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − i ( t − t ) k π s − /µ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ds, so that for t → t the required estimates of the initial E n , D k may be achieved exactly the sameway as in the proof of Proposition 4.5. (cid:3) At the end let us provide a simple corollary.21 orollary 4.7 Let ω t be the Kaden spiral. For any fixed r > , the function t → E r ( ω t ) iscontinuous. Acknowledgements. T. Cie´slak was partially supported by the National Science Centre (NCN), Poland, under grant2013/09/D/ST1/03687. K. Oleszkiewicz was partially supported by the National Science Centre,Poland, project number 2015/18/A/ST1/00553. M. Preisner had a post-doc at WCMCS in War-saw, where he met Kaden’s spirals and started working in a project which led to the present article.He wishes to express his gratitude for support and hospitality. M. Preisner was partially supportedby National Science Centre (NCN), Poland, Grant No. 2017/25/B/ST1/00599. M. Szuma´nska waspartially supported by National Science Centre (NCN), Poland, Grant No. 2013/10/M/ST1/00416 Geometric curvature energies for subsets of the Euclidean space. 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