# Kinetic Energy Represented in Terms of Moments of Vorticity and Applications

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## 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 formula associating the kinetic energy of a very general class of flows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only \(\sigma \)-finite measures (in particular including measures \(\omega \) such that \(\omega (\mathbb {R}^2)=\infty \)), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.

## Keywords

vortex sheet spherical averages Biot–Savart law Kaden spirals## Mathematics Subject Classification

76M40 76B47 28A25## 1 Introduction

*c*and \(\alpha \) are positive constants, yields that local \(H^{-1}\)-norm of \(\omega \) is finite. By the lemma of Schochet [18] it means that the kinetic energy generated by a compactly supported part of such a vortex sheet is locally finite. Property (1.1) is satisfied at least by well-known examples of Kaden and Prandtl, see [3].

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 regular function, velocity is given by the usual Biot–Savart operator. One of the tasks of the present paper is to identify the velocity given by a vorticity being \(\sigma \)-finite measure satisfying (1.1). This is discussed in Sect. 2. We shall say more on it also at the end of Introduction.

*v*, which is defined by

Our main theorem states that the local kinetic energy \(E_r(\omega )\) of a nonnegative \(\sigma \)-finite measure of vorticity \(\omega \) satisfying (1.1) undergoes a precise estimate from below and above.

### Theorem 1.1

*c*being a constant appearing in (1.1), we have

### Theorem 1.2

The above formula is a completely new way of viewing the relation between velocity and vorticity. Hopefully, besides being a useful tool in a current paper, formula (1.10) is of independent interest and may find other applications.

### Remark 1.3

Let us emphasize that Theorems 1.1 and 1.2 state results holding for measures which are only \(\sigma \)-finite, in particular it applies also to measures \(\omega \) such that \(\omega (\mathbb {R}^2)=\infty \) (provided they satisfy the assumptions).

Moreover, we study Kaden’s approximations, see [9], and examine their properties. In particular time 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 energy contained in any ball surrounding the origin of the Kaden approximation tends to the minimal possible value given by the left-hand side of (1.3), while for small times Kaden’s spiral’s kinetic energy approaches the upper bound in (1.3). Actually, we even compute the limiting objects reached by the divergence-free velocities related to the Kaden spiral when *t* approaches 0 as well as when *t* tends to infinity. The results concerning time evolution are obtained using our moment formula (1.10) applied to the difference of two unbounded measures. Such a difference does not have to be a signed measure. Thus, extension of (1.10) requires a precise definition of some new objects.

First of all, let us notice that when we consider a difference of two nonnegative measures \(\omega _1\) and \(\omega _2\), such that \(\omega _1(\mathbb {R}^2) = \omega _2(\mathbb {R}^2)=\infty \), then \(\tilde{\omega }:=\omega _1-\omega _2\) cannot be defined as a signed measure. Indeed, one has a problem to decide what is the value \(\tilde{\omega }(A)\), where *A* is such that \(\omega _i(A)=\infty \) for \(i=1,2\). Such technical difficulties are the main reason for which we work with the following objects, which we shall call *vorticities*. By definition, a *vorticity* is a distribution \(\tilde{\omega }\) which can be represented as a difference of \(\omega _1\) and \(\omega _2\), such that for \(i=1,2\) nonnegative \(\sigma \)-finite measures \(\omega _i\) satisfy (1.9).

In Sect. 3 we extend (1.3) to vorticities \(\tilde{\omega }=\omega _1-\omega _2\), for which \(\omega _i\) satisfy (1.9). Complex moments \(m_{r,n}\) and \(M_{r,n}\), appearing in (1.3), as linear in \(\omega \) are extended to \(\tilde{\omega }\) in a natural way.

We also show an extension of the Biot–Savart law to vorticities from a wide subclass of \(\sigma \)-finite measures, in particular our results are applicable also to measures \(\omega \) such that \(\omega (\mathbb {R}^2)=\infty \). Actually, again our theorem works for quite a wide class of objects. Moreover, for \(\sigma \)-finite measures satisfying (1.1) with \(\alpha \in (0,1)\) it is shown that velocities obtained via the Biot–Savart law are in \(L^2_{loc}\), so that the kinetic energy is finite.

Let us explain ourselves from the slightly non-orthodox structure of the paper. Namely, the technical core with the proof of Theorem 1.2 is a content of Sect. 3.2. It is self-contained. Moreover, some of main results which we prove with the help of Theorem 1.2 are presented in earlier sections. We use there results proven later in Sect. 3.2. This way technical computations are postponed.

At the end of Introduction let us state an easy fact concerning the assumptions we provided for measures \(\omega \). We show that measures on \({\mathbb {C}}\) satistying (1.1) with \(\alpha \in (0,1)\), even those, for which only the upper bound in (1.1) holds, satisfy also the assumption (1.9). Indeed, we have the following proposition.

### Proposition 1.4

Let us fix \(\alpha \in (0,1)\). Let a nonnegative \(\sigma \)-finite measure \(\omega \) satisfy formula \(\mu (B(0,r)) \le cr^\alpha \) for some \(c>0\). Then (1.9) is also satisfied.

### Proof

In view of Proposition 1.4, the class of measures satisfying (1.9) is much wider than those fulfiling (1.1). For instance, one can check that measures considered in [8] are also fine.

**Notation.** We need to fix a convention which we use to speak about 2d vorticity. By definition, \((x_1,x_2)^\perp = (-x_2, x_1)\). The partial derivatives in \(\mathbb {R}^2\) are denoted by \(\partial _1\) and \(\partial _2\), while a vorticity of a vector field \(v=(v_1,v_2)\) is defined as \(\mathrm {curl} (v)=\partial _1v_2-\partial _2v_1\). When it is convenient, we shall use complex notation for \(\mathbb {R}^2\).

## 2 Vorticity and Velocity: Biot–Savart Law

*v*is recovered from vorticity \(\omega \) by the Biot–Savart formula, i.e.,

*v*is given by (2.2). However it is not known whether the Biot–Savart law is still valid in the case of a vorticity which is not a bounded measure. Yet, a very important class of vortex sheets are the so-called self-similar spirals of vorticity (one of them, the Kaden spiral is studied later on in the present paper). For instance measures satisfying (1.1) are immediately unbounded, indeed

*v*related to vortex sheet \(\omega \), i.e. \({\mathrm {curl}}\, (v) =\omega \), provided \(\omega \) satisfies (1.9). Consequently, in view of Proposition 1.4, the Biot–Savart law holds in particular for \(\sigma \)-finite measures satisfying (1.1) with \(\alpha \in (0,1)\). We show that then the integral in (2.2) for

*v*(

*x*) is well-defined a.e., divergence-free in the sense of distributions as well as \(\mathrm {curl}\, (v) =\omega \) holds in the sense of distributions.

Notice that our result is not trivial since the procedure described at the beginning of the present section to derive the Biot–Savart law seems to require strong regularity assumptions. Indeed, solving (2.1) with \(\omega \) being only \(\sigma \)-finite measure, in particular possibly \(\omega (\mathbb {R}^2)=\infty \), seems not trivial. Notice that due to the contribution from infinity of both \(\omega \) and a fundamental solution of Laplace operator, \(\psi \) might not exist, and so one cannot tell that \(v=-^\perp \nabla \psi \). Our result holds in a more general situation, when vorticity is a nonnegative measure which might not possess a stream function. A reader might check that actually stream function does not exist for Kaden’s spirals (that are introduced in Sect. 4). Nevertheless, we show that the velocity field can still be expressed by the Biot–Savart formula (2.2) if a measure \(\omega \) satisfies (1.1) with \(\alpha \in (0,1)\). This way we validate the Biot–Savart law even for vorticities being \(\sigma \)-finite measures satisfying a condition stated already in [3] and being a consequence of Prandtl’s similitude laws (see [3] and the references therein).

As to the proof of Theorem 2.1, the only non-standard part is a justification of the use of Fubini’s theorem in (2.6). This requires a sort of potential theory type estimate. The required result 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 \(\alpha \in (0,1)\). Corollary 3.6 is proven later, its proof is independent on the results of the current section. We hope it does not confuse the reader.

### Theorem 2.1

*v*(

*x*) is given by

### Proof

*v*given by (2.3) satisfies \(v\in L^1_{loc}\), in particular,

*v*is finite a.e. On the other hand, the same Corollary 3.6, again applied to the measure \(\omega (x)\), allows us to use Fubini’s theorem in the integral

Let us conclude this section with a remark concerning higher integrability of velocity *v* given by (2.2) if \(\omega \) satisfies (1.1) with \(\alpha \in (0,1)\). First, observe that in view of Proposition 1.4, (1.1) with \(\alpha \in (0,1)\) implies (1.9). Next, a consequence of Theorem 1.1, which we prove in Sect. 3, is a higher integrability of *v* associated with \(\omega \) via (2.2).

## 3 Kinetic Energy

This section is devoted to the proof of the main result. We prove Theorem 1.2, which yields a formula representing radial averages of the square of the divergence-free velocity associated with the vorticity being nonnegative \(\sigma \)-finite measure satisfying (1.9). Moreover, we show how to infer Theorem 1.1 from our moment representation formula. As a consequence we obtain a precise estimate (from below and above) of the kinetic energy contained in a ball *B*(0, *r*) carried by a vorticity satisfying (1.1).

Finally, we also state a variational problem related to the local kinetic energy estimates and find its lower and upper bounds. Moreover, we identify the measures at which the maximal and minimal values are taken.

The next theorem provides estimates required to obtain Theorem 1.1 as a consequence of Theorem 1.2. It also gives very precise constants in the bounds of local kinetic energy which will be used further to identify minimizers as well as maximizers of a variational problem leading to the local kinetic energy estimates.

### Theorem 3.1

### Proof

We finish this section with the proof of Theorem 1.1 provided that Theorem 1.2 holds. The latter is proven in Sect. 3.2.

### Proof of Theorem 1.1

We recall (1.5) and see that the kinetic energy \(E_r(\omega )\) is given as \(2\pi \int _0^r sA_s(\omega )ds\). Hence, integrating the bounds in (3.1) in *r*, we obtain the claim of Theorem 1.1. \(\square \)

### 3.1 Variational Formulation and its Minimizer and Maximizer

Before proceeding with the argument let us notice that in the case of slightly more regular velocities such a functional was used in literature to construct steady states of the incompressible 2d Euler system, see [1] in the case of regular solutions and [21] in the case of vortex patches. It is related to the hamiltonian structure of the Euler system. It is not clear to us whether the same approach could work in the case of steady vortex sheets (like those introduced in [5]).

### 3.2 Proof of Theorem 1.2

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 \(z = x+iy \in {\mathbb {C}}= \mathbb {R}^2\). The goal of the section is to prove Theorem 1.2. Assume that a nonnegative measure \(\omega \) on \({\mathbb {C}}\) is given such that (1.9) is satisfied.

*v*is finite a.e. Moreover, we have seen in Sect. 2 that Corollary 3.6 is an important factor of the proof of the Biot–Savart formula for vortex sheets satisfying (1.9).

### Lemma 3.2

Let \(\omega \) be a nonnegative \(\sigma \)-finite measure satisfying (1.9). Then the Lebesgue measure of \(G^{c} = (0,\infty ) {\setminus } G\) is zero.

### Proof

The next lemma is crucial in our investigation. It uncovers the essential cancellations separating the outer and inner contribution to the spherical averages of kinetic energy.

### Lemma 3.3

### Proof

Let us notice that Lemma 3.3 can also be proved by the residue theorem. In a special case \(v=\overline{ u }\) we get the following.

### Corollary 3.4

Moreover, using the Cauchy-Schwarz inequality, Corollary 3.4, and the definition of the set *G*, we arrive at the following.

### Corollary 3.5

Corollary 3.5 is essential in the proof of Theorem 1.2. It justifies an application of the Fubini theorem in a crucial moment.

Before proceeding with the proof of Theorem 1.2, let us state the next corollary, which on the one hand, guarantees that *v*, as defined in (3.6), is finite a.e., on the other hand gives a strong estimate which is used in Sect. 2 to state a general version of the Biot–Savart law.

### Corollary 3.6

*v*defined by (3.6), satisfies \(v\in L^1_{loc}\) and so \(v<\infty \) a.e..

### Proof

### Proof of Theorem 1.2

### 3.3 Extension to Vorticities

For future issues let us at this point extend the moment formula to *vorticities*. It will be useful when studying limits of Kaden’s spiral when time approaches 0 or \(\infty \). It seems to us that it can be applicable in many other situations concerning convergence or simply computation of the difference of velocities based on the difference of their vorticities.

Next, proceeding in the same way as in the proof of Corollary 3.5, we arrive at

### Corollary 3.7

Corollary 3.7 allows us to use Fubini’s theorem in the present context and this way extend Theorem 1.2 to its version for *vorticities*. First, notice that due to linearity in \(\omega \) of the Biot–Savart operator as 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 \(\tilde{\omega }\). The following holds.

## 4 Applications: Kaden Spirals

*t*and cumulative vorticity \(\Gamma \) by \(Z(\Gamma , t)\). Then the following equation is satisfied (we refer an interested reader to the handbook [12] for details).

*P*is the point inside the circle, then the velocity in

*P*generated by the vorticity is zero, while for any point lying outside the circle the velocity is the same as the velocity generated by the point vortex placed at the centre of the circle with the mass/strength/vorticity equal to the total vorticity of the circle. Using the above approach we assume that the part of the spiral further away to the origin than considered point \(z(\gamma )\) does not influence the velocity in the point, while the part closer to the origin than \(z(\gamma ) \) extorts the velocity equal to \( \frac{i\gamma }{2 \pi \cdot \overline{z(\gamma )}}. \)

*r*and \(\theta \), namely

*Z*|

### Proposition 4.1

For each \(t>0\) and \(\mu \in (1/2,1)\) the Kaden spiral \(\mathfrak {c}_t\) restricted to a ball *B*(0, *r*), \(r>0\), has infinite length.

### Proof

*B*(0,

*r*) is given by

### Lemma 4.2

### Proof

We end this section with a simple observation concerning kinetic energy of the considered velocity field.

### Proposition 4.3

*B*(0,

*r*),

### Proof

### 4.1 End Point Estimates of the Energy

*t*goes either to zero or to infinity. The constants \(\alpha \) and \(\mu \) are always related by

The results presented below show the applicability of our main Theorem 1.2 in the examination of Kaden’s spiral. On the one hand we show that the velocity field related to Kaden’s spiral, by the results of Theorem 2.1 of Sect. 2 such exists and is an element of \(L^2_{loc}\), dissipates the energy in any ball surrounding the origin of a spiral. Indeed, we prove below that when time approaches zero, kinetic energy contained in a ball centered in an origin of Kaden’s spiral tends to the maximal possible value of local kinetic energy carried by the vorticity satisfying (1.1) with \(\alpha \in (0,1)\). When time tends to \(\infty \), then the kinetic energy in a ball centered in an origin approaches the minimal possible value. This means that in the meantime the energy is pushed out from any ball surrounding the origin of the Kaden approximation, the latter indicates a sort of viscosity in the center of the spiral.

On the other hand, we show that velocity field associated with Kaden’s spiral converges in \(L^2_{loc}\) to (3.5) with \(c=1\) when time tends to 0. This convergence is interesting in view of the problem of uniqueness of Delort’s solutions of 2d Euler equation constructed in [4]. Such solutions have vorticities being compactly supported nonnegative Radon measures. Whether they are unique is still an open question. If the requirement that vorticities are measures is relaxed, it is known that there 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 even measures. Numerical simulations suggest that spirals of vorticity could be the counterexamples to the uniqueness problem in the Delort’s class of solutions with vorticities being measures. It is observed in the computations that such spirals detach from the steady solution of the form similar to that given by (3.5), see for instance [11, 14]. Hence our result concerning the convergence of Kaden’s spiral with time approaching 0 is of interest in this respect, in particular since Kaden’s spiral is a nonnegative measure.

Finally, let us notice that the evolution of the Kaden spiral is a path in the class of nonnegative \(\sigma \)-finite measures linking the object with maximal value of the energy functional with the one with minimal value (see Sect. 3.1).

### Proposition 4.4

*u*(

*t*) and \(u_\infty \) as divergence-free velocity fields related to \(\omega _t\) and \(\omega _\infty \), respectively, via the Biot–Savart law. Then

### Proof

### Proposition 4.5

*u*(

*t*) and \(u_0\) be divergence-free velocity fields associated with \(\omega _t\) and \(\omega _0\), respectively, by the Biot–Savart operator (2.2). Then

### Proof

At the end let us remark that Kaden’s spiral is continuous in a certain sense. The proof is very similar to the proof of Proposition 4.5, so we only provide a sketch.

### Proposition 4.6

*u*(

*t*) and \(u(t_0)\) be the divergence-free velocity fields associated to \(\omega _t\) and \(\omega _{t_0}\), respectively, by the Biot–Savart operator (2.2). Then

### Proof

At the end let us provide a simple corollary.

### Corollary 4.7

Let \(\omega _t\) be the Kaden spiral. For any fixed \(r>0\), the function \(t\rightarrow E_r(\omega _t)\) is continuous.

## Notes

### Acknowledgements

T. Cieślak was partially supported by the National Science Centre (NCN), Poland, under Grant 2013/09/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 Warsaw, 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 supported by National Science Centre (NCN), Poland, Grant No. 2017/25/B/ST1/00599. M. Szumańska was partially supported by National Science Centre (NCN), Poland, Grant No. 2013/10/M/ST1/00416 *Geometric curvature energies for subsets of the Euclidean space.* The second and fourth authors include double affiliations since a part ot the work have been conducted during their leave from University of Warsaw to IMPAN in the academic year 2016/17.

### Compliance with ethical standards

### Conflict of interest

The authors declare no conflict of interest.

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