Global regularity and convergence of a Birkhoff-Rott-\alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

# Global regularity and convergence of a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations

Claude Bardos, Jasmine S. Linshiz and Edriss S. Titi
###### Abstract

We present an -regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler- equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler- equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR- equation, remains Lipschitz for all times, (ii) an initially Hölder , , chord arc curve remains in for all times, and finally, (iii) an initially Hölder , closed chord arc curve remains so for all times. In all these cases the weak Euler- and the BR- descriptions of the vortex sheet motion are equivalent.

Université Denis Diderot and Laboratory J. L. Lions

[0pt] Université Pierre et Marie Curie, Paris, France

[0pt] bardos@ann.jussieu.fr

[0pt] Department of Computer Science and Applied Mathematics

[0pt] Weizmann Institute of Science

[0pt] Rehovot 76100, Israel

[0pt] jasmine.tal@weizmann.ac.il

[0pt] Department of Mathematics and

[0pt] Department of Mechanical and Aerospace Engineering

[0pt] University of California

[0pt] Irvine, CA 92697-3875, USA

[0pt] etiti@math.uci.edu and edriss.titi@weizmann.ac.il

Keywords: inviscid regularization of Euler equations; Euler-; Birkhoff-Rott; Birkhoff-Rott-; vortex sheet.

Mathematics Subject Classification: 76B03, 35Q35, 76B47.

## 1 Introduction

The -regularization of the Navier-Stokes equations (NSE) is one of the novel approaches for subgrid scale modeling of turbulence. The inviscid Euler- model was originally introduced in the Euler-Poincaré variational framework in [39, 38]. In [13, 14, 15, 31, 32] the corresponding Navier-Stokes- (NS-) [also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes- (LANS-)] model, was obtained by introducing the appropriate viscous term into the Euler- equations. The extensive research of the -models (see, e.g., [34, 16, 42, 18, 43, 77, 17, 32, 31, 14, 15, 13, 63, 49, 48, 7, 51, 34, 35, 36, 11, 2, 40, 10, 20, 52]) stems, on the one hand, from the successful comparison of their steady state solutions to empirical data, for a large range of huge Reynolds numbers, for turbulent flows in infinite channels and pipes [14, 15, 13]. On the other hand, the -models can also be viewed as numerical regularizations of the original, Euler or Navier-Stokes, systems [52, 11, 7, 44]. The main practical question arising is that of the applicability of these regularizations to the correct predictions of the underlying flow phenomena.

In this paper we present some analytical results concerning the -regularization of the two-dimensional (2D) Euler equations in the context of vortex sheet dynamics. The incompressible Euler equations are

 ∂v∂t+(v⋅∇)v+∇p=0,∇⋅v=0,v(x,0)=vin(x), (1.1)

where the fluid velocity field and , the pressure are the unknowns, and is the given initial velocity. A vortex sheet is a surface of codimension one (a curve in the plane) in inviscid incompressible flow, across which the tangential component of the velocity has a jump discontinuity, while the normal component is continuous. The flow outside the sheet is irrotational. The evolution of the vortex sheet can be described by the Birkhoff-Rott (BR) equation [8, 67, 68]. This is a nonlinear singular integro-differential equation, which can be obtained formally from the Euler equations assuming that the evolution of a vortex sheet retains a curve-like structure:

 ∂¯z∂t(Γ,t)=12πi  p.v.∫∞−∞dΓ′z(Γ,t)−z(Γ′,t),

here is the complex position of the sheet and represents the circulation, that is, is the vorticity density along the sheet. However, the initial data problem for the BR equation is ill-posed due to the Kelvin-Helmholtz instability [8, 69]. Numerous results show that an initially real analytic vortex sheet (curve) can develop a finite time singularity in its curvature. This singularity formation was studied with asymptotic techniques in [64, 23] and numerically in [62, 46, 23]. Specific examples of solutions were constructed in [29, 9], where the development, in a finite time, of curvature singularity from initially analytic data was rigorously proved. After the appearance of the first singularity the solution becomes very irregular. This is a consequence of the elliptic nature of the Birkhoff-Rott equations: if solutions have a certain minimal regularity, then they are actually analytic [79, 80, 50]. An open problem is the determination of this threshold of regularity that will imply analyticity. It was shown in [50] that any solution consisting of a closed chord arc vortex sheet that near a point belongs to , must be analytic. The conclusion is maintained if the vortex sheet is required to be a Lipschitz chord arc curve [79, 80].

The problem of the evolution of a vortex sheet can also be approached, in the general framework of weak solutions (in the distributional sense) of the Euler equations, as a problem of evolution of the vorticity, which is concentrated as a measure along a surface of codimension one. This approach was pioneered by DiPerna and Majda in [26, 27, 28]. The general problem of existence for mixed-sign vortex sheet initial data remains an open question. However, in 1991, Delort [25] proved a global in time existence of weak solutions of the 2D incompressible Euler equations for the vortex sheet initial data with initial vorticity being a Radon measure of a distinguished sign, see also [30, 58, 53, 71, 72, 59]. This result was later obtained as an inviscid limit of the Navier-Stokes regularizations of the Euler equations [58, 71], and as a limit of numerical vortex methods [53, 72, 54]. The Delort’s result [25] was also extended to the case of mirror-symmetric flows with distinguished sign vorticity on each side of the mirror [57]. It is worth mentioning that uniqueness of solutions of the 2D Euler equations was obtained by Yudovich [81] for initially bounded vorticity, see, also, [76] for an improvement with vorticity in a class slightly larger than , and [75] for review of relevant two-dimensional results. This does not include vortex sheets, which admit measure-valued vorticity. There is also a non-uniqueness result for velocity in [24, 70, 73]. However, the problem of uniqueness of a weak solution with a fixed sign vortex sheet initial data is still unanswered, numerical evidences of non-uniqueness can be found, e.g., in [66, 55]. Furthermore, the structure of weak solutions given by Delort’s theorem is not known, while the Birkhoff-Rott equations assume a priori that a vortex sheet remains a curve at a later time. A proposed criterion for the equivalence of a weak solution of the 2D Euler equations with vorticity being a Radon measure supported on a curve, and a weak solution of the Birkhoff-Rott equation can be found in [56]. Also, another definition of weak solutions of Birkhoff-Rott equation has been proposed in [79, 80]. For a recent survey of the subject, see [4].

The Euler- model [15, 39, 38, 37, 61, 21] is an inviscid regularization of the Euler equations (1.1) given by

 ∂v∂t+(u⋅∇)v+∑jvj∇uj+∇π=0,v=(1−α2Δ)u,∇⋅u=∇⋅v=0,u(x,0)=uin(x). (1.2)

Here represents the “filtered” fluid velocity vector, is the “filtered” pressure, and is a regularization lengthscale parameter representing the width of the filter.

The question of global existence of weak solutions for the three-dimensional Euler- equations is still an open problem. On the other hand, the 2D Euler- equations were studied in [65], where it has been shown that there exists a unique global weak solution to the Euler- equations with initial vorticity in the space of Radon measures on , with a unique Lagrangian flow map describing the evolution of particles. In particular, it follows that the vorticity, initially supported on a curve, remains supported on a curve for all times.

In this paper we relate the weak solutions of Euler- equations with distinguished sign vortex sheet initial data to those of the 2D Euler equations, by proving their convergence, as the length scale . This produces a variant of the result of Delort [25], by obtaining a weak solution of Euler equations as a limit of an inviscid regularization of Euler equations, in addition to approximations obtained by smoothing the initial data, viscous regularization, or numerical vortex methods [25, 58, 71, 59, 53, 72, 54]. Since a weak solution of Euler equations with vortex sheet is unlikely to be unique, a different regularization could produce a different weak solution.

We also present an analytical study of the -analogue of the Birkhoff-Rott equation, the Birkhoff-Rott- (BR-) model, which is induced by the 2D Euler- equations. The BR- results that were reported in a short communication [3] are presented here with full details. The BR- model was implemented computationally in [41], where a numerical comparison between the BR- regularization and the existing regularizing methods, such as a vortex blob model [19, 45, 22, 53, 1], has been performed. In the BR- case the singular kernel of the Biot-Savart law determining the velocity in terms of the vorticity is smoothed by a convolution with a smoothing function , which is the Green function associated with the Helmholtz operator . The function is a modified Bessel function of the second kind of order zero. This is similar to vortex blob methods, however, unlike the standard vortex blob methods [19, 6, 45, 47, 22, 1] (and, in particular, the proof of convergence of vortex blobs methods to a weak solution of 2D Euler equations [53]), the BR-alpha smoothing function is unbounded at the origin. Also, unlike the vortex blob methods that regularize the singular Biot-Savart kernel, the Euler- model regularizes the Euler equations themselves to obtain a smoother kernel.

Section 2 contains the preliminaries about the 2D Euler- equations. In Section 3 we investigate the convergence of solutions of the Euler- equations for vortex sheet initial data to those of the 2D Euler equations, as the regularization length scale tends to zero. Specifically, we prove that for the vortex sheet initial data with initial vorticity of a distinguished sign Radon measure one can extract subsequences of weak solutions of the Euler- equations which converge weak- in , as , to a weak solution of the 2D Euler equations. The space denotes the space of finite Radon measures on .

In Section 4 we describe the BR- equation. Section 5 studies the linear stability of a flat vortex sheet with uniform vorticity density for the 2D BR- model. The linear stability analysis shows that the BR- regularization controls the growth of high wave number perturbations, which is the reason for the well-posedness. This is unlike the case for the original BR problem for Euler equations that exhibits the Kelvin-Helmholtz instability, the main mechanism for its ill-posedness. In Section 6 we show global well-posedness of the 2D BR- model in the space of Lipschitz functions and in the Hölder space , , which is the space of -times differentiable functions with Hölder continuous derivative. Specifically, we show that (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR- equation, remains Lipschitz for all times, (ii) an initially Hölder , , chord arc curve remains in for all times, and finally, (iii) an initially Hölder , closed chord arc curve remains in for all times. Notice that for we request to be strictly larger than zero and the curve to be closed. In all these cases the weak Euler- and the BR- descriptions of the vortex sheet motion are equivalent. The convergence of BR- solutions to the solutions of the original BR system on the short interval of existence of solutions will be reported in a forthcoming paper.

## 2 Euler-α equations

In two dimensions, the incompressible Euler equations in the vorticity form are obtained by taking the curl of (1.1) and are given by

 ∂q∂t+(v⋅∇)q=0,v=K∗q,q(x,0)=qin(x), (2.1)

where , is the fluid velocity field, is the vorticity, and is the given initial vorticity. Delort [25] proved a global in time existence of weak solutions of the 2D Euler equations for the vortex sheet initial data with fixed sign initial vorticity in . The space is the space of finite Radon measures on with the norm

 ∥μ∥M=sup{∣∣∣∫R2φdμ∣∣∣:φ∈C0(R2),∥φ∥L∞≤1},

is the space of continuous functions vanishing at infinity. The space denotes the dual of the Sobolev space . The localized Sobolev space , is the set of all distributions such that for any , see, e.g., [33].

A vorticity , , is called a weak solution of (2.1), if for every test function

 W(q;ψ)≡∫T0∫R2∂tψ(x,t)dq(x,t)dt+∫T0∫R2∫R2Hψ(x,y,t)dq(y,t)dq(x,t)dt=0, (2.2)

where

 Hψ(x,y,t)=14π(x−y)⊥⋅(∇ψ(x,t)−∇ψ(y,t))|x−y|2. (2.3)

The initial value is and it makes sense since . The kernel is bounded, continuous outside the diagonal and vanishes at infinity. This weak vorticity formulation is well-defined, since the vorticity has no discrete part (i.e., for all ), which implies that the diagonal has -measure zero, see [71, 25]. Thorough discussions of Delort’s theorem, its extension and different proofs of the result can be found in [25, 59, 12, 30, 58, 53, 71, 72].

Taking the curl of (1.2) yields the vorticity formulation of the 2D Euler- model

 ∂q∂t+(u⋅∇)q=0,u=Kα∗q,q(x,0)=qin(x). (2.4)

Here represents the “filtered” fluid velocity, and is a regularization length scale parameter, which represents the width of the filter. At the limit , we formally obtain the Euler equations (2.1). The smoothed kernel is , where is the Green function associated with the Helmholtz operator , given by

 Gα(x)=1α2G(xα)=1α212πK0(|x|α), (2.5)

here and is a modified Bessel function of the second kind of order zero [78]. To see this relationship in one can take a Fourier transform of , and obtain as the inverse Fourier transform of . Notice that

 (2.6)

where

 Ψα(r)=12π[K0(rα)+logr], (2.7) DΨα(r)=dΨαdr(r)=12π[−1αK1(rα)+1r],

and denotes a modified Bessel functions of the second kind of order one. For details on Bessel functions, see, e.g., [78].

A weak solution of (2.4) is satisfying

 Wα(q;ψ)≡∫T0∫R2∂tψ(x,t)dq(x,t)dt+∫T0∫R2∫R2Hαψ(x,y,t)dq(x,t)dq(y,t)dt=0, (2.8)

for all test functions . The initial value is and it makes sense since . The kernel is a continuous vanishing at infinity function given by

 Hαψ(x,y,t)=12DΨα(|x−y|)(x−y)⊥⋅(∇ψ(x,t)−∇ψ(y,t))|x−y|. (2.9)

Oliver and Shkoller [65] showed global well-posedness of the Euler- equations with initial vorticity in .

###### Theorem 2.1.

(Oliver and Shkoller [65]) For initial data , there exists a unique global weak solution of Euler- equations (2.4) in the sense of (2.8).
Let denote the group of all homeomorphism of , which preserve the Lebesgue measure and let denote the Lagrangian flow map induced by (2.4), i.e., which obeys the equation
,  . Then the unique Lagrangian flow map exists globally and the vorticity is transported by the flow, i.e., .

Notice that the original BR equations assume a priori that a vortex sheet remains a curve at a later time, however, in the 2D Euler- case, it follows as a consequence of the existence of the unique Lagrangian flow map, that the vorticity that is initially supported on a curve remains supported on a curve for all times.

## 3 Convergence of a fixed sign Euler-α vortex sheet to an Euler vortex sheet

Let the initial vorticity be of a fixed sign, , and compactly supported. In this section we show that there is a subsequence of the solutions of 2D Euler- model with initial data , guaranteed by Theorem 2.1, that converge to a weak solution of 2D Euler equations in the sense of (2.2). This produces a variant of the result of Delort [25], by obtaining a weak solution of Euler equations as a limit of solutions of inviscid regularization of Euler equations, namely, the Euler- equations. The above regularization method is different from the various existing regularizations that are obtained, for instance, by smoothing the initial data, viscous regularization or numerical vortex methods [25, 58, 71, 59, 53, 72, 54]. Since a weak solution of Euler equations with vortex sheet is unlikely to be unique, a different regularization could produce a different weak solution of Euler equations.

In order to prove the convergence of the solutions of the Euler- equations (2.4) to a weak solution of Euler equations (2.1) we follow ideas similar to those reported in [25, 59, 58, 71]. However, due to the structure of the Euler- equations one needs to deal with various technical estimates concerning the “filtered” vorticity and . Specifically, we show in Lemma 3.2 and Lemma 3.3, respectively, that have a uniform decay in small disks, , and the contribution of converges to zero, as .

###### Theorem 3.1.

Let be the solutions of the weak vorticity formulation of Euler- equations (2.8), guaranteed by Theorem 2.1, with initial data , and compactly supported and let . Then there exists a subsequence that weak- converges to in and in for each fixed , as , and is a weak solution of the Euler equations (2.1) in the sense of (2.2) with initial data .

The weak- convergence in means that

 limαj→∞∫T0∫R2φ(x,t)dqαj(x,t)dt=∫T0∫R2φ(x,t)dq(x,t)dt,

for all .

We denote the velocity and the “filtered” velocity by and , respectively, and their corresponding vorticities by and .

Given , we define a linear continuous functional acting on every by

 ⟨ωα,φ⟩=∫R2((1−α2Δ)−1φ)dqα, (3.1)

where is defined as the unique, vanishing at infinity, solution of , given by

 (1−α2Δ)−1φ=∫R21α212πK0(|y|α)φ(x−y)dy, (3.2)

the function is a modified Bessel function of the second kind of order zero, , , see, e.g., [78]. From the above its follows that .

We observe that if then is a nonnegative linear functional. Indeed, let , , then

 (1−α2Δ)−1φ=∫R21α212πK0(|y|α)φ(x−y)dy≥0,

and hence by (3.1) . Also,

 |⟨ωα,φ⟩|≤∥qα∥M∥∥(1−α2Δ)−1φ∥∥L∞≤∥qα∥M∥φ∥L∞.

Therefore, by the Riesz representation theorem (see, e.g., [33, Chapter 7] ) the functional can be represented by a unique nonnegative Radon measure, which we also denote by , and

 ∥ωα∥M≤∥qα∥M. (3.3)

Again, by the Riesz representation theorem, a linear functional defined by

 (3.4)

for every , can be identified with a Radon measure, which we also denote by . Observe that, since for every

 α2Δ(1−α2Δ)−1φ=(1−α2Δ)−1φ−φ,

we have

that is,

We note that by Theorem 2.1 the solution of Euler- equations (2.8) is transported by the flow, that is, , , hence for all

 ∥qα(⋅,t)∥M=∥∥qin∥∥M. (3.5)

In addition, if , then for all times, and therefore also for all times.

The kernel appearing in the non-linear term of (2.2) is discontinuous on the diagonal , so, following [26, 59], to prove the convergence of the non-linear term we need the following estimate, which shows uniform decay of the “filtered” vorticity in small disks.

###### Lemma 3.2.

Let be the solutions of (2.8) with initial data , and compactly supported. Then for defined by (3.1), there exists a constant , such that for all , , and we have

 ∫|x−x0|
###### Proof.

Recall that for all times. The idea of the proof, which is shown in details below, is to convolve the initial data with a standard mollifier to obtain a sequence of solutions of the Euler- equations that has a uniform decay of the circulation on small disks

 ∫|x−x0|≤Rωα,ε(x,t)dx≤C(T)|logR|−1/2,

, , , and then the weak- limit in of a subsequence when , which is the solution of Euler- equations with initial data , satisfies a similar bound.

We observe that, similarly to the Euler equations, any smooth radially symmetric vanishing at infinity vorticity defines a stationary solution of Euler- equations (2.4) with the corresponding velocity . This could be seen using the vorticity stream function formulation for Euler- equations, which is

 qt+J(φ,Δψ)=0, q=Δψ,

where is the Jacobian, is the velocity stream function, , and is the “filtered” stream function, . Since and are rotationally invariant, we have that the corresponding , and are also radially symmetric, therefore and hence defines a stationary solution of Euler- equations.

Let be a standard mollifier, for example,

 ρ(x)=⎧⎨⎩Cexp(1/(|x|2−1))if |x|<1,0if |x|≥1,

, . Smoothing the initial data by a mollification with , , we have that for all the smoothed initial vorticities satisfy , (since is compactly supported), . Following [26, 59] for the 2D Euler case we decompose the velocity into a combination of a stationary bounded velocity plus a time dependent velocity with finite total energy. Let be any smooth radially symmetric function with compact support, such that . Define , and . Notice, that by direct calculation and . Since , and has compact support we have that . Also, due to the fact that with compact support, and hence, for , the smooth are uniformly bounded in with a common compact support and are uniformly bounded in , and since is independent of , we have that are uniformly bounded in , for .

Observe, that the stationary part

satisfies

 ∥¯u∥L∞≤∥¯v∥L∞, (3.7) ∥∇¯u∥L∞≤∥∇¯v∥L∞, ∥∥∂2¯u∥∥L∞≤12π1α∥∇¯v∥L∞,

since and its derivative are smooth functions outside of the origin, satisfying , and rapidly decaying at infinity.

Consider the partial differential equation

 ∂∂t~vα,ε +(~uα,ε⋅∇)~vα,ε+∑j(~vα,ε)j∇(~uα,ε)j (3.8) +(~uα,ε⋅∇)¯v+∑j¯vj∇(~uα,ε)j +(¯u⋅∇)~vα,ε+∑j(~vα,ε)j∇¯uj+∇~πα,ε=0, ~vα,ε= (1−α2Δ)~uα,ε.

This evolution equation is similar to the Euler- equations. Moreover, if is the solution of the equation (3.8) with initial data , then is the solution of the 2D Euler- equations (1.2) with initial data

Similarly to the Euler case (see, e.g., [59]) this equation has a unique global infinitely smooth solution, since, as in 2D Euler case, we have an a priori uniform control over the norm of the , which implies the global existence, as in the proof of the Beale-Kato-Majda criterion [5]. The solution is in for all , and hence, by Sobolev embedding theorem, and, consequently, are also in for all .

Moreover, the solution is in due to the following a priori estimate. Taking the inner product of (3.8) with we have (omitting the subindices and )

 12ddt(|~u|2L2+α2|∇~u|2L2) =α2((¯u⋅∇)Δ~u,~u)−∑j(~vj∇¯uj,~u) =I1−I2.

Since , for we have

 I1 =−α2∑i,j,k∫¯ui∂2~uj∂x2k∂∂xi~uj =α2∑i,j,k∫∂¯ui∂xk∂~uj∂xk∂~uj∂xi+α2∑i,j,k∫¯ui(∂2∂xi∂xk~uj)∂∂xk~uj.

Since the second term on the right is zero, we obtain that

 |I1|≤Cα2∥∇¯u∥L∞∥∇~u∥2L2.

Now we estimate

 I2 =∑i,j,k∫~uj∇¯uj⋅~u−α2∫Δ~uj∇¯uj⋅~u =I21+I22.

We have

 ∣∣I21∣∣≤C∥~u∥2L2∥∇¯u∥L∞

and

 I22=α2∑i,j,k∫∂~uj∂xk∂¯uj∂xi∂~ui∂xk+α2∑i,j,k∫∂~uj∂xk∂2¯uj∂xk∂xi~ui,

hence

 ∣∣I22∣∣≤Cα2∥∇~u∥2L2∥∇¯u∥L∞+Cα2∥∇~u∥L2∥~u∥L2∥∥∂2¯u∥∥L∞.

To conclude, we obtain

 12ddt(∥~u∥2L2+α2∥∇~u∥2L2)≤C(α2∥∇¯u∥L∞∥∇~u∥2L2+∥~u∥2L2∥∇¯u∥L∞+α∥∇~u∥L2∥~u∥L2α∥∥∂2¯u∥∥L∞).

Hence, thanks to (3.7),

 12ddt(∥~u∥2L2+α2∥∇~u∥2L2)≤C∥∇¯v∥L∞(α2∥∇~u∥2L2+∥~u∥2L2),

and by Grönwall inequality

 ∥~u(⋅,t)∥2L2+α2∥∇~u(⋅,t)∥2L2 ≤eC∥∇¯v∥L∞t(∥~u(⋅,0)∥2L2+α2∥∇~u(⋅,0)∥2L2) ≤eC∥∇¯v∥L∞t∥~v(⋅,0)∥2L2.

Hence we have that for all , , the solution of Euler- equations with the smoothed initial data satisfies (we now put back the subindices and )

 ∥uα,ε(⋅,t)∥L2(B(x0,1)) ≤∥~uα,ε(⋅,t)∥L2(B(x0,1))+∥¯u∥L2(B(x0,1)) ≤∥~uα,ε(⋅,t)∥L2(R2)+π∥¯u∥L∞(R2) ≤C(T),

where . This is enough to show uniform decay of the vorticity in small disks (see [71], we remark that here the fixed sign of the vorticity comes into place111In [71] to prove the uniform decay of the vorticity in small circles one defines for Then . We have