Communications in Partial Differential Equations Volume 22, 2001, pp. 295–314

THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID

Marcel Oliver^{1} Mathematisches Institut Universität Tübingen 72076 Tübingen Germany

Steve Shkoller^{2} Department of Mathematics University of California Davis, CA 95616 USA

Abstract. We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M(ℝ^{2}). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M(ℝ^{2}) such as point-vortex data, and show that the associated co-adjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations.

1991 Mathematics Subject Classification. Primary 35Q35, 65M99; Secondary 76C05, 76A05.

1E-mail address: oliver@member.ams.org

2E-mail address: shkoller@math.ucdavis.edu

1. Introduction

The starting point of our investigation is the somewhat surprising fact that the equations of motion for an inviscid non-Newtonian fluid of second grade, Chorin’s vortex blob algorithm with a particular choice of cut-off or blob function, and the so-called Lagrangian averaged Euler equations (or Euler-α equations) are, at least formally, equivalent.

The velocity field u = u(x, t) of a second grade fluid, under the assumptions of observer objectivity and material frame-indifference, satisfies the unique equation

(1 − α^{2}Δ)∂_{t}u − νΔu + curl(1 − α^{2}Δ)u × u = − grad p ,  (1.1a)
div u = 0 ,  (1.1b)
u(0) = u_{0} ,  (1.1c)

where p = p(x, t) is the pressure function which is determined (modulo constants) by the velocity field. See [1] and references therein for a discussion of the constitutive theory of second grade fluids, and [2, 3, 4, 5] for well-posedness of the viscous second-grade fluid equations. In this context, the constant α > 0 is a material parameter which represents the elastic response of the fluid. This paper concerns only the inviscid case, i.e. ν = 0.

When α is interpreted as a spatial scale, (1.1) are known as the isotropic Lagrangian averaged Euler (or Euler-α) equations which model the flow of an ideal incompressible fluid on spatial scales larger than α. A formal derivation was given in [6] in case that the fluid domain is all of ℝ^{n}; a different approach, based on averaging the Lagrangian flow map on the space of volume-preserving maps of fluid domains with boundary, was given in [7]. In this context, it has been suggested that the averaged Euler equations could be also viewed as a subgrid-scale model for the small scale motion with built-in material frame indifference.

The analysis and rich geometry of these equations has received much recent attention [8, 9, 4]. In particular, solutions of (1.1) on a compact n-dimensional Riemannian manifold (M, g) are geodesics on the group of H^{s}-class volume preserving diffeomorphisms D_{μ}^{s}, s > (n/2) + 1, with respect to a new weak right invariant metric, given at the identity element e ∈ D_{μ}^{s} by

⟨u, v⟩_{e} = (u, v)_{L^{2}} + 2α^{2}(Def u, Def v)_{L^{2}}  (1.2)

where Def u = (∇u + ∇u^{t})/2 is the (rate of) deformation tensor [4].

In two dimensions, taking the curl of equation (1.1a) and setting q = (1 − α^{2}Δ) curl_{2D} u yields the vorticity equation

∂_{t}q + u · grad q = 0 ,  (1.3a)
u = K^{α} ∗ q ,  (1.3b)
q(0) = q_{0} ,  (1.3c)

where q = q(x, t) is called the potential vorticity, and K^{α} is the integral kernel of the inverse of (1 − α^{2}Δ) curl_{2D}, defined so that the divergence condition (1.1b) is satisfied.

The vortex blob method was introduced by Chorin [10] as a regularization of the point vortex algorithm for ideal hydrodynamics, and can be understood as follows. Consider the vorticity form of the Euler equations on ℝ^{2},

∂_{t}ω + u · grad ω = 0 ,  (1.4a)
u = K ∗ ω,  (1.4b)
ω(0) = ω_{0} .  (1.4c)

Here K(x, y) = −1/(2π) ∇^{⊥} log |x − y| and ω = ω(x, t) is the physical vorticity of the flow. When the velocity field is sufficiently regular—u is at least continuous in t and quasi-Lipschitz in x, uniformly over finite intervals of time—we may define the Lagrangian flow map η_{t} = η( · , t) by

∂_{t}η(x, t) = u(η(x, t), t) ,  (1.5)

or, equivalently, by

∂_{t}η_{t} = u_{t} ∘ η_{t} .  (1.6)

For each t, the map η( · , t) is in G, the group of all homeomorphisms ϕ of ℝ^{2} which preserve the Lebesgue measure. The pointwise conservation of vorticity under the Euler flow is thus expressed by ω_{t} ∘η_{t} = ω_{0}; combining (1.5), (1.4b), and the initial condition η( · , 0) = e, we obtain the ordinary differential equation

∂_{t}η(x, t) = ∫_{ℝ^{2}} K(η(x, t), η(y, t)) ω_{0}(y) dy .  (1.7)

Letting δ denote the Dirac measure and substituting the point vortex ansatz

ω(x, t) = ∑_{i=1}^{N} Γ_{i} δ(x − x_{i}(t)) ,  (1.8)

into (1.7), we obtain a finite dimensional system of ordinary differential equations for the vortex centers x_{1}, …, x_{N}. However, the induced velocity field has 1/|x|-type singularities at the vortex centers. Hence, the point vortex system is neither numerically well-behaved (the exact solution of the point vortex system may even collapse in a finite time for small sets of initial data [11]), nor does it approximate physically relevant velocity fields very well.

The idea of the vortex blob method is to smooth the Dirac measure by a cut-off or blob function χ that decays at infinity and whose mass is mostly supported in a disc of diameter α. This leads to the following equation for the Lagrangian flow:

∂_{t}η^{α}(x, t) = ∫_{ℝ^{2}} K^{α}(η^{α}(x, t), η^{α}(y, t)) ω_{0}(y) dy ,  (1.9a)

where

K^{α} = ∇^{⊥}G^{α} ,  (1.9b)
−ΔG^{α}(x, y) = χ^{α}(|x − y|) ≡ \frac{1}{α^{2}} χ(\frac{|x − y|}{α}) .  (1.9c)

Many researchers have investigated the convergence properties of this scheme [12, 13, 14, 15, 16]. In particular, for certain smooth cut-off functions, such as Bessel functions, the order of accuracy with respect to the regularization parameter α depends only on the smoothness of the Euler flow (“infinite order accuracy”).

It is now easy to see that the equation of a second grade fluid (1.3) and the vortex blob method coincide when χ(x) = −1/(2π) K_{0}(x), where K_{0} is the modified Bessel function of the second kind which is the Green’s kernel for the Helmholtz operator (1 − Δ). Thus far, this relationship has only been formally established, as it remains to be proven that the point-vortex ansatz (1.8) makes sense as data for the partial differential equation (1.3); moreover, it is not a priori clear if solutions to the vortex blob method converge to true Euler solutions. Our results are the following.

We show that the Lagrangian flow formulation of the blob method (1.9) with cut-off function K_{0} is well-posed for initial potential vorticities q_{0} in M(ℝ^{2}), the space of Radon measures on ℝ^{2}. In particular, this includes point-vortex initial data. Such a result does not hold for the Euler equations, where the flow map of the point vortex system (1.7) is not known to be well-defined.

Second, we consider the matter of greatest practical importance: the convergence of solutions of the vortex blob method to solutions of the Euler equations as the blob diameter α → 0. We prove this convergence result under the rather mild assumption that the initial Euler vorticity field ω_{0} is in the Yudovich space L^{∞}(ℝ^{2})∩L^{1}(ℝ^{2}), and is approximated by a sequence of weakly converging measures in M(ℝ^{2}) that have uniformly bounded total variation.

Theorem 1. For initial data q_{0} ∈ M(ℝ^{2}), there exists a unique global weak solution to (1.3) with

η^{α} ∈ C^{1}(ℝ; G) , u^{α} ∈ C^{0}(ℝ; C_{div}^{0}(ℝ^{2}, ℝ^{2})) , and q ∈ C^{0}(ℝ; M(ℝ^{2})) , (1.10)

where the subscript div denotes divergence-free.

Remark 1. The solution that we construct may not necessarily have finite energy, i.e., the velocity field u^{α} may not be in L^{2}. None of our results, however, relies on energy type estimates. Furthermore, as is the case for the Euler equations, the initial potential vorticity can be decomposed into a radially symmetric and a mean-zero part, with a corresponding velocity field u in the affine space u_{stationary} + L^{2}(ℝ^{2}, ℝ^{2}). For details see DiPerna and Majda [17].

Remark 2. An immediate consequence of the uniqueness of the solution and the time-reversibility of the equation is that the vortex blob system cannot collapse in finite time, i.e., two or more vortex centers cannot merge into one in finite time. For non-regularized Euler vortex dynamics, on the other hand, it is known that vortex collapse occurs on small sets of initial configurations [11] although, for particular approximating sequences of initial vortex distributions, collapse is prevented for small enough choice of the approximation parameter [18].

Remark 3. The kernel K^{α} which corresponds to a second grade fluid (or the isotropic averaged Euler equation) is the least regular kernel (modulo possible sub-logarithmic corrections) for which uniqueness of point vortex solutions can be shown. An equivalent uniqueness result based on Sobolev space methods, and for bounded domains, is given in [19].

Theorem 2. Let η be the flow map of the Euler equation (1.7) with initial vorticity ω_{0} ∈ L^{1}(ℝ^{2}) ∩ L^{∞}(ℝ^{2}). Suppose that ω_{0} is approximated by a sequence of measures q_{0}^{n} in M(ℝ^{2}) such that q_{0}^{n} ⇀ ω_{0} weakly in M(ℝ^{2}) and ∥q_{0}^{n}∥_{M} → ∥ω_{0}∥_{L^{1}}. Then for every T > 0, there exists a sequence {α_{n}} converging to zero as n → ∞ such that when η^{α_{n}} denotes the flow map of the vortex method with α = α_{n} and initial data q_{0}^{n},

lim_{n→∞} sup_{t∈[0,T]} sup_{x∈ℝ^{2}}|η^{α_{n}}(x, t) − η(x, t)| = 0 .  (1.11)

Remark 4. The idea of analyzing the vortex method as a PDE posed on some space of distributions was already used by Marchioro and Pulvirenti [16] and Cottet [14]. Cottet’s result requires stronger assumptions on the cut-off function and hence smoothing kernel, stronger regularity assumptions on the underlying Euler flow, and his approximation of the Euler initial vorticity field requires a uniform grid. The trade-off, however, is that these more stringent constraints give an improved (algebraic) convergence in α. There is, in general, a trade-off between the order of convergence on the one hand, and the assumptions placed on K^{α}, ω_{0}, and the approximation at time t = 0 on the other. The result which, to our knowledge, comes closest to Theorem 2 is given in Marchioro and Pulvirenti [16]. The authors, however, assume that K^{α} is Lipschitz, which again excludes kernels corresponding to the equations of second grade fluids.

Remark 5. Stronger results can be proved for kernels K^{α} with a higher degree of smoothing. For example, by replacing 1 − α^{2}Δ with (1 − α^{2}Δ)^{s}, one obtains a hierarchy of regularizations of the Euler equations which coincide with geodesic flow on the volume-preserving diffeomorphism group with respect to the H^{s} metric. Other choices of K^{α} may introduce nonlocal pseudo-differential operators into equation (1.3), but the analysis can still proceed as before.

Remark 6. In three dimensions, the formal connection between second grade fluids (averaged Euler equations) and particular vortex filament methods still holds and is the subject of a forthcoming article. In this setting, one looks at the set of vorticity distributions of the following form. Let γ be a curve in ℝ^{3} extending to infinity in both directions, and let δ_{γ} be the Dirac distribution given by integration along γ with respect to arc length. Let ω_{γ} be the 2-form along γ defined by i_{T} dx ∧ dz, where T is the unit tangent vector to γ. Then if Γ is any constant, Γω_{γ}δ_{γ} is the vorticity corresponding to γ with strength Γ. See [20].

Theorem 1 allows us to rigorously classify the co-adjoint orbits characterized by point-vortex initial data. Let us explain what we mean by this. The configuration space for ideal incompressible hydrodynamics is the volume-preserving diffeomorphism group, and for s > (n/2) + 1, D_{μ}^{s} is a C^{∞} Hilbert manifold and a smooth topological group. While D_{μ}^{s} is not a Lie group (left composition and inversion are only C^{0} and the group exponential map does not cover a neighborhood of the identity), it behaves similar to a Lie group, because of the smooth properties of the Riemannian exponential map (see [21] and [8, 4]). The Eulerian phase space for the fluid motion is the single fiber T_{e}D_{μ}^{s} consisting of H^{s}-class divergence free vector fields on the fluid container, and this vector space can be formally thought of as the “Lie algebra” of D_{μ}^{s}. The cotangent space at the identity is given by H^{s}(Λ^{1})/dH^{s}(Λ^{0}), the H^{s}-class differential 1-forms modulo exact 0-forms. Using the fact that the exterior derivative d: T_{e}^{∗}D_{μ}^{s} → H^{s−1}(Λ^{2}) is an isomorphism, and the fact that we may identify H^{s−1}(Λ^{2}) with H^{s−1}(Λ^{0}), the role of the dual of the “Lie algebra” for 2D hydrodynamics is played by the H^{s−1}-class vorticity functions. The representation of D_{μ}^{s} on this “Lie algebra” is provided by the co-Adjoint action, so that for η ∈ D_{μ}^{s} and ω ∈ H^{s−1}(Λ^{0}), Ad_{η}^{∗}(ω) = ω ∘ η, and the invariance of the co-Adjoint orbit is merely the pointwise conservation of vorticity which is fundamental to 2D hydrodynamics.

If one temporarily ignores the topology and works formally, then it is possible to classify certain interesting and important co-Adjoint orbits. Specifically, it is a result of Marsden and Weinstein [20] that point-vortex initial data (1.8) define the co-Adjoint orbit on which point-vortex dynamics evolve. This is clearly a formal result as Dirac measures are not elements of H^{s−1} for s > 2; consequently, the problem is to supply a candidate topology for the “Lie algebra” which is general enough to contain the Dirac measures, and weaken the regularity of the configuration space so that its “representation” is well-defined. In doing so, one can establish a rigorous classification of the orbit. By using G for the configuration space and M(ℝ^{2}) for the “Lie algebra,” and by defining a new notion of weak co-adjoint action which coincides with the notion of a weak solution, we are able to establish the orbit classification for point-vortex initial data, and prove that our particular vortex blob method leaves such weak co-adjoint orbits invariant.

Remark 7. Theorem 1 provides a C^{1} flow for initial vorticity in the class of Radon measures M(ℝ^{2}). In the case that the initial Euler vorticity is in H^{s−1}(ℝ^{2}) for s > 2, the flow map is smooth, i.e., η ∈ C^{∞}(ℝ, D_{μ}^{s}) and has C^{∞} dependence on the initial data [4]. It would be interesting to see if this smoothness property can be obtained for the weak class of initial data that we have considered.

Remark 8. Since for our choice of smoothing kernel, the vortex method is exactly the system of inviscid second-grade fluids (or the Lagrangian averaged Euler equations), and since these PDEs are well-posed when supplemented with either the no-slip, free-slip, or mixed boundary conditions [4], we believe that the K_{0} smoothing kernel produces natural vortex methods on domains with boundary. In future work, we shall explore these ideas in the study of boundary layers arising when fluid flows past an obstacle.

2. Kernel estimates

The crucial ingredients for the proof of our theorems are quasi-Lipschitz estimates on the Euler kernel K and the regularized kernel K^{α}. For x ∈ ℝ^{2} we define the function

φ(x) = \begin{cases}|x| (1 − ln |x|) & for |x| < 1 , \\ 1 & for |x| ≥ 1 . \end{cases}  (2.1)

Lemma 3. For ω ∈ L^{1}(ℝ^{2}) ∩ L^{∞}(ℝ^{2}),

∫_{ℝ^{2}} |K(x, y) − K(x′, y)| |ω(y)| dy ≤ c φ(x − x′) (∥ω∥_{L^{1}} + ∥ω∥_{L^{∞}}) . (2.2)

The proof is standard and can be found, for example, in McGrath [22]. Somewhat less standard is the following estimate, still for the Euler kernel, which is similar to estimates in Benedetto et al. [23].

Lemma 4. Let ω ∈ L^{1}(ℝ^{2}) ∩ L^{∞}(ℝ^{2}) and let ϕ be an area preserving measurable transformation on ℝ^{2}. Then

|∫_{ℝ^{2}}(K(x, y) − K(x, ϕ(y))) ω(y) dy|

≤ c sup_{x∈ℝ^{2}} φ(x − ϕ(x)) (∥ω∥_{L^{1}} + ∥ω∥_{L^{∞}}) . (2.3)

Proof. Set r = sup_{x}|x − ϕ(x)|; as in the proof of Lemma 3, the interesting case is when r < 1. We split the integral in (2.3) into two parts. First, consider

∫|x−y|≤2r|K(x, y) − K(x, ϕ(y))| |ω(y)| dy

≤ \frac{1}{2π} ∫_{|x−y|≤2r} \frac{|ω(y)|}{|x − y|} dy + ∫_{|x−y|≤2r} \frac{|ω(y)|}{|x − ϕ(y)|} dy

≤ \frac{1}{2π} ∫_{|x−y|≤2r}(\frac{1}{|x − y|} + \frac{1}{|x − ϕ(y)|})dy ∥ω∥_{L^{∞}}

≤ \frac{1}{π} ∫_{|x−y|≤2r} \frac{1}{|x − y|} dy ∥ω∥_{L^{∞}} ≡ c r ∥ω∥_{L^{∞}} .  (2.4)

The last inequality above holds because ϕ is area preserving, and among all such transformations, the symmetric rearrangement ϕ = e maximizes the integral over |x − ϕ(y)|^{−1}.

Next, we consider the case when |x − y| ≥ 2r. Observe that

|x − ϕ(y)| ≥ |x − y| − |y − ϕ(y)| ≥ |x − y| − r ≥ \frac{1}{2} |x − y| ,  (2.5)

so that

∫|x−y|≥2r|K(x, y) − K(x, ϕ(y))| |ω(y)| dy

≤ \frac{1}{2π} ∫_{|x−y|≥2r} \frac{|y − ϕ(y)|}{|x − y| |x − ϕ(y)|} |ω(y)| dy

≤ \frac{1}{π} ∫_{|x−y|≥2r} \frac{r}{|x − y|^{2}} |ω(y)| dy

= \frac{r}{π} (∫_{2r≤|x−y|≤2} \frac{|ω(y)|}{|x − y|^{2}} dy + ∫_{|x−y|≥2} \frac{|ω(y)|}{|x − y|^{2}} dy)

≤ \frac{r}{π} (∫_{2r}^{2} \frac{dρ}{ρ} ∥ω∥_{L^{∞}} + \frac{1}{4} ∥ω∥_{L^{1}})

≤ c φ(r) (∥ω∥_{L^{1}} + ∥ω∥_{L^{∞}}) .  (2.6)

By combining the two estimates we complete the proof.

Finally, we give the corresponding result for the vortex method kernel.

Lemma 5. There exists a constant c_{2} which is independent of α, such that

sup_{y∈ℝ^{2}}|K^{α}(x, y) − K^{α}(x′, y)| ≤ \frac{c_{2}}{α} φ(\frac{x − x′}{α} ) .  (2.7)

Proof. Note that on ℝ^{2}, K^{α}(x, y) = K^{α}(|x − y|) = ∇^{⊥}G^{α}(|x − y|), where

G^{α}(r) = −\frac{1}{2π} K_{0}(\frac{r}{α}) − \frac{1}{2π} ln r  (2.8)

and K_{n} denotes the nth order modified Bessel function of the second kind [24]. For simplicity, we take α = 1 and compute

\frac{dG^{α}}{dr} (r) = \frac{1}{2π} (K_{1}(r) − \frac{1}{r}) = \frac{1}{4π} r ln r + O(r) ,  (2.9)
\frac{d^{2}G^{α}}{dr^{2}} (r) = \frac{1}{2π} (\frac{1}{r^{2}} − K_{0}(r) − \frac{1}{r} K_{1}(r)) = \frac{1}{4π} |ln r| + O(1) , (2.10)

as r → 0. Set r ≡ |x − x′| and assume, without loss of generality as K^{α} is bounded, that r < 1.

If |x − y| < 2r, then |x′ − y| ≤ |x′ − x| + |x − y| < 3r, so that

|K^{α}(x, y) − K^{α}(x′, y)| ≤ |∇^{⊥}G^{α}(|x − y|)| + |∇^{⊥}G^{α}(|x − y|)|

≤ |\frac{dG^{α}}{dr} (|x − y|)| + |\frac{dG^{α}}{dr} (|x − y|)|

≤ \frac{2}{π} r |ln r| + O(r) .  (2.11)

Since dG^{α}/dr is continuous and decays at infinity, this implies a bound of the form

|K^{α}(x, y) − K^{α}(x′, y)| ≤ c φ(|x − x′|) .  (2.12)

If, on the other hand, |x − y| ≥ 2r, we use the mean value theorem to estimate

|K^{α}(x, y) − K^{α}(x′, y)| ≤ sup_{x′′∈B(x,r)} |∇K^{α}(x′′, y)| |x − x′|

≤ sup_{x′′∈B(x,r)}|\frac{d^{2}G^{α}}{dr^{2}} (|x′′ − y|)| r

≤ \frac{1}{4π} r |ln r| + O(r) ,  (2.13)

which again implies a bound of the form (2.12). In the last step we have used (2.10) in conjunction with |x′′ − y| > r.

To recover the scaling of the estimate in α, divide (2.12) by α, rescale x, x′, and y by α^{−1}, and note that K^{α}(r) = K^{α=1}(r/α)/α.

Corollary 6. For q ∈ M(ℝ^{2}),

∫_{ℝ^{2}} |K^{α}(x, y) − K^{α}(x′, y)| |q(y)| dy ≤ \frac{c_{2}}{α} φ(\frac{x − x′}{α} ) ∥q∥_{M} . (2.14)

3. Well-posedness

We can now prove the existence of unique global weak solutions to the Lagrangian flow equation (1.9).

Proof of Theorem 1. Due to the quasi-Lipschitz condition for K^{α}, we can adopt the method that Kato developed for the Euler equations in [25], by simply replacing the kernel estimates in L^{1} by the corresponding estimates in L^{∞}. Our presentation follows to some extent that of Marchioro and Pulvirenti [11].

For simplicity, we assume α = 1 throughout this proof. We introduce a sequence of approximate solutions

∂_{t}η^{n}(x, t) = u^{n}(η^{n}(x, t), t) ,  (3.1a)
η^{n}(x, 0) = x ,  (3.1b)
η^{0}(x, t) = x ,  (3.1c)
q^{n}(η^{n}(x, t), t) = q_{0}(x) ,  (3.1d)
u^{n}(x, t) = ∫_{ℝ^{2}} K^{α}(x, y) q^{n−1}(y, t) dy ,  (3.1e)

for n ∈ ℕ. The proof now proceeds in several steps.

Step 1. Prove that η^{n} ∈ C^{1}([0, ∞); G) for every n ∈ ℕ.

We proceed inductively. Notice that for every n the vector field u^{n} is quasi-Lipschitz in space and continuous in time. This is a consequence of Lemma 5 as

|u^{n}(x, t) − u^{n}(x′, t)|
= |∫_{ℝ^{2}}[K^{α}(x, η^{n−1}(y, t)) − K^{α}(x′, η^{n−1}(y, t))] q_{0}(y) dy|
≤ c φ(x − x′) ∥q_{0}∥_{M} ,  (3.2)

and

|u^{n}(x, t) − u^{n}(x, t′)|

= |∫_{ℝ^{2}}[K^{α}(x, η^{n−1}(y, t)) − K^{α}(x, η^{n−1}(y, t′))] q_{0}(y) dy|

≤ sup_{y∈ℝ^{2}}|K^{α}(x, η^{n−1}(y, t)) − K^{α}(x, η^{n−1}(y, t′))| ∥q_{0}∥_{M}

≤ c sup_{y∈ℝ^{2}} φ(η^{n−1}(y, t) − η^{n−1}(y, t′)) ∥q_{0}∥_{M}

≤ c sup_{y∈ℝ^{2}} sup_{s∈[t,t′]} φ(|η̇^{n−1}(y, s)| |t − t′|) ∥q_{0}∥_{M}

= c sup_{y∈ℝ^{2}} sup_{s∈[t,t′]} φ(|u^{n−1}(y, s)| |t − t′|) ∥q_{0}∥_{M} .  (3.3)

This implies uniform continuity in time, as u^{n} is bounded for every n:

|u^{n}(x, t)| = |∫_{ℝ^{2}} K^{α}(x, η^{n−1}(y, t)) q_{0}(y) dy|

≤ sup_{y∈ℝ^{2}}|K(x, y)| ∥q_{0}∥_{M} ≡ c ∥q_{0}∥_{M} . (3.4)

Since u^{n} is continuous in time and quasi-Lipschitz in space, the vector field generates a local flow η^{n} ∈ C^{1}([0, T); C(ℝ^{2})) for some T > 0—see, e.g., Chapter 2, Lemma 3.2 in Marchioro and Pulvirenti [11]. Because of the global bound (3.4), the right side of (3.1a) is bounded and the flow exists globally in time.

Step 2. Show that there exists a limiting flow map η ∈ C([0, ∞); G).

We first prove that the sequence η^{n} is Cauchy in C([0, T]; G) for some T > 0. To simplify notation, we shall drop the explicit time dependence of u and η, and estimate

|η^{n}(x, t) − η^{n−1}(x, t)|

≤ ∫_{0}^{t}|u^{n}(η^{n}) − u^{n−1}(η^{n−1})|ds

≤ ∫_{0}^{t}|∫_{ℝ^{2}}[K^{α}(η^{n}(x), η^{n−1}(y)) − K^{α}(η^{n−1}(x), η^{n−1}(y))]q_{0}(y) dy|ds

+ ∫_{0}^{t}|∫_{ℝ^{2}}[K^{α}(η^{n−1}(x), η^{n−1}(y)) − K^{α}(η^{n−1}(x), η^{n−2}(y))]q_{0}(y) dy|ds

≤ c ∫_{0}^{t} φ(η^{n}(x) − η^{n−1}(x)) ds ∥q_{0}∥_{M}

+ c ∫_{0}^{t} φ(η^{n−1}(x) − η^{n−2}(x)) ds ∥q_{0}∥_{M} .  (3.5)

By taking the supremum over x on both sides, we obtain

sup_{x∈ℝ^{2}}|η^{n}(x, t) − η^{n−1}(x, t)| ≤ c ∥q_{0}∥_{M} ∫_{0}^{t}[φ( sup_{x∈ℝ^{2}}|η^{n}(x) − η^{n−1}(x)|)

+ φ( sup_{x∈ℝ^{2}}|η^{n−1}(x) − η^{n−2}(x)|)]ds . (3.6)

Defining

ρ^{N}(t) ≡ sup_{n≥N} sup_{x∈ℝ^{2}}|η^{n}(x, t) − η^{n−1}(x, t)| ,  (3.7)

we can simplify the previous estimate, and obtain

ρ^{N}(t) ≤ c ∫_{0}^{t} φ(ρ^{N−1}(s))ds .  (3.8)

It is well known that this implies

lim_{N→∞} ρ^{N}(t) → 0 ,  (3.9)

uniformly on [0, T] for T sufficiently small. Since T depends only on α and the M-norm of q_{0}, the result can be extended to arbitrarily large times. Thus, the contraction mapping theorem implies the assertion of Step 2.

Step 3. Show that the Lagrangian flow equation (1.9) is satisfied in the limit, and that η ∈ C^{1}(ℝ; G).

We define the limiting potential vorticity q and the limiting velocity u in the obvious way, and check by direct estimation that

q^{n} ⇀ q ≡ q_{0} ∘ η^{−1}  (3.10)

weakly in M(ℝ^{2}), and

u^{n} → u ≡ K^{α} ∗ q  (3.11)

in C(ℝ^{2}); both limits are uniform over finite intervals of time.

To prove that η, u, and q solve the limit problem (1.9), we consider its integrated version

η(x, t) − ∫_{0}^{t} u(η(x, s), s) ds

= η(x, t) − ∫_{0}^{t} u(η(x, s), s) ds − η^{n}(x, t) + ∫_{0}^{t} u^{n}(η^{n}(x, s), s) ds

≤ |η(x, t) − η^{n}(x, t)| + ∫_{0}^{t}|u^{n}(η^{n}(x, s), s) − u(η^{n}(x, s), s)| ds
+ ∫_{0}^{t}|u(η^{n}(x, s), s) − u(η(x, s), s)| ds
→ 0 uniformly in x as n → ∞ .  (3.12)

Thus, the left side must be zero. Since u(η(x, s), s) is continuous in x, we can differentiate with respect to t, and find that η satisfies (1.9) and that ∂_{t}η is in fact continuous. Due to the time-reversibility of the equation, the result extends to negative times as well.

Moreover, one can show—first by formal calculation for smooth function, and then extending by the usual density argument—that the weak solution q defined through (3.10) satisfies

∫_{ℝ} ∫_{ℝ^{2}}(∂_{t}ϕ + u · grad ϕ) q dx dt = 0  (3.13)

for every ϕ ∈ C_{0}^{∞}(ℝ × ℝ^{2}). This shows that solutions of the vortex method, and hence the equations of second-grade non-Newtonian fluids and the Lagrangian averaged Euler equations, preserve the (weak) coadjoint action as defined in the following section.

Step 4. Prove that the solution is unique.

Uniqueness is shown by a direct estimate on the difference of two flow maps. This leads to another log-Gronwall inequality, which can be treated in the same way as the previous ones; we omit all details.

Remark 9. The homeomorphisms that we consider have the vector space ℝ^{2} as the range; we may thus subtract two elements of this class. For homeomorphisms of a compact domain Ω of ℝ^{2}, one must isometrically embed the set of measure-preserving homeomorphisms of Ω into the vector space L^{2}(Ω, ℝ^{2}), and take differences in this large space. Similarly, the difference u^{α} ∘ η^{α} − u ∘ η is not an intrinsic operation, but rather relies on the trivial identification of vector spaces induced by the trivial geometry of T ℝ^{2}. On the other hand, when the configuration space is D_{μ}^{s}(M), s > 2 and M is a compact Riemannian manifold, the map u^{α} ∘ η^{α} is an element of the fiber T_{η^{α}}D_{μ}^{s} while u∘η is in T_{η}D_{μ}^{s}; thus, in order to compare the two maps, we must parallel transport u ∘ η into T_{η^{α}}D_{μ}^{s} along the Riemannian connection.

4. Weak co-adjoint action and reduction

As we described, classical solutions of the two-dimensional averaged Euler equations are geodesics on the Hilbert-class volume-preserving diffeomorphism group D_{μ}^{s} for s > 2. We identify the space of classical vorticity solutions H^{s−1}(M) with the reduced space T_{e}D_{μ}^{s} = T D_{μ}^{s}/D_{μ}^{s} (symmetry reduction by the massive particle relabeling symmetry group D_{μ}^{s} of hydrodynamics), and note that this space is the union of the D_{μ}^{s}- co-adjoint orbits.

In the case that M = ℝ^{2}, and for the purpose of studying weak solutions to (1.3) we shall substantially relax the regularity requirements on the configuration space, and use G in place of D_{μ}^{s}; correspondingly, we shall use the vector space of Radon measures on ℝ^{2}, which we denote by M(ℝ^{2}), for the reduced space (the Eulerian phase space) of vorticity functions in place of the space of H^{s−1} functions.

Recall that the co-Adjoint action of D_{μ}^{s} on H^{s−1}(ℝ^{2}) is given by

Ad_{η}^{∗}(q) = q ∘ η .  (4.1)

We shall need to define the notion of weak co-Adjoint action of G on M(ℝ^{2}). First, note that the operation Ad^{∗}: G × M(ℝ^{2}) → M(ℝ^{2}) given by Ad_{η}^{∗}(q) = q ∘ η is well-defined. Next, define the weak co-Adjoint action w-Ad^{∗}: G × M(ℝ^{2}) → M(ℝ^{2}) by

∫_{ℝ} ∫_{ℝ^{2}} w-Ad_{η}^{∗}(q) ϕ dx dt = ∫_{ℝ} ∫_{ℝ^{2}} q (ϕ ∘ η) dx dt  (4.2)

for all ϕ ∈ C_{0}^{∞}(ℝ × ℝ^{2}).

It follows that if η_{t} is a C^{1} curve in G such that e = η_{0} and u = (d/dt)|_{t=0}η_{t}, then we may—computing the time derivative of w-Ad_{η_{t}}^{∗}(q) at t = 0—define the weak analogue of the algebra co-adjoint action by

∫_{ℝ} ∫_{ℝ^{2}} w-ad_{u}^{∗}(q) ϕ dx dt = ∫_{ℝ} ∫_{ℝ^{2}} q (∂_{t}ϕ + u · grad ϕ) dx dt  (4.3)

for every ϕ ∈ C_{0}^{∞}(ℝ × ℝ^{2}). Recall that the classical co-adjoint action is defined by ad_{u_{t}}^{∗}(q_{t}) = (d/dt)|_{t=0}(η_{t}^{∗}q_{t}) where (d/dt)|_{t=0}(η_{t}^{∗}q_{t}) = ∂_{t}q_{t} + L_{u_{t}}q_{t}. In two dimensions, the Lie derivative term L_{u_{t}}q_{t} reduces to u_{t} · grad q_{t}.

Theorem 7. For any q_{0} ∈ M(ℝ^{2}), let O_{q_{0}} denote the co-Adjoint orbit {q: q = q_{0} ∘ η, η ∈ G}. The weak co-adjoint action of C_{div}^{0}(ℝ^{2}) on M(ℝ^{2}) is well-defined, and solutions of the second grade fluids (or averaged Euler equations or Chorin’s vortex blob method) with initial data q_{0} leave O_{q_{0}} invariant.

Proof. The result immediately follows from the fact that the vanishing of the weak co-adjoint action is equivalent to the weak formulation of (1.3). Theorem 1, giving global well-posedness of weak solutions, then concludes the argument.

5. Convergence

We can now prove convergence of the flow of the vortex blob method to the flow of the Euler equations. This is done in two steps. First we show that the averaged Euler equation, or vortex method PDE, approximates the Euler equation as α → 0 for bounded vorticity fields. In the second step, we prove that bounded solutions of the averaged Euler equation can be approximated by measure-valued ones. These two results together imply Theorem 2.

Lemma 8. Let q_{0} ≡ ω_{0} ∈ L^{1}(ℝ^{2}) ∩ L^{∞}(ℝ^{2}). Then for every T > 0 there exists a positive constant C(T) such that

sup_{t∈[0,T]} sup_{x∈ℝ^{2}}|η^{α}(x, t) − η(x, t)| ≤ C(T) α^{e^{−T}} .  (5.1)

Proof. We estimate the difference of the Euler and Euler-α flow maps:

|η^{α}(x, t) − η(x, t)| ≤ ∫_{0}^{t}|u^{α} ∘ η^{α} − u ∘ η| ds

≤ ∫_{0}^{t} ∫_{ℝ^{2}} |K^{α}(η^{α}(x), η^{α}(y)) − K(η^{α}(x), η^{α}(y))| |ω_{0}(y)| dy ds

+ ∫_{0}^{t} ∫_{ℝ^{2}} |K(η^{α}(x), η^{α}(y)) − K(η(x), η^{α}(y))| |ω_{0}(y)| dy ds

+ ∫_{0}^{t} ∫_{ℝ^{2}} |K(η(x), η^{α}(y)) − K(η(x), η(y))| |ω_{0}(y)| dy ds
≡ ∫_{0}^{t}(I_{1} + I_{2} + I_{3}) ds  (5.2)

To estimate I_{1}, we note that on ℝ^{2} the difference of the kernels is explicitly given by |K^{α}(r) − K(r)| = 1/(2π) K_{1}(r/α)/α, so that

I_{1} = \frac{1}{2π} ∫_{ℝ^{2}} \frac{1}{α} K_{1}(\frac{|x − y|}{α}) |ω^{α}(y, s)| dy

≤ ∫_{0}^{∞} \frac{1}{α} K_{1}(\frac{r}{α}) r dr ∥ω^{α}(s)∥_{L^{∞}}

≤ c α ∥ω_{0}∥_{L^{∞}}  (5.3)

The other two integrals can be estimated by using the quasi-Lipschitz conditions, Lemma 3 and Lemma 4, respectively. One finds that

I_{2} ≤ c φ(η^{α}(x, s) − η(x, s)) (∥ω_{0}∥_{L^{1}} + ∥ω_{0}∥_{L^{∞}}) ,  (5.4)

and

I_{3} = ∫_{ℝ^{2}} |K(η(x, s), y) − K(η(x, s), η^{α} ∘ η^{−1}(y, s))| |ω(y, s)| dy

≤ c sup_{x∈ℝ^{2}} φ(x − η^{α} ∘ η^{−1}(x, s)) (∥ω(s)∥_{L^{1}} + ∥ω(s)∥_{L^{∞}})

= c sup_{x∈ℝ^{2}} φ(η(x, s) − η^{α}(x, s)) (∥ω_{0}∥_{L^{1}} + ∥ω_{0}∥_{L^{∞}}) .  (5.5)

By inserting the bounds for I_{1} to I_{3} back into (5.2) and taking the supremum on both sides, we obtain the log-Gronwall inequality

sup_{x∈ℝ^{2}}|η^{α}(x, t) − η(x, t)| ≤ ∫_{0}^{t}[α C_{1} + C_{2} sup_{x∈ℝ^{2}} φ(η^{α}(x, s) − η(x, s))] ds . (5.6)

To obtain explicit bounds that are valid on any finite interval of time [0, T], we set

ρ(t) = sup_{x∈ℝ^{2}}|η^{α}(x, t) − η(x, t)| ,  (5.7)

and use the tangent approximation of the concave function φ; namely, for any ε ∈ (0, 1),

φ(r) ≤ φ(ε) + φ′(ε) r = (− ln ε) r + ε .  (5.8)

This makes the right-hand-side of (5.8) linear in r. For notational simplicity, we also rescale α and t such that C_{1} = C_{2} = 1. We substitute (5.8) into (5.6) and obtain the usual Gronwall inequality; it follows that ρ must satisfy the differential inequality

ρ̇ ≤ (− ln ε) ρ + ε + α ,  ρ(0) = 0 .  (5.9)

Setting ε = e^{−1}α^{exp(−t)} and integrating (5.9) with this choice of ε(α), we find that

ρ(t) ≤ \frac{e^{t} − 1}{e} α^{e^{−t}} + e^{t} \frac{α^{e^{−t}} − α}{− ln α} .  (5.10)

Thus, ρ = O(α^{exp(−T)}) uniformly on [0, T].

In the following we will consider α as fixed and approximate bounded data by measure valued data. Let η, q, and u denote quantities corresponding to a solution of the Euler-α equation with initial data q_{0} ∈ L^{∞}(ℝ^{2}), and let η^{n}, q^{n}, and u^{n} denote a sequence of solutions to the Euler-α equation with initial data q_{0}^{n} ∈ M(ℝ^{2}) for every n ∈ ℕ. Then the following is true.

Lemma 9. Let q_{0} ∈ L^{1}(ℝ^{2}) ∩ L^{∞}(ℝ^{2}), and suppose that q_{0} is approximated by a sequence of measures in M(ℝ^{2}) such that q_{0}^{n} ⇀ q_{0} weakly in M(ℝ^{2}), and ∥q_{0}^{n}∥_{M} → ∥q_{0}∥_{L^{1}}. Then, for every T > 0,

lim_{n→∞} sup_{t∈[0,T]} sup_{x∈ℝ^{2}}|η^{n}(x, t) − η(x, t)| = 0 .  (5.11)

Proof. As in the proof of Lemma 8, we estimate

|η(x, t) − η^{n}(x, t)|

≤ ∫_{0}^{t}|∫_{ℝ^{2}}[K^{α}(η(x, s), η(y, s)) q_{0}(y) − K^{α}(η^{n}(x, s), η^{n}(y, s)) q_{0}^{n}(y)] dy| ds

≤ ∫_{0}^{t}|∫_{ℝ^{2}} K^{α}(η(x), η(y)) (q_{0}(y) − q_{0}^{n}(y)) dy| ds

+ ∫_{0}^{t} ∫_{ℝ^{2}} |K^{α}(η(x), η(y)) − K^{α}(η(x), η^{n}(y))| |q_{0}^{n}(y)| dy ds

+ ∫_{0}^{t} ∫_{ℝ^{2}} |K^{α}(η(x), η^{n}(y)) − K^{α}(η^{n}(x), η^{n}(y))| |q_{0}^{n}(y)| dy ds

≡ ∫_{0}^{t}(J_{1} + J_{2} + J_{3}) ds  (5.12)

We find, after a change of variables, that

sup_{x∈ℝ^{2}} J_{1} = sup_{x∈ℝ^{2}} |∫_{ℝ^{2}} K^{α}(x, y) (q_{0} − q_{0}^{n})(η^{−1}(y)) dy| .  (5.13)

By Lemma 10 below with ϕ(x − y) = K^{α}(x, y) and q_{n}(y) = (q_{0} − q_{0}^{n})(η^{−1}(y)), this expression converges to zero as n → ∞. Moreover, by Lemma 5,

J_{2} ≤ sup_{x∈ℝ^{2}} sup_{y∈ℝ^{2}}|K^{α}(η(x), η(y)) − K^{α}(η(x), η^{n}(y))| ∥q_{0}^{n}∥_{M}
≤ sup_{y∈ℝ^{2}} \frac{c}{α} φ(\frac{η(y) − η^{n}(y)}{α}) ∥q_{0}^{n}∥_{M}  (5.14)

and

J_{3} ≤ \frac{c}{α} φ(\frac{η(x) − η^{n}(x)}{α}) ∥q_{0}^{n}∥_{M} .  (5.15)

By inserting these estimates back into (5.12) and taking the supremum in x on both sides, we obtain an integral inequality that can be solved with the log-Gronwall inequality exactly as in the proof of Lemma 8. The result then follows.

Lemma 10. Let {q_{n}} be a sequence of measures in M(ℝ^{2}) converging weakly to zero with uniformly bounded total variation and uniform decay at infinity. Further assume that ϕ is a continuous test function with ϕ → 0 as |x| → ∞. Then

lim_{n→∞} sup_{x∈ℝ^{2}} ∫_{ℝ^{2}} ϕ(x − y) q_{n}(y) dy = 0 .  (5.16)

Proof. Set M = sup_{n}∥q_{n}∥_{M} and M′ = sup_{x}|ϕ(x)|. Let ε > 0 be fixed. By assumption on the {q_{n}}, there exists an R > 0 such that for every n ∈ ℕ,

∫_{|y|>R}|q_{n}(y)| dy < \frac{ε}{6M′} .  (5.17)

Moreover, there exists an R′ > 0 such that |ϕ(x)| < ε/(2M) for |x| > R′. Since ϕ is uniformly continuous on compact sets, there exists δ > 0 such that |ϕ(x)−ϕ(x′)| < ε/(3M) for all x, x′ ∈ B(0, 2R+R′) with |x−x′| < δ. Cover B(0, R + R′) with finitely many balls of radius δ and denote the centers of these balls by x_{i}, i ∈ I. Choose N large enough such that for n ≥ N,

max_{i∈I} |∫_{ℝ^{2}} ϕ(x_{i} − y) q_{n}(y) dy| < \frac{ε}{3} .  (5.18)

Then for |x| < R + R′ there exists an i ∈ I such that |x − x_{i}| < δ, and

|∫_{ℝ^{2}}ϕ(x − y) q_{n}(y) dy|

≤ ∫_{ℝ^{2}} |ϕ(x − y) − ϕ(x_{i} − y)| |q_{n}(y)| dy + |∫_{ℝ^{2}} ϕ(x_{i} − y) q_{n}(y) dy|

≤ sup_{|y|≤R}|ϕ(x − y) − ϕ(x_{i} − y)| ∥q_{n}∥_{M}

+ 2 sup_{x∈ℝ^{2}}|ϕ(x)| ∫_{|y|>R} |q_{n}(y)| dy + \frac{ε}{3}

≤ \frac{ε}{3M} M + 2M′ \frac{ε}{6M′} + \frac{ε}{3} = ε .  (5.19)

On the other hand, if |x| ≥ R + R′, then

|∫_{ℝ^{2}}ϕ(x − y) q_{n}(y) dy|

≤ sup_{|x|≥R′}|ϕ(x)| ∫_{|y|<R} |q_{n}(y)| dy + sup_{x∈ℝ^{2}}|ϕ(x)| ∫_{|y|<R} |q_{n}(y)| dy

≤ \frac{ε}{2M} M + M′ \frac{ε}{6M′} < ε .  (5.20)

This completes the proof.

Acknowledgments

The authors thank Thomas Beale, Darryl Holm, Robert Krasny, David Levermore, Christian Lubich, Jerry Marsden, Tudor Ratiu, and Edriss Titi for interesting discussions. SS was partially supported by the NSF-KDI grant ATM-98-73133 and the Alfred P. Sloan Foundation Research Fellowship. MO was partially supported through German Science Foundation SFB 382.

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