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Three-Dimensional Stability of Burgers Vortices: the Low Reynolds Number Case

In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change in the circulation number (which characterizes the limiting vortex completely.) We also give a rigorous proof of the existence and stability of non-axisymmetric Burgers vortices provided the Reynolds number is sufficiently small, depending on the asymmetry parameter.


Introduction
Numerical simulations of turbulent flows have lead to the general conclusion that vortex tubes serve as important organizing structures for such flows -in the memorable phrase of [9] they form the "sinews of turbulence". After the discovery by Burgers [1] of the explicit vortex solutions of the three-dimensional Navier-Stokes equation which now bear his name, these solutions have been used to model various aspects of turbulent flows [17]. It was also observed in numerical computations of fluid flows that the vortex tubes present in these simulations usually did not exhibit the axial symmetry of the explicit Burgers solution, but rather an elliptical core region. This lead to a search for non-axisymmetric vortices [13], [9], [6]. While no rigorous proof of their existence was available until recently, perturbative calculations and extensive numerical simulations have lead to the expectation that stationary vortical solutions of the three-dimensional Navier-Stokes equation do exist for any Reynolds number and all values of the asymmetry parameter (which we define below) between zero and one.
When addressing the stability of Burgers vortices, it is very important to specify the class of allowed perturbations. If we consider just two-dimensional perturbations (i.e., perturbations which do not depend on the axial variable), then fairly complete answers are known. Robinson and Saffman [13] computed perturbatively the eigenvalues of the linearized operator at the Burgers vortex and proved its stability for sufficiently small Reynolds numbers. Numerical computations of these eigenvalues were performed by Prochazka and Pullin [10], and no instability was found up to Re = 10 4 . A similar conclusion was drawn for non-symmetric vortices [11]. The first mathematical work is [5], where we proved that the axisymmetric Burgers vortex is globally stable with respect to integrable, two-dimensional perturbations, for any value of the Reynolds number. Decay rates in time of spatially localized perturbations were also computed, explaining partially the numerical results of [10]. Building on this work the existence and local stability of slightly asymmetric vortices with respect to two dimensional perturbations was proved in [4] for arbitrary Reynolds numbers.
The stability issue is much more difficult if we allow for perturbations which depend on the axial variable too, and very few results have been obtained so far in this truly three-dimensional case. One early study by Leibovich and Holmes [7] concluded that one could not prove global stability for any Reynolds number solely by means of energy methods. Using a kind of Fourier expansion in the axial variable, Rossi and Le Dizès [14] showed that the point spectrum of the linearized operator is associated with purely twodimensional perturbations. Crowdy [2] obtained a formal asymptotic expansion of the eigenfunctions in the axial variable. In an important recent work, Schmid and Rossi [16] rewrote the linearized equations in a form which allowed them to compute numerically the evolution of various Fourier modes, from which they concluded that eventually all perturbative modes will be damped out.
In this paper we address rigorously the existence of non-axisymmetric vortices and the stability with respect to three-dimensional perturbations of both the symmetric and non-symmetric vortex solutions. More precisely we will prove that, for all values of the asymmetry parameter between zero and one, non-axisymmetric vortices exist at least for small Reynolds numbers. In addition, we show that this family of vortex solutions is, in the language of dynamical systems theory, asymptotically stable with shift. That is to say, if we take initial conditions that are small perturbations of a vortex solution, the resulting solution of the Navier-Stokes equation will converge toward a vortex solution, but not, in general, the one which we initially perturbed. We also give a formula for computing the limiting vortex toward which the solution converges.
Under reasonable assumptions which will be satisfied for the solutions we consider, the rotational part U of the velocity can be recovered from the vorticity Ω by means of the Biot-Savart law: For notational simplicity we begin by discussing the axisymmetric case where γ 1 = γ 2 = −γ 3 /2. In this situation, it is well-known [1] that (3) has a family of explicit stationary solutions of the form Ω = ΓΩ B , where Γ ∈ R is a parameter and Here x ⊥ = (x 1 , x 2 ), |x ⊥ | 2 = x 2 1 + x 2 2 , and γ ≡ γ 3 > 0. The velocity field corresponding to ΓΩ B is ΓÛ B , whereÛ These solutions are called the axisymmetric Burgers vortices. Observe that R 2Ω B dx ⊥ = 1, so that the parameter Γ represents the circulation of the velocity field ΓÛ B at infinity (in the horizontal plane x 3 = 0). Following [9], we define the Reynolds number associated with the Burgers vortex ΓÛ B as Our principal result concerns the evolution of solutions of (3) with initial conditions that are close to a Burgers vortex. Unlike in much previous work the perturbations we consider do not merely depend on the transverse variables x ⊥ , but also on x 3 . We prove that any solution of (3) starting sufficiently close to the Burgers vortex with circulation Γ converges as t → +∞ toward a Burgers vortex with circulation Γ ′ close to Γ, and we give an explicit formula for computing the difference Γ ′ −Γ in terms of the initial perturbation.
We now introduce some function spaces to measure the size of our perturbations. Roughly speaking, we require the perturbations to decay as inverse powers of x ⊥ as |x ⊥ | → ∞, but need only boundedness in x 3 . To be specific, we use the following weight function b(x ⊥ ) = (1 + γx 2 Given any m ≥ 0, we define L 2 (m) = {ω : In other words, a function ω belongs to L 2 (m) if and only if ω, |x 1 | m ω, |x 2 | m ω, and |x 1 x 2 | m ω are square integrable over R 2 . For later use, we observe that L 2 (m) is contin- Our main space X 2 (m) will be the set of all ω : for all x 3 ∈ R, and such that the map x 3 → ω(·, x 3 ) is bounded and continuous from R into L 2 (m). As is easily verified, is a vector field whose components are elements of . A similar abuse of notation will occur for other function spaces too.
Consider initial conditions for the vorticity equation (3) which are a perturbation of the Burgers vortex: with Γ ∈ R and ω 0 ∈ X 2 (m) 3 . Define Just to make sure the notation is clear, in the integrand ω 0 3 refers to the third component of the initial perturbation ω 0 .

Remark 1.3
Here and in the sequel, all constants are independent of the strain γ and the viscosity ν. In fact, both parameters will shortly be eliminated by a rescaling.
The proof of Theorem 1.2 uses ideas from our analysis of the stability of the twodimensional Oseen vortices in [5]. The main observation is that, if we linearize equation (3) at the Burgers vortex ΓΩ B for small Γ, we obtain a small perturbation of a nonconstant coefficient differential operator for which we can explicitly compute an integral representation of the associated semigroup. This semigroup decays exponentially when acting on functions ω ∈ X 2 (m) 3 provided ω 3 ∈ X 2 0 (m), where Thus an important step in the proof consists in decomposing the perturbation as ω( By constructionω 3 ∈ X 2 0 (m), henceω(x, t) will decay exponentially to zero by the remark above. Now, the crucial point is that ϕ(x 3 , t) satisfies the amazingly simple equation which can be solved explicitly, see (51) below. From the solution formula we see that ϕ(x 3 , t) converges uniformly on compact sets to the value δΓ as t → ∞, and (13) follows. In other words, ϕ is a "zero mode" which is responsible for the fact that the family of Burgers vortices is only asymptotically stable with shift.
While the axisymmetric vortex solution has been extensively studied because of the explicit formulas for its velocity and vorticity fields, numerical experiments on turbulent flows seem to indicate that the vortex tubes that are prominent in these flows are not symmetric, but rather elliptical in cross section. A natural way to obtain such vortices is to assume that the straining flow is not axisymmetric, namely where γ > 0 and λ ∈ [0, 1) is an additional parameter which measures the asymmetry of the strain. While no explicit formulas for the vortex are known when λ > 0, extensive perturbative and numerical investigations indicate that there do exist stationary solutions of (3) for 0 ≤ λ < 1, which for λ close to zero are small perturbations of the vorticity field of the axisymmetric Burgers vortex [13], [9], [11]. As in the symmetric case, there is in fact a family of vortices for each value of λ ∈ [0, 1) parametrized by the total circulation Γ, but when λ > 0 these solutions are not just multiples of one another. In Section 2 we give a simple but rigorous proof of the existence of these nonaxisymmetric vortex solutions for all values of λ ∈ [0, 1), provided the circulation Reynolds number of the flow is sufficiently small (depending on λ). A complementary result is obtained in [4] where we prove that, if λ > 0 is sufficiently small, non-axisymmetric vortex solutions exist for all values of the Reynolds number.
Remark 1.6 Theorem 2 shows that the asymmetric Burgers vortex Ω B (x ⊥ ; ρ, λ) decays rapidly as |x ⊥ | → ∞, since the parameter m > 3/2 is arbitrary (note, however, that the constants R 1 , K 1 depend on m). In fact, proceeding as in [4], it is possible to show that Ω B has a Gaussian decay as |x ⊥ | → ∞. Moreover, Ω B is also a smooth function of x ⊥ , see Remark 2.3 below.
Finally, we prove that these families of non-symmetric vortices are stable in the same sense as the symmetric Burgers vortices are. Theorem 1.7 Fix m > 3/2, λ ∈ [0, 1), and assume that (γ 1 , γ 2 , γ 3 ) is given by (15) with γ = 1. For any µ ∈ (0, 1 2 (1−λ)), there exist R 2 (λ) > 0 and then the solution Ω(x, t) of (17) with initial data Ω 0 converges as t → +∞ to the vortex solution (12). More precisely, for any compact interval I ⊂ R, we have The symmetric case λ = 0 is included in Theorem 1.7, which therefore subsumes Theorem 1.2. Note however that the assumptions on the initial data are more restrictive when λ > 0, because we then need a condition on ∂ 3 ϕ 0 . This is due to the fact that non-axisymmetric Burgers vortices with different circulation numbers are not multiples of one another.
The rest of the text is organized as follows. In Section 2, we prove the existence of non-axisymmetric Burgers vortices for small Reynolds numbers. The core of the paper is Section 3, where we show that these families of vortices are asymptotically stable with shift. Section 4 is an appendix where we collect various estimates on the semigroup associated to the linearized vorticity equation, together with a few remarks concerning the Biot-Savart law.

Existence of non-axisymmetric Burgers vortices
The properties of non-axisymmetric Burgers vortices seem first to have been studied by Robinson and Saffman [13] who used perturbative methods to investigate their existence for small values of the Reynolds number. There were many further investigations in the intervening years -we mention particularly the perturbative study of the large Reynolds number limit of these vortices by Moffatt, Kida and Ohkitani [9], and the numerical work of Prochazka and Pullin [11]. However, as far as we know there has been no rigorous proof of the existence of these types of solutions and so in this section we present a simple argument which proves the existence of non-symmetric vortices in the case of small Reynolds number.
Fix λ ∈ [0, 1) and assume that γ 1 , γ 2 , γ 3 are given by (15) with γ = 1. Motivated by the perturbative calculations of [13] we look for stationary solutions of (17) of the form for some ρ ∈ R (recall that |ρ| is the Reynolds number). Since Ω B depends only on the horizontal variable x ⊥ = (x 1 , x 2 ) and has only the third component nonzero, the associated velocity field U B depends only on x ⊥ and has only the first two components nonzero. Thus U B is naturally identified with a two-dimensional velocity fieldŪ B which can be computed using the two-dimensional version of the Biot-Savart law: Inserting these expressions into (17), we see that Ω B satisfies the scalar equation where L ⊥ and M are the differential operators Here we have used the natural notations ∇ ⊥ = (∂ 1 , ∂ 2 ) and ∆ ⊥ = ∂ 2 1 + ∂ 2 2 . We shall solve (24) in the weighted space L 2 (m) defined by (9) (with γ = ν = 1). Our approach rests on the fact that the spectrum of the linear operator L ⊥ + λM in L 2 (m) can be explicitly computed, see Section 4.2. If m > 1/2, this operator turns out to be invertible on the invariant subspace L 2 0 (m) defined by This allows to rewrite (24) as a fixed point problem which is easily solved by a contraction argument. As a preliminary step, letV λ (x ⊥ ) be the two-dimensional velocity field obtained from G λ (x ⊥ ) by the Biot-Savart law (23). Using (18) and (25) one can easily verify that where ω ∈ L 2 0 (m) andū is the velocity obtained from ω by the Biot-Savart law (23). With these notations, finding a solution to (24) is equivalent to solving Note that (ρV λ +ū) · ∇ ⊥ (ρG λ + ω) = ∇ ⊥ · ((ρV λ +ū)(ρG λ + ω)) sinceV λ andū are divergence-free. Thus the right-hand side of the (28) has zero mean as expected. The next proposition ensures that the operator L ⊥ + λM is invertible on L 2 0 (m) and that (L ⊥ + λM) −1 ∇ ⊥ defines a bounded operator from L p (m) into L 2 0 (m), where L p (m) is the weighted space defined in (72).
Moreover, if p ∈ (1, 2], there exists C(m, λ, p) > 0 such that, for all g ∈ L p (m), Proof: Let T λ (t) denote the strongly continuous semigroup generated by L ⊥ + λM, the properties of which are studied in the Section 4.2. If f ∈ L 2 0 (m), we know from (73) that Thus L ⊥ + λM is invertible on L 2 0 (m) and we have the Laplace formula Combining (31), (32), we easily obtain (29). Assume now that f = ∂ i g for some i ∈ {1, 2} and some g ∈ L p (m). Using (32) and (74) or (75), we obtain an estimate of the form Since p > 1, the singularity in the integral at t = 0 is integrable and (30) follows.
We can now rewrite (28) as ω = F λ,ρ (ω), where F λ,ρ : For any r > 0, let B m (0, r) denote the closed ball of radius r centered at the origin in L 2 0 (m). The main result of this section is: and ω λ,ρ is a smooth function of both λ and ρ.
Choosing r = 2R 1 , we obtain the existence and uniqueness claim in Proposition 2.2. Then . Finally, the smoothness property is a immediate consequence of the implicit function theorem. Indeed, the map , and the partial differential is invertible at ω = ω λ,ρ , and the implicit function theorem implies that ω λ,ρ is a smooth function of both λ and ρ.
where ω λ,ρ is as in Proposition 2.2, and we denote bȳ U B (x ⊥ ; ρ, λ) the two-dimensional velocity field obtained from Ω B by the Biot-Savart law (23). Then is a stationary solution of (17) which has all the desired properties. In particular, since ω λ,ρ ∈ L 2 0 (m), we have while the fact that ω λ,ρ ∈ B m (0, K 1 ρ 2 ) implies that (20) holds. For later use, we observe that there exists C(λ, m) > 0 such that, for |ρ| ≤ R 1 , for any k ∈ N. This shows that the asymmetric Burgers vortex (the space of all continuous and bounded functions on R 2 ) and we have the analogue of (38): 3 Stability with respect to three-dimensional perturbations We now prove that the family of vortices constructed in the previous section is asymptotically stable with shift, provided the circulation Reynolds number is sufficiently small, depending on the asymmetry parameter λ ∈ [0, 1). In particular, our result applies to the classical family of symmetric Burgers vortices (λ = 0). Throughout this section we fix some λ ∈ [0, 1). For |ρ| sufficiently small we denote by Ω B (x ⊥ ; ρ), U B (x ⊥ ; ρ) the asymmetric vortex (36) with total circulation ρ (to simplify the notation, we omit the dependence on λ). As we mentioned in the introduction, if we slightly perturb the vortex Ω B (·; ρ) the solution of the vorticity equation will converge toward another vortex with a possibly different circulation. This means that we must allow the parameter ρ to depend on time. Also, since the perturbations we consider may depend on the axial variable x 3 , it turns out to be convenient to approximate the solutions by vortices with different circulation numbers in different x 3 sections. In other words, we will consider solutions of (17) of the form In view of (37), it is obvious that any pertubation of Ω B (·; ρ) that is integrable with respect to the transverse variables x ⊥ can be decomposed in a unique way as in (41). Similarly, we write the rotational part of the velocity field as whereŨ B (x, t; ρ, ϕ) is the velocity field obtained from the vorticity Ω B (x ⊥ ; ρ + ϕ(x 3 , t)) by the Biot-Savart law (4). It will be shown in Proposition 4.11 thatŨ B (x, t; ρ, ϕ) is a small perturbation of U B (x ⊥ ; ρ + ϕ(x 3 , t)) if ϕ varies slowly in the x 3 direction, namely there exists C(λ) > 0 such that Let ω = (ω 1 , ω 2 , ω 3 ) T and u = (u 1 , u 2 , u 3 ) T denote the remainder terms in (41) and (42) respectively. By construction, u is the velocity field obtained from ω by the Biot-Savart law (4).
In broadest terms, our strategy is to show that ω(x, t) and ∂ 3 ϕ(x, t) converge to zero as time tends to infinity, so that the vorticity Ω(x, t) approaches one of the vortices Ω B (·; ρ ′ ) constructed in Section 2. With that in mind, we now write out the evolution equations for ω and ϕ.
• The linear operator L is the leading order part of the equation, which takes into account the diffusion and the effects of the background strain: Here γ 1 , γ 2 , γ 3 are given by (15) with γ = 1, and • The term P ϕ ω = ∇ × (Ũ B × ω + u × Ω B ) describes the linear interaction between the perturbation and the modulated vortex, namely: Here and in the sequel, to simplify the notation, we write Ω B instead of Ω B (·; ρ + ϕ) and U B instead ofŨ B (·; ρ, ϕ).
A simple calculation shows that H i (ϕ) = ∂ 3 (Ũ B i Ω B ) for i = 1, 2. The third component of H(ϕ) has a more complicated expression: where L ⊥ and M are defined in (25). Equation (44) governs the evolution of both ϕ and ω. To separate out the evolution equation for ϕ, we recall that ω satisfies the constraint R 2 ω 3 (x ⊥ , x 3 , t) dx ⊥ = 0. If we integrate the third component of the vectorial equation (44) with respect to the transverse variables x ⊥ , the first three terms in the right-hand side give no contribution, as can be seen from the formulas (45), (46), (47). So we must impose As is clear from (24), the first term in the right-hand side of (48) has zero mean with respect to x ⊥ , and so does the second term because it is explicitly in divergence form. On the other hand, differentiating (37) with respect to ρ, we obtain the identities R 2 ∂ ρ Ω B dx ⊥ = 1 and R 2 ∂ 2 ρ Ω B dx ⊥ = 0. Thus (49) gives the evolution equation for ϕ: Remarkably, this equation is linear and completely decoupled from the rest of the system. As is easily verified, the solution of (50) with initial data ϕ(x 3 , 0) = ϕ 0 (x 3 ) is given by the explicit formula where The following simple estimates will be useful: , the solution of (50) with initial data ϕ 0 satisfies If moreover ∂ 3 ϕ 0 ∈ L ∞ (R), we also have Proof: Since G t L 1 = 1, it follows immediately from (51) that ϕ(·, t) L ∞ ≤ ϕ 0 L ∞ . If ∂ 3 ϕ 0 ∈ L ∞ (R), the same argument gives (54), because To prove the second estimate in (53), we use the last expression in (55) and observe that

Remark 3.2 Proposition 3.1 shows in particular that (51) defines a semigroup of bounded linear operators on
, the space of all bounded and continuous functions on R equipped with the L ∞ norm. It is easy to verify that this semigroup is not strongly continuous in time, due to the dilation factor e −t in (51) which in turn originates in the unbounded advection term x 3 ∂ 3 in (50). However, if we equip C 0 b (R) with the (weaker) topology of uniform convergence on compact sets, then (51) defines a continuous function of time.
We now return to the evolution equation for ω. Using (48), equation (50) for ϕ, and equation (24) satisfied by the asymmetric vortex Ω B , we obtain for the inhomogeneous term H(ϕ) the simpler expression where as usual U B = U B (·; ρ + ϕ). Before starting the rigorous analysis, let us briefly comment here on why we expect solutions of (44) to go to zero as t goes to infinity. Given m > 3/2, we assume that ω i ∈ X 2 (m) for i = 1, 2, 3, where X 2 (m) is the space defined in (10). By construction, ω 3 then belongs to the subspace X 2 0 (m) given by (14). As we show in Section 4.3, the linear operator L has spectrum that lies in the half-plane {z ∈ C | Re z ≤ − 1 2 (1−λ)} when acting on X 2 (m)×X 2 (m)×X 2 0 (m). Thus, the semigroup generated by this operator can be expected to decay like exp(− 1 2 (1−λ)t). The remaining linear terms in the equation, namely P ϕ (ω), contain a factor of the vortex solution which is proportional to ρ + ϕ (see (39) and (100)) and hence, for small Reynolds number, they will be a small perturbation of L and will not destroy the exponential decay. The same is true for the nonlinear terms N(ω) provided we restrict ourselves to sufficiently small perturbations. Finally, the inhomogeneous term H(ϕ) decays at least like e −t by (43) and Proposition 3.1, so we expect the solution ω(x, t) of (44) to converge exponentially to zero if the initial data are sufficiently small.
We now put these heuristic arguments into a rigorous form. Let X(m) be the Banach space X 2 (m)×X 2 (m)×X 2 0 (m) equipped with the norm ω X(m) = ω 1 X 2 (m) + ω 2 X 2 (m) + ω 3 X 2 (m) . As is shown in Proposition 4.6, the linear operator L is the generator of a semigroup e tL of bounded operators on X 2 (m), hence the same is true for the operator L acting on X(m). A natural idea is then to use Duhamel's formula to rewrite (44) as an integral equation: which can then be solved by a fixed point argument. A problem with this approach is that the semigroup e tL fails to be strongly continuous on X 2 (m), essentially for the reason mentioned in Remark 3.2. To restore continuity, it is thus necessary to equip X 2 (m) with a weaker topology. For any n ∈ N * we define the seminorm and we denote by X 2 loc (m) the space X 2 (m) equipped with the topology defined by the family of seminorms (58) for n ∈ N * , i.e. the topology of the Fréchet space C 0 (R, L 2 (m)). In other words, a sequence ω k converges to zero in X 2 loc (m) if and only if |ω k | X 2 n (m) → 0 as k → ∞ for all n ∈ N * , namely if ω k (x 3 ) converges to zero in L 2 (m) uniformly on compact sets in x 3 . We define the product space X loc (m) in a similar way. Then Proposition 4.6 shows that the semigroup e tL is strongly continuous on X loc (m), and the integrals in (57) can be defined as X loc (m)-valued Riemann integrals, see Corollary 4.7 and Remark 4.8.

Appendix
In this appendix we collect a number of technical estimates used in the main body of the paper. They relate mostly to the behavior of the semigroup generated by the linearization of the vorticity equation around the Burgers vortex. We also prove some estimates relating the vorticity field to the corresponding velocity field defined by the Biot-Savart law.

The one-dimensional Fokker-Planck operator
Fix α > 0, and consider the one-dimensional linear equation where x ∈ R and t ≥ 0. If ω(x, t) =ω( √ αx, αt), then ∂ tω = L 1ω hence we could assume without loss of generality that α = 1. However for our purposes it is more convenient to keep α > 0 arbitrary. The linear operator L α is formally conjugated to the Hamiltonian of the harmonic operator in quantum mechanics: As is well-known, the spectrum of L α in L 2 (R) is a sequence of simple eigenvalues: σ(L α ) = − nα 2 n = 0, 1, 2, . . . , and the associated eigenfunctions are the Hermite functions h n (x) = e αx 2 /8 ∂ n x e −αx 2 /4 . This observation, however, is not sufficient to determine the whole spectrum of L α because we want to consider this operator acting on a space of functions with algebraic (rather than Gaussian) decay at infinity.
For any m ≥ 0 and p ≥ 1 we define the space L p (m) = {f ∈ L p (R) | w m f ∈ L p (R)}, where w(x) = (1 + x 2 ) 1/2 . This Banach space is equipped with the natural norm The parameter m determines the decay rate at infinity of functions in L p (m). For instance, it is easy to verify that L 2 (m) ֒→ L 1 (R) if (and only if) m > 1/2, because in that case w −m ∈ L 2 (R) so that any f ∈ L 2 (m) satisfies by Hölder's inequality. If m > 1/2, we thus define This closed subspace of L 2 (m) is clearly invariant under the evolution defined by (66).
In ( [3], Appendix A) it is shown that the spectrum of L α in L 2 (m) is Thus, in addition to the discrete spectrum of the harmonic oscillator, the operator L α also has essential spectrum due to the slow spatial decay of functions in L 2 (m). Note however that this essential spectrum can be pushed far away from the imaginary axis by taking m ≥ 0 sufficiently large. Therefore, if m is large, the relevant part of the spectrum of L α is still given by the first few eigenvalues of L α . In particular zero is an isolated eigenvalue of L α if m > 1/2, and the rest of the spectrum is strictly contained in the left-half plane.
If we restrict ourselves to the invariant subspace L 2 0 (m), the spectrum of L α is unchanged except for the zero eigenvalue (which is absent).

The three-dimensional semigroup
This section is devoted to the three-dimensional equation where x ∈ R 3 , t ≥ 0, and α 1 ≥ α 2 > 0. In the particular case where α 1 = 1 + λ and It is important to realize that the evolution defined by (77) is essentially contracting in the transverse variables x ⊥ = (x 1 , x 2 ) and expanding in the axial variable x 3 . This is due to the signs of the advection terms, which in turn originate in our choice of the straining flow (2). For this reason we can assume that the solutions of (77) decay to zero as |x ⊥ | → ∞, but we cannot impose any decay in the x 3 variable (otherwise the solutions will not stay uniformly bounded for all times in the corresponding norm). This motivates the following choice of our function space. For p ≥ 1 and m ≥ 0, we introduce the Banach space equipped with the norm For any n ∈ N * we also define the seminorm |ω| X p n (m) = sup and we denote by X p loc (m) the space X p (m) equipped with the topology defined by the family of seminorms (79) for n ∈ N * . For later use, we observe that the ball {ω ∈ X p (m) | ω X p (m) ≤ R} is closed in X p loc (m) for any R > 0. At least formally, the space X 2 (m) can be thought of as the tensor product C 0 b (R) ⊗ L 2 (m), i.e. the space generated by linear combinations of elements of the form ω(x ⊥ , and g ∈ L 2 (m). In this picture, the linear operator defined by (77) can be decomposed asL where G t is defined in (52), and we know from Section 4.2 that L α 1 ,α 2 generates a strongly continuous semigroup in L 2 (m). Thus we expect thatL α 1 ,α 2 will generate a semigroup {S t } t≥0 in X 2 (m) given by S t = e tL 3 ⊗ e tLα 1 ,α 2 , or explicitly We shall prove that these heuristic considerations are indeed correct in the sense that (81) defines a semigroup of bounded operators in X 2 (m) with the property that ω(t) = S t ω is the solution of (77) with initial data ω ∈ X 2 (m). However, the map t → S t ω is not continuous in the topology of X 2 (m), but only in the (weaker) topology of X 2 loc (m). This is due to the fact that equation (77) has "infinite speed of propagation" in the sense that the advection term in the vertical variable is unbounded, see Remark 3.2.
On the other hand, by (86), we have for all k ∈ N: where ε k = |z|≥k G(z) dz → 0 as k → ∞. Since |x 3 e −t + c(t) 1/2 z 3 | ≤ n + k whenever |x 3 | ≤ n and |z 3 | ≤ k, we deduce that for all n ∈ N * : This bound implies that S t : B R → X 2 loc (m) is continuous, uniformly in time on compact intervals.
Furthermore, if ω ∈ X 2 (m) and t > 0, we have hence proceeding as above we find for all n, k ∈ N * : The last term goes to zero as t → 0+ because e tLα 1 ,α 2 is a strongly continuous semigroup on L 2 (m) and ω : [−n, n] → L 2 (m) is continuous (hence has compact range). Similarly, for each k ∈ N, the integral goes to zero as t → 0+ by Lebesgue's dominated convergence theorem, because ω : [−n−k, n+k] → L 2 (m) is uniformly continuous. Since ε k → 0 as k → ∞, we conclude that S t ω → ω in X 2 loc (m) as t → 0+. Then, using the semigroup property, we deduce that the map t → S t ω is continuous to the right at any t ≥ 0. Finally, if t > ε > 0, we have S t ω − S t−ε ω = S t−ε (S ε ω − ω), hence by (88) Since |S ε ω − ω| X 2 n+k (m) → 0 as ε → 0+ for all k, n ∈ N * , this shows that t → S t ω is also continuous to the left at any t > 0.
Next, using (86), (80), and the explicit formula for e tLα 1 ,α 2 , it is rather straightforward to verify that, for any ω ∈ X 2 (m), the map ( is smooth and satisfies (77) for t > 0. Thus S t ω is indeed the solution of (77) with initial data ω.
Proof: For any t ∈ [0, T ], we define ψ t : [0, t] → X 2 (m) by ψ t (s) = S t−s f (s). Since f ∈ C 0 ([0, T ], X 2 loc (m)) and since the semigroup S t is continuous on X 2 loc (m) as described in Proposition 4.6, it is easy to verify that the map ψ t : [0, t] → X 2 loc (m) is also continuous. As X 2 loc (m) is a subspace of the Fréchet space C 0 (R, L 2 (m)), the integral t 0 ψ t (s) ds can be defined as in ( [15], Theorem 3.17). However, in the present case, we can also use the following "pedestrian" construction (which agrees with the general one). For any n ∈ N * , we define where χ n denotes the map x 3 → 1 [−n,n] (x 3 ). Clearly χ n ψ t (s) is a continuous function of s ∈ [0, t] with values in the Banach space C 0 ([−n, n], L 2 (m)), hence F n (t) ∈ C 0 ([−n, n], L 2 (m)) can be defined for any t ∈ [0, T ] as a Banach-valued Riemann integral. Using again the continuity of the semigroup S t one finds that F n : [0, T ] → C 0 ([−n, n], L 2 (m)) is continuous and satisfies (Note that t → N m (t) and t → f (t) X 2 (m) are lower semicontinuous, hence measurable.) Now, for each t ∈ [0, T ], it is clear that (F m (t))(x 3 ) = (F n (t))(x 3 ) if |x 3 | ≤ n ≤ m, hence there is a unique F (t) ∈ C 0 (R, L 2 (m)) such that (F (t))(x 3 ) = (F n (t))(x 3 ) whenever |x 3 | ≤ n. By construction, F (t) X 2 (m) ≤ is bounded in X 2 (m) and continuous in X 2 loc (m). In that case, for each n ∈ N * and each t ∈ (0, T ], F n (t) ≡ χ n F (t) is defined by a "generalized" Riemann integral, because the integrand has a singularity at s = t.

The three-dimensional Biot-Savart law
In this final section, we discuss the three-dimensional Biot-Savart law, namely We first prove the analogue of Proposition 4.4 in the spaces X p (m) defined by (78).