Well-posedness of the 2D Euler equations when velocity grows at infinity

We prove the uniqueness and finite-time existence of bounded-vorticity solutions to the 2D Euler equations having velocity growing slower than the square root of the distance from the origin, obtaining global existence for more slowly growing velocity fields. We also establish continuous dependence on initial data.

λ > 0, is a scaled radial cutoff function (see Definition 1.3). Also, we have used the notation, v * · w = v i * w i if v and w are vector fields, A * · B = A ij * B ij if A, B are matrix-valued functions on R 2 , where * denotes convolution and where repeated indices imply summation.
The Biot-Savart law, u = K * ω, recovers the unique divergence-free vector field u decaying at infinity from its vorticity (scalar curl) ω. It does not apply to bounded vorticity, bounded velocity solutions-indeed this is the greatest difficulty to overcome with such solutions-but because of the manner in which a λ cuts off the Biot-Savart kernel in (1.1), we see that all the terms in Serfati's identity are finite. In fact, there is room for growth at infinity both in the vorticity and the velocity, though to avoid excessive complications, we only treat growth in the velocity. Because ∇∇ ⊥ (1 − a λ )K j (x) decays like |x| −3 , we see that as long as |u(x)| grows more slowly than |x| 1/2 , all the terms in (1.1) will at least be finite.
In brief, we will establish uniqueness of solutions having o(|x| 1/2 ) growth along with finitetime existence. We will obtain global existence only for much more slowly growing velocity fields.
To give a precise statement of our results, we must first describe the manner in which we prescribe the growth of the velocity field at infinity and give our formulation of a weak solution, including the function spaces in which the solution is to lie.
In what follows, an increasing function means nondecreasing; that is, not necessarily strictly increasing.
We define four types of increasingly restrictive bounds on the growth of the velocity, as follows: We call h a global well-posedness growth bound if for some such µ,  Definition 1.4 (Lagrangian solution). Fix T > 0 and let h be a growth bound. Assume that u ∈ C(0, T ; S h ), let ω = curl u := ∂ 1 u 2 − ∂ 2 u 1 , and let X be the unique flow map for u. We say that u is a solution to the Euler equations in S h without forcing and with initial velocity u 0 = u| t=0 in S h if the following conditions hold: where ω 0 := curl u 0 ; (2) Serfati's identity (1.1) holds for all λ > 0.
The existence and uniqueness of the flow map X in Definition 1.4 is assured by u ∈ C([0, T ]; S h ) (see Lemma 5.1). It also then follows easily that the vorticity equation, ∂ t ω + u · ∇ω = 0, holds in the sense of distributions-so u is also a weak Eulerian solution. (See also the discussion of Eulerian versus Lagrangian solutions in Section 2.) Our main results are Theorem 1.5 through Theorem 1.10. where J(t, x) = (a h(x) K) * (ω 0 1 − ω 0 2 ) • X −1 1 (t) (x) − (a h(x) K) * (ω 0 1 − ω 0 2 ) (X 1 (t, x)). (1.5) Define , L(t) := u 1 (t, X 1 (t, x)) − u 2 (t, X 2 (t, x)) ζ(x) For all t ≥ 0, we have Uniqueness is an immediate corollary of Theorem 1. Proof. Apply Theorem 1.6 with u 0 1 = u 0 2 = u(0) so that a(T ) = 0 and set ζ = h, noting that ζ/h = 1 and ζh = h 2 are both growth bounds. Corollary 1.8 applies Theorem 1.5 and Corollary 1.7 to two explicit well-posedness growth bounds. The proof is simply a matter of verifying that the two given well-posedness growth bounds actually satisfy the pertinent conditions in Definition 1.1. (Since the proof is very "calculational," we defer it to Appendix A.) We note that the existence of solutions for h 2 was shown in [5] using different techniques. Corollary 1.8. Let h 1 (r) := (1+ r) α for some α ∈ [0, 1/2), h 2 (r) := log 1 4 (e+ r). If u 0 ∈ S h j , j = 1 or 2, then there exists a unique solution to the Euler equations in S h j on [0, T ] as in Definition 1.4 for some T > 0. If u 0 ∈ S h 2 then T can be made arbitrarily large.
Loosely speaking, Corollary 1.8 says that solutions in S h are unique and exist for finite time as long as h 2 is sublinear, while global-in-time solutions exist for velocities growing very slowly at infinity. These slowly growing velocities are somewhat analogous to the "slightly unbounded" vorticities of Yudovich [18], which extends the uniqueness result for bounded vorticities in [17].
If u 0 1 = u 0 2 then a(T ) does not vanish. If u 0 1 , u 0 2 ∈ S h for some well-posedness growth bound h, and u 0 1 − u 0 2 is small in S h , we might expect u 1 (t) − u 2 (t) S h to remain small at least for some time. The S h norm, however, includes the L ∞ norm of ω 1 (t) − ω 2 (t), with each vorticity being transported by different flow maps. Hence, we should expect ω 1 (t) − ω 2 (t) L ∞ to be of the same order as ω j (t) L ∞ , j = 1, 2, immediately after time zero. Thus, it is too much to ask for continuous dependence on initial data in the S h norm. In this regard, the situation is the same as for the classical bounded-vorticity solutions of Yudovich [17], and has nothing to do with lack of decay at infinity. The best we can hope to obtain is a bound on (u 1 (t) − u 2 (t))/ζ in L ∞ -and so, by interpolation, in C α for all α < 1.
To obtain continuous dependence on initial data or control how changes at a distance from the origin affect the solution near the origin (effect at a distance, for short), we can employ the bound on Q in Theorem 1.6 to obtain a bound on how far apart the two solutions become, weighted by ζ. For continuous dependence on initial data, ζ = h is most immediately pertinent; for controlling effect at a distance, ζ ≥ h is better.
The simplest form of continuous dependence on initial data, which follows from Theorem 1.9 applied to Theorem 1.6, shows that if the initial velocities are close in S ζ then they remain close (in a weighted L ∞ space) for some time. Theorem 1.9. Make the assumptions in Theorem 1.6. Then While Theorem 1.9 gives a theoretically meaningful measure of continuous dependence on initial data, the assumption that the initial velocities are close in S ζ is overstrong. For instance, it would not apply to two vortex patches that do not quite coincide. One approach, motivated in part by this vortex patch example, is to make some assumption on the closeness of the initial vorticities locally uniformly in an L p norm for p < ∞, as was done in [16,1]. This assumption is, however, unnecessary (and in our setting somewhat artificial) as shown for the special case of bounded velocity (h ≡ 1) in [7].
In [7], the focus was not on continuous dependence on initial data, per se, but rather on understanding the effect at a distance. Hence, we used a function, ζ(r) = (1 + r) α for any α ∈ (0, 1), in place of h and obtained a bound on (u 1 (t) − u 2 (t))/ζ in terms of (u 0 1 − u 0 2 )/ζ, each in the L ∞ norm. In Theorem 1.10, we obtain a similar bound using very different techniques. Our need to assume bounded velocity (h = 1) arises from our inability to obtain usable transport estimates for non-Lipschitz vector fields growing at infinity. It is not clear whether this is only a technical issue or represents some fundamental new phenomenon causing unbounded velocities to have less stability, in the sense of effect at a distance, than bounded velocities. (See Remark 8.2.) Theorem 1.10. Let u 0 1 , u 0 2 ∈ S 1 and let ζ be a growth bound. Let u 1 , u 2 be the corresponding solutions in S 1 on [0, T ] with initial velocities u 0 1 , u 0 2 . Fix δ, α with 0 < δ < α < 1 and choose any T * > 0 such that where, Remark 1.11. To obtain a bound on a(T ), we iterate the bound in (1.10) N times, where T = N T * (decreasing T * if necessary so as to obtain the minimum possible positive integer N ), applying the bound on Q(t) from Theorem 1.6 after each step. Since T * depends only upon C 0 , which does not change, we can always iterate this way. In principle, the resulting bound can be made explicit, at least for sufficiently small a(T ).
The bound on u 1 − u 2 given by the combination of Theorems 1.6 and 1.10 is not optimal, primarily because we could use an L 1 -in-time bound on J(t, x) in place of the L ∞ -in-time bound. (This would also be reflected in the bound on a(T * ) in (1.10).) This would not improve the bounds sufficiently, however, to justify the considerable complications to the proofs.
The issue of well-posedness of solutions to the 2D Euler equations with bounded vorticity but velocities growing at infinity was taken up recently by Elgindi and Jeong in [8]. They prove existence and uniqueness of such solutions for velocity fields growing linearly at infinity under the assumption that the vorticity has m-fold symmetry for m ≥ 3. We study here solutions with no preferred symmetry, and our approach is very different; nonetheless, aspects of our uniqueness argument were influenced by Elgindi's and Jeong's work. In particular, the manner in which they first obtain elementary but useful bounds on the flow map inspired Lemma 5.2, and the bound in Proposition 4.4 is the analog of Lemma 2.8 of [8] (obtained differently under different assumptions).
Finally, we remark that it would be natural to combine the approach in [8] with our approach here to address the case of two-fold symmetric vorticities (m = 2). The goal would be to obtain Elgindi's and Jeong's result, but for velocities growing infinitesimally less than linearly at infinity. A similar argument might also work for solutions to the 2D Euler equations in a half plane having sublinear growth at infinity. This paper is organized as follows: We first make a few comments on our formulation of a weak solution in Section 2. Sections 3 to 5 contain preliminary material establishing useful properties of growth bounds, estimates on the Biot-Savart law and locally log-Lipschitz velocity fields, and bounds on flow maps for velocity fields growing at infinity. We establish the existence of solutions, Theorem 1.5, in Section 6. We prove Theorem 1.6 in Section 7, thereby establishing the uniqueness of solutions, Corollary 1.7. In Section 8, we prove Theorems 1.9 and 1.10, establishing continuous dependence on initial data and controlling the effect of changes at a distance.
In Section 9, we employ Littlewood-Paley theory to establish estimates in negative Hölder spaces for u ∈ L ∞ (0, T ; S 1 ). These estimates are used in the proof of Theorem 1.10 in Section 8.
Finally, Appendix A contains the proof of Corollary 1.8.

Some comments on our weak formulation
In our formulation of a weak solution in Definition 1.4 we made the assumption that Serfati's identity holds. Yet formally, Serfati's identity and the Euler equations are equivalent, and with only mild assumptions on regularity and behavior at infinity, they are rigorously equivalent. We do not purse this equivalence in any detail here, as it would be a lengthy distraction, we merely outline the formal argument. That a solution to the Euler equations must satisfy Serfati's identity for smooth, compactly supported solutions (and hence also formally) is shown in Proposition 4.1 of [1]. The reverse implication is more difficult. One approach is to start with the velocity identity, where ϕ λ := ∇a λ · K ⊥ ∈ C ∞ c (R 2 ). This identity can be shown to be equivalent to the Serfati identity by exploiting Lemma 4.4 of [11]. It turns out that ϕ λ * is a mollifier: taking λ → 0 yields (after a long calculation) the velocity equation for the Euler equations in the limit.
Assuming Serfati's identity holds, then, introduces some degree of redundancy in Definition 1.4, but this redundancy cannot be entirely eliminated if we wish to have uniqueness of solutions. This is demonstrated in [11], where it is shown that for bounded vorticity, bounded velocity solutions, Serfati's identity must hold-up to the addition of a time-varying, constantin-space vector field. This vector field, then, serves as the uniqueness criterion; its vanishing is equivalent to the sublinear growth of the pressure (used as the uniqueness criterion in [16]).
Bounded vorticity, bounded velocity solutions are a special case of the solutions we consider here, but the technology developed in [11] does not easily extend to velocities growing at infinity. Hence, we are unable to dispense with our assumption that the Serfati identity holds, as we need it as our uniqueness criterion.
Another closely related issue is that we are using Lagrangian solutions to the Euler equations, as we need to use the flow map in our uniqueness argument (with Serfati's identity). We note that it is not sufficient to simply know that the vorticity equation, ∂ t ω + u · ∇ω = 0, is satisfied. This tells us that the vorticity is transported, in a weak sense, by the unique flow map X, but we need that this weak transport equation has ω 0 (X −1 (t, x)) as its unique solution. Only then can we conclude that the curl of u truly is ω 0 (X −1 (t, x)). (For bounded velocity, bounded vorticity solutions, the transport estimate from [3] in Proposition 9.7 would be enough to obtain this uniqueness.) The usual way to establish well-posedness of Eulerian solutions to the 2D Euler equations is to construct Lagrangian solutions (which are automatically Eulerian) and then prove uniqueness using the Eulerian formulation only. Such an approach works for bounded vorticity, bounded velocity solutions, as uniqueness using the Eulerian formulation was shown in [16] (see also [7]). Whether this can be extended to the solutions we study here is a subject for future work.

Properties of growth bounds
We establish in this section a number of properties of growth bounds.
, and the analogous statement holds for h 2 when h is a well-posedness growth bound.
for all x > 0 and a ∈ [0, 1]. We apply this twice with x = r + s > 0 and a = r/(r + s), giving Because h ′ (0) < ∞ and h is concave we also have h(r) ≤ cr + d. The facts regarding h 2 follow in the same way. Lemma 3.3 gives the existence of the function µ promised in Definition 1.1. A µ that yields a tighter bound on E ≤ µ will result in a longer existence time estimate for solutions, as we can see from the application of Lemma 6.1 in the proof of Theorem 1.5. The estimate we give in Lemma 3.3 is very loose; in specific cases, this bound can be much improved.
It follows that r Remark 3.4. Abusing notation, we will often write h(x) for h(|x|), treating h as a map from R 2 to R. Treated this way, h remains subadditive in the sense that Here, we used the triangle inequality and that h : Proof. To prove (3.2), we will show first that for any positive integer n and any r ≥ 0, For n = 1, (3.4) trivially holds, so assume that (3.4) holds for n − 1 ≥ 1. Then because h is subadditive (Lemma 3.2), Thus, (3.4) follows for all positive integers n by induction. If a = n + α for some α ∈ [0, 1) then (Note that the supremum is over n ≥ 1 since we assumed that a ≥ 1.) In Section 7 we will employ the functions, Γ t , F t : [0, ∞) → (0, ∞), defined for any t ∈ [0, T ] in terms of an arbitrary growth bound h by We know that Γ t and so F t are well-defined, because recalling that h(r) ≤ cr + d by Lemma 3.2.
Remark 3.6. If h(0) were zero, then Γ t would be the bound at time t on the spatial modulus of continuity of the flow map for a velocity field having h as its modulus of continuity. Much is known about properties of Γ t (they are explored at length in [10]), and most of these properties are unaffected by h(0) being positive. One key difference, however, is that Γ t (0) > 0 and Γ ′ t (0) < ∞. As we will see in Lemma 3.7, this implies that Γ t is subadditive. This is in contrast to what happens when h(0) = 0, where Γ t (0) = 0, Γ ′ t (0) = ∞, and Γ satisfies the Osgood condition.
Lemma 3.7 shows that F t is a growth bound that is equivalent to h in that it is bounded above and below by constant multiples of h.
Here, we used Lemma 3.5 (we increase c ′ so that c ′ ≥ 1 if necessary) and that h increasing Finally, if h is a global well-posedness growth bound then C(t)µ serves as a bound on the function E of Definition 1.1 for F t .
Lemma 3.8. Assume that h is a growth bound and let g := 1/h. Then g is a decreasing convex function; in particular, |g ′ | is decreasing. Moreover since h > 0 and h ′′ ≤ 0. Thus, g is a decreasing convex function. Then, because g ′ is negative but increasing, |g ′ | is decreasing. Finally, is decreasing, since log is concave and h is concave so log h is concave. Therefore, We will also need the properties of µ (defined in (1.7)) given in Lemma 3.9.

Biot-Savart law and locally log-Lipschitz velocity fields
Proposition 4.1. Let a λ be as in Definition 1.3. There exists C > 0 such that, for all x ∈ R 2 and all λ > 0 we have, Let U ⊆ R 2 have Lebesgue measure |U |. Then for any p in [1,2), Proof. See Propositions 3.1 and 3.2 of [1].
There exists C > 0 such that for all λ > 0, from which the result follows. Proposition 4.3. Let X 1 and X 2 be measure-preserving homeomorphisms of R 2 . Let U ⊂ R 2 have finite measure and assume that δ : Proof. As in the proof of Proposition 3.3 of [1], we have In [1], R 1 p was bounded above by max{1, R}, which gave p = − log δ as the minimizer of the norm as long as δ < e −1 . Keeping the factor of R 1 p we see that the minimum occurs when p = − log(δ/R) as long as δ ≤ e −1 R, the minimum value being This gives the bound for δ ≤ e −1 R; the δ > e −1 R bound follows immediately from (4.2) with p = 1.
In Proposition 4.4, we establish a bound on the modulus of continuity of u ∈ S h . In Lemma 2.8 of [8], the authors obtain the same bound as in Proposition 4.4 for h(x) = 1 + |x|, but under the assumption that the velocity field can be obtained from the vorticity via a symmetrized Biot-Savart law (which they show applies to m-fold symmetric vorticities for m ≥ 3, but which does not apply for our unbounded velocities).

Proposition 4.4.
Let h be a pre-growth bound. Then for all x, y ∈ R 2 such that |y| ≤ C(1 + |x|) for some constant C > 0, we have, for all u ∈ S h , If h ≡ C, we need no restriction on |y|.
Proof. Fix x ∈ R 2 and let ψ be the stream function for u on R 2 chosen so that ψ( Applying Morrey's inequality gives, for any y with |y| ≤ R, and any p > 2, Because ω is compactly supported, u = K * ω. Thus, we can apply the Calderon-Zygmund inequality to obtain When R −1 |y| ≤ e −1 (meaning also that |y| ≤ R, as required), the infimum occurs at p = −2 log( R −1 y ) and we have Having minimized over p for a fixed R, we must now choose R.
where we used the subadditivity of h (see Lemma 3.2) and Lemma 3.5. Hence, we have, for |y| ≤ e −1 h(x), the last equality holding as long as |y| ≤ h(x)e −1 = e −1 R ≤ R.
In proving uniqueness in Section 7, we will need to bound the term in the Serfati identity (1.1) coming from a convolution of the difference between two vorticities. Since the vorticities have no assumed regularity, we will need to rearrange the convolution so as to use an estimate on the Biot-Savart kernel that involves the difference of the flow maps, as in Proposition 4.5. This proposition is a refinement of Proposition 6.2 of [1] that better accounts for the effect of the parameter λ in the cutoff function a λ . Note that although we assume the solutions lie in some S h space, h does not appear directly in the estimates, rather it appears indirectly via the value of δ(t), as one can see in the application of the proposition. Proposition 4.5. Let X 1 and X 2 be measure-preserving homeomorphisms of R 2 and let ω 0 ∈ L ∞ (R 2 ). Fix x ∈ R 2 and λ > 0. Let V = supp a λ (X 1 (s, x) − X 1 (s, ·)) ∪ supp a λ (X 1 (s, x) − X 2 (s, ·)) and assume that (4.6) Then we have The constant, C, depends only on the Lipschitz constant of a.

Flow map bounds
In this section we develop bounds related to the flow map for solutions to the Euler equations in S h on [0, T ]. First, though, is the matter of existence and uniqueness: Lemma 5.1. Let h be a pre-growth bound (which we note includes h(x) = C(1 + |x|)) and assume that u ∈ L ∞ (0, T ; S h ). Then there exists a unique flow map, X, for u; that is, a function X : [0, T ] × R 2 → R 2 for which Proof. Because u is locally log-Lipschitz by Proposition 4.4, this is (essentially) classical.
Lemma 5.2. Let h be a pre-growth bound. Assume that u 1 , u 2 ∈ L ∞ (0, T ; S h ). Let F t be the function defined in (3.6). We have, Hence by Osgood's inequality, Similarly, These bounds yield the result.
Lemmas 3.7 and 5.2 together show that over time, the flow transports a "particle" of fluid at a distance r from the origin by no more than a constant times h(r). This will allow us to control the growth at infinity of the velocity field over time so that it remains in S h (for at least a finite time), as we shall see in the next section. As the fluid evolves over time, however, the flow can move two points farther and farther apart; that is, its spatial modulus of continuity can worsen, though in a controlled way, as we show in Lemma 5.3. (A similar bound to that in Lemma 5.3 holds for any growth bound, but we restrict ourselves to the special case of bounded vorticity, bounded velocity velocity fields, for that is all we will need.) Lemma 5.3. Let u ∈ L ∞ (0, T ; S 1 ) and let X be the unique flow map for u. Let C 0 = u L ∞ (0,T ;S 1 ) . For any t ∈ [0, T ] define the function, Then for all x, y ∈ R 2 , The same bound holds for X −1 .
Proof. The bounds, are established in Lemma 8.2 of [12]. We note, however, that that proof applies only for all sufficiently small |x − y|. A slight refinement of the proof produces the bounds as we have stated them.
The following simple bound will be useful later in the proof of Proposition 9.3:

Existence
Our proof of existence differs significantly from that in [1] only in the use of the Serfati identity to obtain a bound in L ∞ (0, T ; S h ) of a sequence of approximating solutions and to show that the sequence is Cauchy, which is more involved than in [1]. Although velocities in S h are not log-Lipschitz in the whole plane (unless h is constant), they are log-Lipschitz in any compact subset of R 2 . Since the majority of the proof of existence involves obtaining convergence on compact subsets, this has little effect on the proof. Therefore, we give only the details of the bound on L ∞ (0, T ; S h ) using the Serfati identity, as this is the main modification of the existence proof. We refer the reader to [1] for the remainder of the argument.
Proof of existence in Theorem 1.5. Let u 0 ∈ S h and assume that u 0 does not vanish identically; otherwise, there is nothing to prove. Let (u 0 n ) ∞ n=1 and (ω 0 n ) ∞ n=1 be compactly supported approximating sequences to the initial velocity, u 0 , and initial vorticity, ω 0 , obtained by cutting off the stream function and mollifying by a smooth, compactly supported mollifier. (This is as done in Proposition B.2 of [1], which simplifies tremendously when specializing to all of R 2 .) Let u n be the classical, smooth solution to the Euler equations with initial velocity u 0 n , and note that its vorticity is compactly supported for all time. The existence and uniqueness of such solutions follows, for instance, from [14] and references therein. (See also Chapter 4 of [12] or Chapter 4 of [4].) Finally, let ω n = curl u n .
As we stated above, we give only the uniform L ∞ ([0, T ]; S h ) bound for this sequence, the rest of the proof differing little from that in [1].
We have, It follows as in Proposition 4.1 of [1] that the Serfati identity (1.1) holds for the approximate solutions. It is important to note that x and t are fixed in this identity, so λ can be a function both of t and x (though not s). Or, to see this more expicitly, we can write the critical convolution in the derivation of the Serfati identity in Proposition 4.1 of [1] as, and it becomes clear that in moving derivatives from one side of the convolution to another we are in effect integrating by parts, taking derivatives always in the variable y.
In any case, it follows from the Serfati identity that The first convolution we bound using (4.1) and (6.1) as For the second convolution, we have, using Proposition 4.2, The second inequality follows from the subadditivity of h 2 (as in Remark 3.4). Hence,

Dividing both sides by h(x) gives
Now, for any fixed t, we can set which we note nearly minimizes the right-hand side of (6.2). Defining , where f (z) := zH(z). In the third-to-last inequality we used that H is decreasing and h(x) ≥ h(0) > 0. Observe that although this inequality was obtained by choosing λ = λ(t, x) for one fixed t, the inequality itself holds for all t ∈ [0, T ].
Taking the supremum over x ∈ R 2 and squaring both sides, we have where E, µ are as in Definition 1.1. Now we can apply Lemmas 3.3 and 6.1 to conclude that u n ∈ L ∞ (0, T ; S h ) with a norm bounded uniformly over n. Lemma 6.1 also gives global-intime existence (T arbitrarily large) when (1.3) holds.
and for all t ∈ [0, T ] for any fixed T ≥ 1, then Λ ∈ L ∞ loc ([0, ∞)). Proof. Because µ is convex, we can apply Jensen's inequality to conclude that As long as t ≤ 1, Remark 3.10 allows us to write and Osgood's lemma gives (6.6). Now suppose that T > 1. Then Remark 3.10 gives the weaker bound, so that leading to (6.7).
Finally, if (6.8) holds then applying Osgood's lemma to (6.7) shows that Λ is bounded on any interval [0, T ], so that Λ ∈ L ∞ loc ([0, ∞)). We make a few remarks on our proof of Theorem 1.5. Lemma 6.1 allows us to obtain finite-time or global-in-time existence of solutions, but unless we have a stronger condition on µ, neither the finite time of existence nor the bound on the growth of the L ∞ norm that results will be optimal. For both of our example growth bounds in Corollary 1.8 there are stronger conditions; namely, if µ 1 , µ 2 are the function in Definition 1.1 corresponding to h 1 , h 2 then for all a, r ≥ 0, It is easy to see that the condition on µ 2 in fact, implies (6.8), though the condition on µ 1 is too weak to do so. Both conditions improve the bound on the L ∞ norm resulting from Lemma 6.1 and for h 1 , the time of existence.
Moreover, Lemma 3.3 shows that, up to a constant factor, µ(r) := Cr(1 + r) works for all well-posedness growth bounds (and gives µ 2 (ar) ≤ C 0 a 2 µ 2 (r) for all a, r ≥ 0). This suggests that a slight weakening of the condition we placed on growth bounds in (ii) of Definition 1.1 could be made that would still allow finite-time existence to be obtained.

Uniqueness
In this section we prove Theorem 1.6, from which uniqueness immediately follows. Our argument is a an adaptation of the approach of Serfati as it appears in [1]. It starts, however, by exploiting the flow map estimates in Lemma 5.2, inspired by the proof of Lemma 2.13 of [8], which is itself an adaptation of Marchioro's and Pulvirenti's elegant uniqueness proof for 2D Euler in [13], in which a weight is introduced.
Proof of Theorem 1.6. We will use the bound on X 1 and X 2 given by Lemma 5.2 with the growth bound, F T [ζ], defined in (3.6). This is valid since ζ ≥ h. By Lemma 3.7, F T [ζ] is a growth bound that is equivalent to ζ, up to a factor of C(T ); hence, we will use ζ in place of F T [ζ], which will simply introduce a factor C(T ) into our bounds.

It follows that
To bound A 2 , we use the Serfati identity, choosing λ(x) = h(x), to write We write, Making the two changes of variables, z = X 1 (s, y) and z = X 2 (s, y), we can write We are thus in a position to apply Proposition 4.5 to bound A 1 2 . To do so, we set U j := {y ∈ R 2 : |X 1 (s, x) − X j (s, y)| ≤ h(x)} so that V := U 1 ∪ U 2 is as in Proposition 4.5. Then with δ as in (4.6), we have where C 1 = C(T ). Above, we applied Lemma 5.2 in the second inequality and the last inequality follows from repeated applications of Lemma 3.5 to ζ. Hence, Proposition 4.5 gives But by Lemma 3.9 (noting that h(x)/ζ(x) ≤ 1) and (7.2), Hence, We now bound A 2 2 (x). We have, Because ζh is subadditive (being a pre-growth bound), letting w = X 1 (r, x), we have ).
We now obtain the bounds on M (t) and Q(t).
Since we can absorb a constant, this same bound holds for Q(t): Hence, we can easily translate a bound on M to a bound on Q.
Returning, then, to (1.6), we have, for a(T ) sufficiently small, Integrating gives − log log s We were able to use a growth bound ζ larger than h and obtain a result for an arbitrary T because, unlike the proof of existence in Section 6, we are assuming that we already know that u 1 , u 2 lies in S h . Hence, the quadratic term in the Serfati identity can in effect be made linear.
Theorem 1.6 gives a bound on the difference in velocities over time. It remains, however, to characterize a(T ) in a useful way in terms of u 0 1 , u 0 2 , and u 0 1 − u 0 2 and so obtain Theorem 1.10. This, the subject of the next section, is not as simple as it may seem.

Continuous dependence on initial data
In this section, we prove Theorems 1.9 and 1.10, bounding a(T ) of (1.4). The difficulty in bounding a(T ) lies wholly in bounding J/ζ, with J = J(t, x) as in (1.5). We can write . Both J 1 /ζ and J 2 /ζ are easy to bound, as we do in Theorem 1.9, if we assume that ω 0 1 − ω 0 2 is close in L ∞ , an assumption that is physically unreasonable, however, as discussed in Section 1.
Proof of Theorem 1.9. We have Combined, these two bounds easily yield the bound on a(T ).
More interesting is a measure of a(T ) in terms of u 0 1 − u 0 2 without involving ω 0 1 − ω 0 2 . Now, J 1 is fairly easily bounded in terms of u 0 1 − u 0 2 (using Lemma 8.5) since X 1 (s, x) has no effect on the L ∞ norm. But in J 2 , the composition of the initial vorticity with the flow map complicates matters considerably. What we seek is a bound on a(T ) of (1.4) in terms of (u 0 1 −u 0 2 )/ζ L ∞ and constants that depend upon u 0 1 S 1 , u 0 2 S 1 . That is the primary purpose of Theorem 1.10, which we now prove, making forward references to a number of results that appear following its proof. These include Propositions 9.3 and 9.7 and Lemma 9.9, which employ Littlewood-Paley theory and Hölder spaces of negative index, and which we defer to Section 9, where we introduce the necessary technology.
Remark 8.1. In the proof of Theorem 1.10, we make use for the first time in this paper of Hölder spaces, with negative and fractional indices. We are not using the classical definition of these spaces, but rather one based upon Littlewood-Paley theory. For non-integer indices, they are equivalent, but the constant of equivalency (in one direction) blows up as the index approaches an integer (see Remark 9.2). Because we will be comparing norms with different indices, it is important that we use a consistent definition of these spaces. In this section, the only fact we use regarding Hölder spaces (in the proof of Lemma 8.6) is that div v C r−1 ≤ C v C r for any r ∈ (0, 1) for a constant C independent of r. For that reason, we defer our definition of Hölder spaces to Section 9. Proof of Theorem 1.10. Let ω 0 1 , ω 0 2 be the initial vorticities, and let ω 1 , ω 2 and X 1 , X 2 be the vorticities and flow maps of u 1 , u 2 .
To bound a(T ), let ω 0 = ω 0 1 − ω 0 2 = curl(u 0 1 − u 0 2 ). Then, since h ≡ 1, we can write, , Applying Lemma 3.5, Lemma 5.2, and Lemma 8.5, we see that the bound holding uniformly over s ∈ [0, T ]. Bounding J 2 is much more difficult, because the flow map appears inside the convolution, which prevents us from writing it as the curl of a divergence-free vector field. Instead, we apply a sequence of bounds, starting with Here we used Lemma 5.2 and applied Proposition 9.3 with f = ω 0 • X −1 1 (s). Applying Proposition 9.7 with α = δ t and β = 2C(δ), followed by Lemma 8.6 gives Note that the condition on δ T * in Proposition 9.7 is satisfied because of our definition of T * and because u LL ≤ C u S 1 , which follows from Proposition 4.4. We also used, and use again below, that Φ α is increasing in its second argument. We apply Lemma 9.9 with r = 1 − δ, obtaining where we used the identity, curl(f u) = f curl u − ∇f · u ⊥ , and that 1/ζ is Lipschitz (though 1/ζ / ∈ C 1 (R 2 ) unless ζ is constant, because ∇ζ is not defined at the origin). Therefore, We conclude that for all 0 ≤ s ≤ T * . Since the bound on J 1 /ζ in (8.1) is better than that on J 2 /ζ, this completes the proof.
Remark 8.3. In the application of Proposition 9.7 and Lemma 9.4 (which was used in the proof of Proposition 9.3) the value of C 0 = u 1 L ∞ (0,T ;S 1 ) enters into the constants. A bound on u 1 L ∞ (0,T ;S 1 ) comes from the proof of existence in Section 6. While we did not explicitly calculate it, for S 1 it yields an exponential-in-time bound, as in [15,1]. Hence, our bound on a(T ) is doubly exponential (it would be worse for unbounded velocities). It is shown in [9] (extending [19]) for bounded velocity, however, that u 1 L ∞ (0,T ;S 1 ) can be bounded linearly in time, which means that C 0 actually only increases singly exponentially in time.
Whether an improved bound can be obtained for a more general h is an open question: If it could, it would extend the time of existence of solutions, possibly expanding the class of growth bounds for which global-in-time existence holds.
is not just a matter of moving the curl from one side of the convolution to the other. Using both that Z is divergence-free and that a is radially symmetric, however, (8.2) is proved in Lemma 4.4 of [11].
The following is a twist on Proposition 4.6 of [2].
Lemma 8.5. Let ζ be a pre-growth bound and suppose that Z ∈ S 1 . For any λ > 0, where δ is the Dirac delta function, since a λ (0) = 1. Hence, Here, we have used that ϕ λ L 1 = 1, as can easily be verified by integrating by parts. (In fact, ϕ λ * is a mollifier, though we will not need that.) Using (8.4), we have Taking the supremum over x ∈ R 2 , we have .

Hölder space estimates
In this section, we make use of the Littlewood-Paley operators ∆ j , j ≥ −1. A detailed definition of these operators and their properties can be found in chapter 2 of [4]. We note here, only that ∆ j f = ϕ j * f , where ϕ j (·) = 2 2j ϕ(2 j ·) for j ≥ 0, ϕ is a Schwartz function, and the Fourier transform of ϕ is supported in an annulus. We can write ∆ −1 f as a convolution with a Schwartz function χ whose Fourier transform is supported in a ball.
We will also make use of the following Littlewood-Paley definition of Holder spaces.
Definition 9.1. Let r ∈ R. The space C r * (R 2 ) is defined to be the set of all tempered distributions on R 2 for which that the C r * norm is equivalent to the classical Hölder space C r norm when r is a positive non-integer: f C r ≤ a r f C r * , f C r * ≤ b r f C r for constants, a r , b r > 0, though a r → ∞ as r approaches an integer. See Remark 8.1. Proposition 9.3. Let ζ be a pre-growth bound and let u ∈ L ∞ (0, T ; S 1 ) with X its associated flow map. Let t ∈ [0, T ] and set η = ζ • X −1 (t). For any α > 0, λ > 0, and f ∈ L ∞ (R 2 ), where C = C(T, ζ, λ), and Φ α is defined in (1.11) (using C 0 = u L ∞ (0,T ;S 1 ) ).
Proof. Define g = 1/η. For fixed N ≥ −1 (to be chosen later), write a λ (y)K(y)η(x − y)(∆ j (f /η))(x − y) dy =: I + II. We first estimate I. Exploiting Definition 9.1, where ζ is a pre-growth bound, Lemma 9.4 implies that where we used boundedness of g. Substituting this estimate into (9.3), we conclude that We now estimate II by introducing a commutator and utilizing the Holder continuity of η, writing We rewrite III as a convolution, noting that the Littlewood-Paley operators commute with convolutions, and apply Bernstein's Lemma and Lemma 9.5 to give To estimate IV , we apply Lemma 9.6 and (4.1) to obtain Substituting the estimates for III and IV into (9.5) gives Finally, substituting the estimates for I and II into (9.2) yields (a λ K) * f (x) η(x) ≤ C (1 + f L ∞ ) 2 αN f /η C −α + 2 −N e −C 0 t g(x) , (9.6) where C depends on λ. Now, we are free to choose the integer N ≥ −1 any way we wish, but if we choose N as close to N * := 1 α + e −C 0 t log 2 1 f /η C −α = − 1 α + e −C 0 t log 2 f /η C −α as possible, we will be near the minimizer of the bound in (9.6), as long as N * ≥ −1 (because such an N * balances the two terms). Calculating with N = N * gives since g is bounded. Rounding N * up or down to the nearest integer will only introduce a multiplicative constant no larger than C2 α , so this yields as long an N * ≥ −1.
Proposition 9.7. Let u ∈ L ∞ (0, T ; S 1 ). Assume f ∈ C([0, T ]; L ∞ ) solves the transport equation For fixed δ ∈ (−1, 0), there exists a constant C = C(δ) such that for any β > C, if Remark 9.8. Proposition 9.7 is proved in greater generality in [3]. The authors assume, for example, that f belongs to the appropriate negative Hölder space. Here, we apply Proposition 9.7 with f =ω 0 ζ • X −1 1 , which clearly belongs to C([0, T ]; L ∞ ) and therefore to all negative Hölder spaces. In addition, the authors integrate a quantity in (9.10) which differs from u LL but is bounded above and below by C u LL for a constant C > 0. Lemma 9.9. For any u ∈ S 1 (R 2 ) and r ∈ (0, 1), Proof. We apply Definition 9.1 and write where we used Bernstein's Lemma to get the second inequality. But, using Lemma 4.2 of [6], which yields the result.