Flows of vector fields with point singularities and the vortex-wave system

The vortex-wave system is a model for the evolution of 2D incompressible fluids in which the vorticity is split into a finite sum of Dirac masses plus an Lp part. Existence of a weak solution for this system was recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for p>2, but their result left open the existence and basic properties of the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the) linear transport problem associated to the vortex-wave system. To this end, we study the flow associated to a two-dimensional vector field which is singular at a moving point. We also present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector-fields.


Introduction
The purpose of this article is to study the flow associated to a particular class of vector fields that contain a point singularity, which arise as weak solutions of the vortex-wave system. For a smooth vector field b : [0, T ] × R 2 → R 2 , the flow of b is the unique map X : [0, T ] × R 2 → R 2 defined by It turns out that, in some cases, even if b is not smooth, it is still possible to define an extended notion of flow for b, nowadays called regular Lagrangian flow (see, e.g., Definition 1.1 below). In their pioneering work, DiPerna and Lions [7] proved the existence and uniqueness of the flow for vector fields belonging to L 1 (W 1,1 loc ) with suitable decay at infinity and with bounded divergence (see assumptions (H 1 ) and (H 3 ) below). The Sobolev-type regularity assumptions on b were later relaxed by Ambrosio [1], allowing for BV vector fields (see assumption (H 2 )). There is a wide literature devoted to this issue, see e.g. [2,3] and references therein for additional or related results. The problem we address here is that the vector field associated to the vortex-wave system is not BV .
We will focus on the case where b is given by b(t, x) = v(t, x) + H(t, x), (1.2) where the field v enters the class of vector fields considered in the theory of DiPerna and Lions and Ambrosio, and where H is a special vector field which is singular along a curve in space time. More precisely, we assume that the first component v satisfies the same assumptions as in [1]: , The result of Ambrosio [1] ensures existence and uniqueness of the regular Lagrangian flow associated to such fields. In addition, in our context, we require the following assumption: Next, we define our singular part H as follows. We consider a given Lipschitz trajectory in R 2 : We introduce the map Then H satisfies (H 1 ) and (H 3 ): actually, it is divergence free. It does not satisfy (H 2 ) therefore such a field is not covered by the result of Ambrosio [1]. However note that H is smooth off of the set {(t, z(t)), t ∈ [0, T ]}. The structure described by (1.2) includes that of solutions of the vortexwave system in the special case of a single vortex together with compactly supported L p vorticity, p > 1.
Next we recall, following DiPerna and Lions [7] and Ambrosio [1], the definition of regular Lagrangian flow. We denote by L 2 the Lebesgue measure on R 2 . Definition 1.1 (Regular Lagrangian flow). We say that a map X : [0, T ] × R 2 → R 2 is a regular Lagrangian flow for the vector field b if: (i) There exists an L 2 -negligible set S ⊂ R 2 such that for all x ∈ R 2 \ S the map t → b(t, X(t, x)) belongs to L 1 ([0, T ]), and (ii) For all R > 0 there exists L R > 0 such that In Section 2 we combine the abstract theory by Ambrosio [2] (Theorem 2.1 below), see also [3], exploiting the link between the ODE and the continuity and transport equations (see (2.1)-(2.2)), with an extension of a renormalization result by Lacave and Miot [9] to show existence and uniqueness for the regular Lagrangian flow of b. Moreover, we prove the additional property that for L 2 -a.e. x ∈ R 2 the trajectory starting from the point x does not collide with the singularity point. More precisely, we prove the following theorem: and where H is given by (1.4). Then there exists a unique regular Lagrangian flow. Moreover, for L 2 -a.e. x ∈ R 2 we have Observe that, by the very definition of regular Lagrangian flow, the absolute continuity of the measure X(t, ·) # L 2 with respect to L 2 implies, by Fubini's theorem, that for L 2 -a.e. x ∈ R 2 we have X(t, x) = z(t) for L 1 -a.e. t ∈ [0, T ]. The main point of Theorem 1.2 is that collisions between the Lagrangian trajectories and the singularity point are avoided for all t ∈ [0, T ]. Indeed Proposition 2.3 yields a quantitative control of the amount of Lagrangian trajectories getting closer than ε to the point singularity: the proof of this proposition uses the additional assumption (H 4 ). We mention that an analogous control on the trajectories was performed in the setting of the Vlasov-Poisson equation with singular fields by Caprino, Marchioro, Miot and Pulvirenti [5].
In the second part of this work we present an effective construction of the regular Lagrangian flow by an approximation argument. In contrast with the point of view adopted in the first part, this construction does not rely on the link between the ODE and the PDE. Moreover, we provide a quantitative rate of convergence, by extending to our setting the estimates performed by Crippa and De Lellis [6] for vector fields without singular part. We restrict ourselves to vector fields satisfying the stronger assumptions: . In Section 3 we define a suitable smooth approximation (b n ) n∈N of b, and we denote by X n the unique corresponding classical flow. We prove the following theorem: the following estimate holds: To conclude this introduction, we describe the vortex-wave system and the connection between the present work and this system. In two-dimensional incompressible fluids, we consider a flow with initial vorticity consisting of the superposition of a diffuse part ω 0 ∈ L p for some p ≥ 1 and a point vortex located at z 0 ∈ R 2 , with unit strength. The evolution of vorticity can be described by a system of equations called the vortex-wave system (with one single point vortex): This system was introduced by Marchioro and Pulvirenti [11,12]. There are two natural notions of weak solution for this system, one is a solution in the sense of distributions, called Eulerian solution, while the other is a solution for which the diffuse part of the vorticity is constant along the trajectories of the flow, called Lagrangian solution, see [8,9] for precise definitions. (By 'trajectories of the flow' we mean the flow associated to the vector field b = v + H above.) In [11,12], Marchioro and Pulvirenti established global existence of a Lagrangian solution with ω ∈ L ∞ (L 1 ∩L ∞ ). In [8], Lopes Filho, Miot and Nussenzveig Lopes established global existence of an Eulerian solution with vorticity ω ∈ L ∞ (L 1 ∩ L p ), with p > 2. For any p > 2 Lagrangian solutions to the vortex-wave system are Eulerian. The converse was left open in [8]. The issue of the Lagrangian formulation is the natural requirement that flow trajectories should not collide with the point vortex. When p = +∞, almost-Lipschitz regularity for the velocity (1/2π)K * ω enables to define flow trajectories in the classical sense, which do not intersect with the point vortex, starting from any x = z 0 [11,12]. For p < +∞ this property is unclear. In Section 4, we use the results established in Sections 2 and 3 to show that any Eulerian solution with ω ∈ L ∞ (L 1 ∩L p ), p > 2, gives rise to a regular Lagrangian flow such that ω is constant along the flow trajectories, which do not, generically, collide with the point vortex. As it happens, when p > 2, the assumptions ( are all satisfied and, in particular, the point vortex trajectory t → z(t) is Lipschitz.

Proof of Theorem 1.2
In the theory of DiPerna and Lions and Ambrosio [1,2,7], the existence, uniqueness and the stability properties of the flow associated to a field b are linked to the well-posedness of the corresponding continuity equation and transport equation Note that one passes formally from the ODE to the continuity and transport equations by noticing that if X solves (1.1) then X(t, ·) # u 0 solves (2.1), and u 0 •X(t, ·) −1 solves (2.2). In the non-smooth case, we consider distributional solutions to (2.1) and (2.2). Such distributional formulations make sense as soon as bu and u div (b) belong to L 1 loc . As a matter of fact, we have the following general abstract result due to Ambrosio [2], somewhat extending this connection to the non smooth context: And, besides, existence and uniqueness for (2.1) hold for vector fields satisfying the assumptions ( 7]. Now, in the case where b is given by (1.2), the PDE well-posedness results cannot be applied directly because of the singular field H. However, the following holds: . Let u 0 ∈ L 1 ∩ L r , with r > 2. Then (2.1) has a unique solution u ∈ L ∞ L 1 ∩ L r .
Proof. First, existence of a distributional solution follows in both cases from standard regularization arguments.
The argument for uniqueness is strictly analogous to the one of Lacave and Miot [9]. We give the main lines for the reader's convenience. First, using the by now standard methods introduced in [1,7], it suffices to show that any solution u satisfies the renormalization property: The arguments leading to (2.3) put together computations as in [7] for case (2), and [1] for case (1), with the renormalization property for the single vector field H in both cases, i.e.
Combining Part (1) of Proposition 2.2 and Theorem 2.1 we obtain the existence and uniqueness of the regular Lagrangian flow X in Theorem 1.2. Therefore we only have to prove that for L 2 -a.e. x ∈ R 2 no collision between X(t, x) and z(t) occurs on [0, T ]. This is a direct consequence of the following Proposition 2.3. For 0 < ε < 1 and R > 0, let where S is as in Definition 1.1. Then Proof. We adapt the strategy introduced in [5] for the Vlasov-Poisson equation. Here, we set α = 1 − 2/q > 0. We introduce where 0 < λ < 1 is a parameter to be determined later, and where We set We first consider the case 2 < q < +∞. We set t i |v(s, X(s, x))| q ds ≤ ε −α , ∀i ∈ {0, . . . , N } and for i ∈ {0, . . . , N } we set By Chebyshev's inequality and Fubini's theorem we have |v(s, X(s, x))| q dx ds.
Using Property (ii) in Definition 1.1 for X(s, ·) and (H 4 ) we get We can assume that s 0 ∈ (t i , t i+1 ) for some i ∈ {0, . . . , N }. Let X(t, x)). Now we observe that, even though b is not uniformly bounded, the modulus |X(t, x) − z(t)| is Hölder continuous in time on the set A. Indeed, for L 1 -a.e. t ∈ [s 0 , s 1 ) such that X(t, x) = z(t) we get, using that Hence for all t ∈ [s 0 , s 1 ] we have by Hölder inequality Finally, by definition of the set A and by definition of β ≥ 1 we get For this choice of λ we obtain |X(s 1 , x) − z(s 1 )| < 2ε, which contradicts the definition of s 1 and shows that we must have s 1 = t i+1 . It follows that Therefore in view of (ii) in Definition 1.1, and finally Combining (2.5), (2.7) and using the definition of λ we obtain Since 2 − β = α, this yields the conclusion.
We now study the case where q = ∞, which is easier and does not require to introduce the sets B i and A. Indeed, let x ∈ P (ε, R). Coming back to (2.6) and proceeding similarly as before we obtain for s ∈ [s 0 , s 1 ) and the conclusion then follows as before.

Proof of Theorem 1.3
We start by defining the smooth approximation involved in Theorem 1.3. Let (ρ n ) n∈N be the usual sequence of Friedrichs mollifyers. Let v n = ρ n * v and let which defines a globally bounded, divergence free and smooth vector field on R 2 . We finally set b n (t, x) = v n (t, x) + K n (x − z(t)).
We first remark that (|X n (t, x) − z(t)|) n∈N is uniformly Lipschitz in time even though b n / ∈ L ∞ . Indeed, by the same computation leading to (2.6), using that K n (y) · y = 0 and (H ′ 1 ), we have (3.1) In particular, we have the local equiboundedness property On the other hand, since div (b n ) = ρ n * div (v) we infer from (H ′ 3 ) that In particular it follows from the standard theory on Jacobians that Part of our subsequent analysis is borrowed from [6]: we introduce We consider the positive quantity From now on C will denote a positive constant depending only on R, T , Before proving Lemma 3.1 we show how it implies Theorem 1.3. In the following we will sometimes write δ instead of δ(m, n).
Proof of Theorem 1.3 with Lemma 3.1.
We fix η > 0 to be determined later. By Chebychev's inequality and Lemma 3.1 we can find a set K ⊂ B R such that L 2 (B R \ K) ≤ η and (3.6) sup Using (3.2), it follows that where we have used (3.6) in the last inequality. We finally optimize the choice of the parameter η as follows. We set η ≡ 2C | ln δ| 1/3 , so that exp(C| ln δ| 2/3 /η) = exp(| ln δ|/2) = δ −1/2 . This yields In particular, we infer that (X n ) n∈N is a Cauchy sequence converging to some Y : [0, T ] × R 2 → R 2 in the space L 1 loc R 2 , L ∞ ([0, T ]) . Finally, the fact that Y is the (unique) regular Lagrangian flow associated to b is standard, and we omit the proof. We finally give the Proof of Lemma 3.1.
Let ε > 0 be a small parameter to be chosen later. We consider the set Using Proposition 2.3 applied to X n , with q = ∞, and thanks to (3.4), we obtain where C depends only on T , L 0 , v L ∞ (L ∞ ) , ż L ∞ , and ∇v L 1 (L p ) . Next, By (3.2) and (3.8), (3.9) B n,m ≤ C| ln δ|ε.
We next estimate the second part, for which we can adapt the proof of Theorem 2.9 in [6] for Sobolev vector fields since H is regular off the set {(t, z(t)), t ∈ [0, T ]}. We have we further obtain G n,m ≤ G 1 n,m + G 2 n,m , where By definition of R and by (3.4) and (3.2) we obtain We now estimate G 2 m,n . Let 0 ≤ χ ε ≤ 1 be a smooth function such that χ ε = 0 on B(0, ε/2) and χ ε = 1 on B(0, ε) c and let In the following M f denotes the maximal function of f . Using the classical estimate of the difference quotient of a function in terms of the maximal function of the derivative (see e.g. Lemma A.3 in [6]) we find By using (3.2) and (3.4) we get In view of (H ′ 2 ) and of the expression of H m,ε we get We gather (3.9), (3.10) and (3.11), obtaining g m,n ≤ C ε| ln δ| + ε −2 .

Lagrangian solutions to the vortex-wave system
We finally comment on the applications of the previous results to the vortex-wave system (1.5). Two notions of weak solution for the vortex-wave system have been introduced: Eulerian solutions and Lagrangian solutions, see [8,9]. These notions coincide when the vorticity ω belongs to L ∞ (L 1 ∩ L ∞ ) [11,12,9]. In [8] the authors establish global existence of an Eulerian solution with ω belonging to L ∞ (L 1 ∩ L p ) for p > 2. We claim that to this Eulerian solution corresponds a unique regular Lagrangian flow and that ω is constant along the flow trajectories. Indeed, if p > 2 then v = 1 2π K * ω satisfies all assumptions (H there exists a unique regular Lagrangian flow associated to the divergence free velocity field b = v + H. Moreover, it can be readily checked (adapting, e.g., the proof of Theorem 1.3 in [9]), that the function ω = X(t, ·) # ω 0 is a distributional solution in L ∞ (L 1 ∩ L p ) of the PDE ∂ t ω + (v + H) · ∇ ω = 0, ω(0) = ω 0 . Now, invoking the uniqueness part of Proposition 2.2 we obtain ω = ω, which establishes our claim.
Finally, we mention that Theorems 1.2 and 1.3 can be extended to vector fields H containing several point singularities