Differentiability properties of the flow of 2d autonomous vector fields

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.


Introduction
We consider bounded vector fields b ∈ L ∞ (0, T ) × R d ; R d . Although the analysis of this paper is limited to the case of autonomous vector fields with d = 2, we introduce the relevant notions and the related results in the general setting. The following notion of regular Lagrangian flow is an appropriate extension for merely locally integrable vector fields of the classical flow associated to Lipschitz vector fields. (1) for L d -a.e. x ∈ R d the map t → X(t, x) is absolutely continuous, X(0, x) = x and for L 1 -a.e. t ∈ (0, T ) it holds ∂ t X(t, x) = b(t, X(t, x)); (2) for every t ∈ [0, T ) it holds X(t, ·) L d ≤ LL d , for some L > 0. Regular Lagrangian flows have been introduced in a different form in [DL89], where the authors proved their existence and uniqueness for vector fields b ∈ L 1 t W 1,p x with p ≥ 1 and bounded divergence. The theory has been extended to vector fields b ∈ L 1 t BV x with bounded divergence in [Amb04]. Uniqueness of regular Lagrangian flows was finally achieved in the more general class of nearly incompressible vector fields with bounded variation in [BB20], introduced in the study of the hyperbolic system of conservation laws named after Keyfitz and Kranzer (see [DL07]).
Definition 1.2. A vector field b ∈ L 1 loc ((0, T ) × R d ; R d ) is called nearly incompressible if there exist C > 0 and ρ ∈ C 0 ([0, T ); L ∞ w (R d )) solving the continuity equation Several results about the differentiability properties of regular Lagrangian flows are available now. By the contributions in [LBL04,AM07], it follows that regular Lagrangian flows associated to vector fields b ∈ L 1 t W 1,1 x are differentiable in measure (see [AM07] for the definition of this notion). The same regularity property has been obtained recently in [BDN20] for nearly incompressible vector fields with bounded variation. The stronger property of approximate differentiability was obtained in [ALM05] for regular Lagrangian flows associated to vector fields b ∈ L 1 t W 1,p x with p > 1. A quantitative version of the same regularity property was provided in [CDL08], where the authors proved a quantitative Lusin-Lipschitz regularity of the flow.
The optimality of the regularity estimates obtained in [CDL08] is discussed in [Jab16]. In particular the author provided through a random construction an example of time dependent divergence-free Sobolev vector field in R 2 such that the regular Lagrangian flow has not bounded variation.
The author has been supported by the SNF Grant 182565.
1.1. 2d autonomous vector fields. The analysis in the setting of 2d autonomous vector fields is facilitated by the following Hamiltonian structure: if b ∈ L ∞ (R 2 ; R 2 ) with div b = 0, then there exists a Lipschitz Hamiltonian H : (1.2) At least formally the Hamiltonian is preserved by the flow, so that the trajectories of the flow are contained in the level sets of H. In the series of papers [ABC14b, ABC13,ABC14a], the authors reduced the uniqueness problem for the continuity equation to a family of one-dimensional problems on the level sets of H. With this approach they were able to characterize the Hamiltonians for which the uniqueness for (1.1) holds in the class of L ∞ solutions, and therefore the uniqueness for the regular Lagrangian flow, including in particular the case of BV vector fields. It is worth to mention that, before the general result in [BB20] was available, the approach introduced above allowed to obtain in [BBG16] a simpler and more direct proof of the uniqueness of regular Lagrangian flow for nearly incompressible vector fields with bounded variation; see also [BG16] for the intermediate step of steady nearly incompressible vector fields, namely vector fields satisfying Def. 1.2 with ρ constant in time.
The approximate differentiability of the flow has been obtained for autonomous divergence free vector field b ∈ BV(R 2 ; R 2 ) in [BM19], as a consequence of a suitable Lusin-Lipschitz property.
In the present paper we investigate under which assumptions the regular Lagrangian flow inherits the Sobolev or BV regularity of the vector field. The first result is a local estimate for nearly incompressible vector fields. Proposition 1.3. Let b ∈ BV(R 2 ; R 2 ) be a bounded nearly incompressible vector field and let Ω ⊂ R 2 be an open ball of radius R > 0 such that there exist δ > 0 and e ∈ S 1 for which b · e > δ a.e. in Ω. Let Ω ⊂ Ω be an open set andt > 0 be such that dist(Ω , ∂Ω) > b L ∞t. Then The following global result is stated for divergence-free vector fields and we additionally assume that the vector field b ∈ BV(R 2 ; R 2 ) is continuous. Since we are going to consider bounded vector fields, by finite speed of propagation, it is not restrictive to assume that b has compact support. In particular there exists a unique Hamiltonian H ∈ C 1 c (R 2 ) satisfying (1.2) and it is straightforward to check that the set of critical values S := {h ∈ R : ∃x ∈ R 2 (H(x) = h and b(x) = 0)} is closed. Therefore the set of regular values R := H(R) \ S and Ω = H −1 (R \ S) = H −1 (R) are open.
Theorem 1.4. Let b ∈ BV(R 2 ; R 2 ) be a continuous divergence-free vector field with bounded support and let Ω be defined as above. Then for every t > 0 the regular Lagrangian flow has a representative X(t) ∈ C 0 (Ω) ∩ BV(Ω).
The last result is an example that shows that the existence of δ > 0 as in Proposition 1.3 cannot be dropped, as well as the restriction to Ω in Theorem 1.4. Proposition 1.5. There exists a divergence-free vector field b : R 2 → R 2 such that b ∈ W 1,p loc (R 2 ; R 2 ) for every p ∈ [1, ∞), b(z) · e 1 > 0 for L 2 -a.e. z ∈ R 2 and for every time t > 0 the regular Lagrangian flow The construction of the Hamiltonian H associated to b in Proposition 1.5 is a suitable modification of the construction in [ABC13] of a Lipschitz Hamiltonian for which the uniqueness of the corresponding regular Lagrangian flow fails. As opposed to the already mentioned result in [Jab16], the proposed construction is deterministic and disproves the Sobolev regularity of the regular Lagrangian flow also for autonomous vector fields.
We finally mention that the question about the Sobolev or BV regularity of the regular Lagrangian flow associated to autonomous planar vector fields was posed to the author by M. Colombo and R.
Tione, motivated by the study of the commutativity property of the flows associated to vector fields with vanishing Lie bracket [CT20].

Local estimate for nearly incompressible vector fields
In this section we prove Proposition 1.3. We begin with two preliminary lemmas about autonomous nearly incompressible vector fields in R d . In the first lemma we show that in the case of autonomous nearly incompressible vector fields we can assume without loss of generality that the existence time T of ρ in Definition 1.2 is arbitrarily large.
Lemma 2.1. Let b : R d → R d be an autonomous nearly incompressible vector field and let ρ : Proof. By Ambrosio's superposition principle (see [AC08]), there exists a Radon measure η on Γ T := C([0, T ]; R d ) such that for every t ∈ [0, T ] it holds where e t (γ) := γ(t) denotes the evaluation map at time t defined on Γ T . We denote by {η x } x∈R d ⊂ P(Γ T ) its disintegration with respect to the evaluation map at time 0, so that Iterating the construction above we obtain a solutionρ : R + × R d → R of (1.1) such that (2.3) holds for every t ≥ T . In particular for every N ∈ N and for every t ∈ [N T, (N + 1)T ] it holds The vector fields for which the function ρ in Definition 1.2 can be chosen independent of t are called steady nearly incompressible. Although not every nearly incompressible autonomous vector field is steady nearly incompressible, we can reduce to the latter case under the assumptions of Proposition 1.3. The proof of the following lemma is an adaptation of the argument in [BBG16].
Lemma 2.2. Let b : R d → R d be an autonomous, bounded, nearly incompressible vector field and let Ω ⊂ R d be an open ball of radius R > 0. Assume that there exist δ > 0 and e ∈ S d−1 for which for L d -a.e. x ∈ Ω it holds b(x) · e ≥ δ. Then b Ω is steady nearly incompressible, namely there exists r : Ω → R andC > 0 such that Proof. Let ρ : (0, T ) × R d → R and C > 0 be as in Definition 1.2. Let η ∈ M(Γ T ) the Radon measure provided by Ambrosio's superposition principle. In particular if we denote bỹ For every γ ∈ Γ we set be the (possibly empty) connected component of I γ containing 0 and similarly let I γ,T = (t + γ , T ] be the connected component of I γ containing T . We denote bỹ and with 0 ≤ρ ≤ ρ. The following standard computation shows that the densityρ satisfies the continuity equation where in the last equality we used that ϕ(0, ·) = ϕ(T, ·) ≡ 0. In particular µ is concentrated on ∂Ω so that Since b · e > δ in Ω, every connected component of I γ has length at most 2R/δ. Up to change the constant C > 0, by Lemma 2.1 we can assume that e. x ∈ Ω, by integrating (2.4) with respect to t, it follows that satisfies div(rb) = 0 in D (Ω). From (2.5) and the definition of r, it follows that for L d -a.e. x ∈ R 2 it holds and this proves the claim withC = 3C.
In the following of this paper we restrict to the case d = 2 and in the remaining part of this section we will always assume that the hypothesis in Proposition 1.3 are satisfied. In particular there exists a Lipschitz Hamiltonian H : Ω → R such that The generic point in R 2 will be denoted by z = (x, y) and we assume without loss of generality that e = e 1 . Being b · e 1 > δ and r We will also denote byf In the following we consider vector fields b with bounded variation. As already mentioned in the introduction, the uniqueness problem for the regular Lagrangian flow associated to rb was solved in [BBG16], where in particular it is proven that the local Hamiltonian is preserved by the flow, namely ∀z ∈ Ω and t < dist(∂Ω, z).
We will consider the representative of the regular Lagrangian flow defined as follows: for every z = (x, y) ∈ Ω and t < dist(∂Ω, z), we set X(t, z) = (X 1 (t, z), f H(z) (X 1 (t, z))) where X 1 (t, z) is uniquely determined byˆX and whereb denotes the precise representative of b, defined at H 1 -a.e. z ∈ R 2 (see for example [AFP00]).
In particular, if we denote by it holds that H 1 -a.e. z ∈ H −1 (h) ∩ Ω is a Lebesgue point of b with valueb(z). In the following we will still denote by b the precise representativeb.
Proposition 2.3. Let b ∈ BV(R 2 ; R 2 ) be a bounded autonomous nearly incompressible vector field. Let Ω ⊂ R 2 be an open ball of radius R > 0 such that there exist δ > 0 and e ∈ S 1 for which b · e > δ a.e.
in Ω. Then there exists g ∈ BV loc (R) and a constant C = C (R, b L ∞ , δ, C) > 0 such that for every z, z ∈ Ω with H(z), H(z ) ∈ R and everyt > 0 for which

10)
where C is the compressibility constant in Definition 1.2 and H is the Hamiltonian introduced in (2.6).
Proof. We denote by h = H(z) and h = H(z ). By (2.9) it follows that ds, (2.11) wheref h andf h are defined in (2.7). Without loss of generality we assume x ≤ x and we also suppose that X 1 (t, z) ≤ X 1 (t, z ), being the opposite case analogous. We first estimate the distance of the horizontal components of the flows. We denote by X 1 (t, z)), I 3 = (X(t, z), X(t, z )).
If I 2 = ∅, since b · e 1 > δ in Ω, then If I 2 = ∅, it follows by (2.11) that , h ))), where I(h, h ) denotes the closed interval with endpoints h and h ; in the last inequality we used that the function v → 1/v is δ −2 -Lipschitz on (δ, +∞). Then we estimate the difference of the vertical components: (2.12) By definition of f h it holds where L denotes the Lipschitz constant of the function f h and is bounded by C 2 b L ∞ δ as in (2.8). By definition of f h we have that (2.14) where we used b 1 ≥ δ, ∂ y H = rb 1 and ∇H L ∞ ≤ C b L ∞ . Plugging (2.13) and (2.14) in (2.12), we finally obtain so that (2.10) holds with Notice that g ∈ BV loc (R) by construction, since Dg = H |Db| is a finite Radon measure.
If b ∈ W 1,p (R 2 ; R 2 ) the same computation leads to (2.10) with It only remains to check that g ∈ W 1,p loc (R). Denoting by E h = {z ∈ Ω : H(z) = h}, by the coarea formula we have that Being |∇H| = |rb| > δ/C, then by Jensen's inequality and co-area formula we get where L is as above. This concludes the proof of the proposition.
In order to conclude the proof of Proposition 1.3, we deduce in the following two lemmas the BV and Sobolev regularity of the flow from the pointwise estimate obtained in Proposition 2.3. Proof. From Proposition 2.3 it is sufficient to check that g • H ∈ BV(Ω ). Let g n be a sequence of smooth functions converging to g in L 1 (R) with Tot.Var. R (g n ) ≤ Tot.Var. R (g). By coarea formula In particular is uniformly bounded. Hence g n • H converges in L 1 (Ω ) to g • H and Corollary 2.5. Let us consider the same setting as in Proposition 2.3 with b ∈ W 1,p (R 2 ; R 2 ). Let Ω ⊂ Ω be an open set andt > 0 be such that dist(Ω , ∂Ω) > b L ∞t. Then X(t) ∈ W 1,p (Ω ).
Proof. From Proposition 2.3 it is sufficient to check that g • H ∈ W 1,p (Ω ). By chain rule and coarea formula we haveˆΩ

Global estimate for divergence free vector fields
In this section we prove Theorem 1.4. In the next lemma we show that we can cover Ω with countably many open sets invariant for the flow and such that |b| is uniformly bounded from below far from 0. The main estimate in the proof of Theorem 1.4 is proven in the following lemma.
Proof. The proof is divided in several steps.
Step 1. By Lemma (3.1) Ω k is compactly contained in the open set Ω k+1 . Since b is continuous and uniformly bounded from below on Ω k+1 , for every L > 0 there existr > 0 and a finite covering (Br of Ω k such that (1) for every i = 1, . . . , N it holds B 4r (z i ) ⊂ Ω k+1 ; (2) for every i = 1, . . . , N there exists e i ∈ S 1 such that b(z) · e i ≥ |b(z)| cos(tan −1 (L)) ∀z ∈ B 4r (z i ). (3.1) We take L > 0 sufficiently small so that cos(tan −1 (L)) > 1/2 and such that for every i = 1, . . . , N and for every h ∈ H(B 3r (z i )) there exist an open interval I i,h ⊂ R and a L-Lipschitz function f i,h : Step 2. We show that the function g : R → R defined by is continuous and with bounded variation.
Since Dg = H |Db| Ω k+1 is a finite measure, the function g has bounded variation. In order to prove that g is continuous it is sufficient to check that for every h ∈ R it holds |Db|(H −1 (h) ∩ Ω k+1 ) = 0.

By
Step 1, the set H −1 (h) ∩ Ω k+1 is the union of finitely many Lipschitz curve of finite length. Being b continuous, the measure |Db| vanishes on all sets with finite H 1 measure (see for example [AFP00]), and this proves the continuity of g.
For every z ∈ R 2 with |z −z| ≤ r we prove that for every j = 1, . . . ,Ñ there exists s j > 0 such that By (3.2) and the definition of (z i ) N i=1 , for every j = 1, . . . ,Ñ there exists a unique pointz j ∈ Ω k+1 in z ∈ B 2r (z i(j) ) such that H(z j ) = h andz j · e i = X(t j ,z) · e i . We immediately have Notice in particular that |X(t j ,z) −z j | ≤r by (3.2). In order to prove the claim, it is sufficient to show that for every j = 1, . . . ,Ñ , there exists s j ≥ 0 satisfying (3.3) and such that X(s j ) =z j . We prove this by induction on j.
Remark 3.3. If we additionally assume that b ∈ W 1,p (R 2 ) in Lemma 3.2, then the statement holds true with |∇H|dz.
In particular we showed in the proof of Proposition 1.3 that g ∈ W 1,p loc (R). Proof of Theorem 1.4. By Lemma 3.1, it is sufficient to prove that X(t) ∈ BV(Ω k ) for every k ∈ N. In the same setting as in Lemma 3.2, ifz ∈ Ω k and z ∈ B r (z), then The argument in the proof of Corollary 2.4 shows that g • H ∈ BV(Ω k ) therefore it follows from (3.8) that X(t) ∈ BV(Ω k ∩ B r (z)) for every z ∈ Ω k . Being Ω k bounded this proves that X(t) ∈ BV(Ω k ).
Finally the continuity of X(t) follows immediately from (3.8) and the continuity of g. If moreover we assume that b ∈ W 1,p (R 2 ), then the same argument proves that X(t) ∈ W 1,p (Ω k ) thanks to Remark 3.3 and Corollary 2.5.
Remark 3.4. By inspection in the argument used to prove Lemma 3.2, we observe that X(t) BV (Ω) (or X(t) W 1,p (Ω)) is locally bounded for t ∈ [0, +∞) and it diverges at most linearly in t as t → ∞.

Example
In this section we prove Proposition 1.5.
4.1. Construction of C n , D n , E n and F n . We consider the following parameters: c n = 1 n 2 2 n , a n = n − 1 2n c n 2 − c n+1 ∼ 1 2 n−1 n 3 , r n = 1 2n c n 2 − c n+1 ∼ 1 2 n−1 n 4 . (4.1) We set C 1 = [0, 1/2] 2 ⊂ R 2 and we inductively define C n+1 for n ≥ 1 as follows: C n+1 ⊂ C n and every connected component R of C n contains two connected components of C n+1 , which are squares of side c n as in Figure 1. For every n ∈ N we also consider the sets D n , E n , F n ⊂ C n as in Figure 1. We observe that for every n ≥ 1 it holds

4.2.
Construction of f n and h n . The function f 0 : R 2 → R is defined by f 0 (x, y) = y. The function f n coincides with f n−1 on R 2 \ C n and its level lines in C n are as in Figure 2. In particular f n coincides with f n−1 on F n . Let R be a connected component of D n , then f n is affine on R and depends only on y, therefore ∇f n = (0, v n ). Let R be a connected component of C n and denote by s n = Osc(f n , R ).  In particular the infinite product is strictly positive and we denote it by σ. We finally get v n = s n+1 c n+1 ∼ c 1 σ n 2 2 n .
Similarly we compute the speed ∇f n = (0, v n ) in the region E n as in the picture. Denoting by R one of its components, we have

4.3.
Estimates on the norms of ∇f n and ∇h n . We first estimate ∇f n L ∞ (Cn) . From Figure 2 we observe that ∂ 2 f n L ∞ (Cn) = v n and the maximal slope of the level sets of f n in C n is cn−8an 4an . Therefore ∇f n L ∞ (C n ) ≤ v n c n − 8a n 4a n + 1 ∼ c 1 σ 16 n 4 2 n .
Being f n affine on each connected component of D n , it follows from the properties of the convolution kernel that h n * ρ n (z) = h n (z) for every z ∈ D n such that dist(z, ∂D n ) > r n . We denote bỹ D n := {x ∈ D n : dist(x, ∂D n ) > r n }, E n := {x ∈ E n : dist(x, ∂E n ) > r n }, F n := {x ∈ D n : dist(x, ∂F n ) > r n }.
Observe that all the sets above are non-empty by the choice of the parameters (4.1). SinceD n = C n+1 , we have in particular that f n =f n on the set C n+1 . Similarly h n * ρ n = h n onẼ n ∪F n . As in Section 4.4, we denote byT 1 n the total amount of time needed by an integral curve of the vector field −∇ ⊥f n−1 to cross a connected component of C n . Since f n−1 =f n−1 on C n , thenT 1 n = T 1 n . We moreover denote bỹ T s n the amount of time needed by an integral curve of the vector field −∇ ⊥f n intersectingD n to cross a connected component of C n . Sincef n = f n inD n ∪F n it is straightforward to check thatT s n ∼ T s n . Similarly, we denoteT f n the amount of time needed by an integral curve of the vector field −∇ ⊥f n intersectingẼ n to cross a connected component of C n . Sincef n = f n inẼ n ∪F n it is straightforward to check thatT f n ∼ T f n . We are now in position to repeat the argument in Sections 4.4 and 4.5 and this proves that for every t > 0 the regular Lagrangian flowX of the vector field −∇ ⊥f satisfies X(t) / ∈ BV loc (R 2 , R 2 ).
This concludes the proof of Proposition 1.5.