Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

We study the regularity of the flow X(t,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{X}}(t,y)$$\end{document}, which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution ρ∈L∞(Rd+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \in L^\infty ({\mathbb {R}}^{d+1})$$\end{document} of the continuity equation ∂tρ+div(ρb)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t \rho + {{\,\textrm{div}\,}}(\rho {\varvec{b}}) = 0, \end{aligned}$$\end{document}with b∈Lt1BVx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{b}}\in L^1_t {{\,\textrm{BV}\,}}_x$$\end{document}. We prove that X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{X}}$$\end{document} is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\varvec{X}}(t,y+rz) - {\varvec{X}}(t,y)}{r} \underset{r \rightarrow 0}{\rightarrow }\ W(t,y) z \quad \text {in measure}, \end{aligned}$$\end{document}where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} W(t, y) = \frac{(D {\varvec{b}})_y(\textrm{d}t)}{J(t-,y)} W(t-, y), \end{aligned}$$\end{document}where (Db)y(dt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D{\varvec{b}})_y(\textrm{d}t)$$\end{document} is the disintegration of the measure ∫Db(t,·)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int D {\varvec{b}}(t,\cdot ) \, \textrm{d}t$$\end{document} with respect to the partition given by the trajectories X(t,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{X}}(t, y)$$\end{document} and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} J(t,y) = ({{\,\textrm{div}\,}}{\varvec{b}})_y(\textrm{d}t) = \textrm{Tr}(D{\varvec{b}})_y(\textrm{d}t). \end{aligned}$$\end{document}The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a BV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{BV}\,}}$$\end{document} vector field.


Introduction
We consider a vector field b : R × R d → R d of class L 1 t BV x , and a solution ρ ∈ C([0, T ], L ∞ w (R d )) to the continuity equation ∂ t ρ + div(ρb) = 0, (t, x) ∈ (0, T ) × R d . (1.1) We assume that b and ρ are compactly supported. From the results of [16], it follows that ρ has a unique representation is terms of characteristics, that is absolutely continuous solutions to the ODE d dt γ (t) = b(t, γ (t)), t ∈ (0, T ).
More precisely, there exists a unique flow X : [0, T ] × R d → R d , defined for ρ(0, ·)L d -almost everywhere y ∈ R d , such that ρ(t, ·) = X(t, ·) (ρ(0, ·)L d ), which means that, for every test function ϕ ∈ C ∞ c ((0, T ) × R d ), For the precise statement, see Theorem 3.5 of Section 3.2. The appropriate notion of flow for ODEs driven by rough (non-Lipschitz continuous) vector fields, introduced in the seminal papers [8,40], is the one of regular Lagrangian flow, which consists of a measurable selection of characteristics such that X(t, ·) L d CL d holds (see, for example, [9,10] for further information).
The main result of this paper is the differentiability in measure of the flow X (in the sense of Ambrosio-Malý, see [13]). Let (Db) y be the rescaled conditional probabilities associated with the disintegration of Db along the trajectories of X; that is, if where F is a σ -compact set where ρ(0, ·) is concentrated, then (up to a negligible (ρ(0, ·)L d )-set) or, equivalently, for every test function ϕ ∈ C c (R + × R d ), Similarly, for the divergence div b, we can write div b F =ˆF (div b) y (dt)L d (dy), (div b) y = Tr(Db) y .
Our main theorem is as follows: In the statement W (t−, y), J (t−, y) are the left limits of W (·, y), J (·, y), whose existence follows from the fact that they solve their respective ODE with measure right-hand side and then are BV functions of time; we also notice that, in general, W (·, y) and J (·, y) are discontinuous due to the singular measure (Db) y . We remark that the convergence in measure expressed by formula (1.2) can be written equivalently as 1 ∧ X(T, y + r z) − X(t, y) r − W (T, y) · z dy dz = 0.

Uniqueness and regularity of the flow associated to a rough velocity field
The study of the well-posedness of transport equations driven by rough velocity fields started with the pioneering paper [40], where DiPerna and Lions introduced the notion of renormalized solution and proved existence and uniqueness for (1.1) in the case of Sobolev W 1, p vector fields (with p ∈ [1, ∞]) with bounded divergence (or divergence in a suitable L p space). Ambrosio extended the theory to BV vector fields with bounded divergence in [8] (see also [32,48]). More recently, Bianchini and Bonicatto proved a uniqueness result in the more general case of nearly incompressible BV vector fields (see [16]), obtaining, as a consequence, a positive answer to Bressan's compactness conjecture (see [27]). We refer also to the more recent [17] for a variation of the strategy in [27].
A locally integrable vector field is called nearly incompressible if there exists a solution C −1 ρ(t, x) C for L d−1 -almost everywhere (t, x) ∈ (0, T ) × R d to the continuity equation (1.1); such assumption is implied by the stronger condition ÷b ∈ L ∞ . We refer the reader to [9,10,37] and the references therein for a more comprehensive overview of this area of research.
In case b ∈ L 1 t W 1,1 x and divergence-free (plus some growth assumptions), in [46], Le Bris and Lions proved that, if X(t, y) is the unique regular Lagrangian flow generated by b, then there exists a limit for the incremental ratio X(t, y + εr ) − X(t, y) ε → ε→0 W (t, y, r ) in measure, and W (t, y, r ) is a renormalized solution to ∂ t W (t, y, r ) = ∇ y b(X(t, y))W (t, y, r ), W (0, y, r ) = r,Ẋ = b(t, X), or, equivalently, any renormalized solution to is given by ϕ(t, X(t, y), W (t, y, r )) = ϕ(0, y, r ).
In [13], Ambrosio and Malý proved that W (t, y, r ) = W (t, y)r , and compared this differentiability in measure to other notions of differentiability. As it turns out (see [13,Section 5]), this property is much weaker than approximate differentiability (see [11,Section 3.6]). Approximate differentiability of regular Lagrangian flows generated by W 1, p vector fields, with p > 1, was first obtained by Ambrosio, Lecumberry and Maniglia in [12]. In [33], Crippa and De Lellis improved this result by proving a quantitative estimate of Lusin-Lipschitz type for the flow generated by a L 1 t W 1, p x vector field with bounded divergence, with p > 1: for every ε, one can remove a set of measure ε and X(t = T ) on the remaining set coincides with a Lipschitz continuous function having Lipschitz constant e C/ε . Their approach is based on a priori estimates for a functional measuring a "logarithmic distance" between two flows associated to the same vector field (see also [21,24,34,44,45,49,[53][54][55]57,58] for related results that rely on this strategy). However, as noted in [31], this approach cannot be used to prove a regularity result for the flow associated to a BV vector field.
A quantitative Lusin-Lipschitz regularity results for the flow X associated to a vector field b implies lower bounds on the mixing scale of passive scalars driven by b through the transport equation (1.1) (see [56]). In particular, extending the result by Crippa and De Lellis to the case p = 1 would give a positive answer to the well-known Bressan's mixing conjecture proposed in [28] (see also [5][6][7]15,29,30,35,36,41,43,47,50,51,59,62] for related results on both transport and advection-diffusion equations).
For the special case of bounded autonomous divergence-free vector fields b ∈ BV(R 2 ; R 2 ) with compact support, in [23], Bonicatto and Marconi proved a Lusin-Lipschitz regularity result and showed that the Lipschitz constant grows at most linearly in time. In this setting, the analysis is facilitated by the Hamiltonian structure of the vector fields (see [2][3][4]18,20,22]).
In the present paper, we establish the differentiability in measure for a nearly incompressible vector field b ∈ L 1 t BV x . Our approach is based on the localization of the problem (which relies on the theory of proper sets introduced in [16]): we exploit the local structure of the vector field b to prove differentiability in measure locally; then, Theorem 1.1 is obtained by suitably combining the local estimates.

Notations
For an integer d 1, the d-dimensional Euclidean real vector space is denoted by R d . We write the component of a d-dimensional point or vector as x = (x 1 , . . . , x d ); we also write x i, j,... to denote the point obtained by removing the coordinate component i, j, . . . from x. The unit vector along the i-coordinate is e i .
The d-dimensional ball in R d of radius r centered at x is written as B d r (x). Given a curve t → γ (t) ∈ R d , we write The relative closure of the set A in the topological space B is denoted by clos(A, B); we also write clos A when the ambient topological space is clear. Similarly, the interior of a set A is written as int A or int(A, B). The boundary is denoted by Fr A or Fr(A, B) or, sometimes, by the standard notation ∂ A. We write A B if clos A is a compact set contained in B.
I is the identity matrix, the minimum between two quantities a, b is denoted by a ∧ b, and the maximum by a ∨ b.
The d-dimensional Lebesgue measure is denoted by L d , and the k-dimensional Hausdorff measure by H k .
If X is a set and A is a σ -algebra on X , we will call (X, A ) a measure space. A measure μ is concentrated on a set C ⊂ X if |μ|(X \C) = 0. Let μ be a measure on (X, A ) and A ∈ A . We define the restriction μ A of μ to A as the measure on A given by μ A (E) := μ(A ∩ E) for any E ⊂ A .
The σ -algebra generated by open sets is called Borel σ -algebra and will be denoted by B(X ). Let X, Y be two metric spaces, μ a measure on (X, B(X )) and f : X → Y a Borel function. We define the push-forward of μ with respect to f as the measure on (Y, B(Y )) given by f μ(B) := μ( f −1 (B)) for all B ∈ B(Y ). In particular, for a Borel map g : Y → R it holds that Y g(y)( f μ)(dy) =ˆX (g • f )(x)μ(dx).
The disintegration of a measure μ with respect to a partition {A α } α is written as where f is the partition function, that is f −1 (α) = A α (see [42,Section 452]).
The Lebesgue spaces L p (X, μ; Y ) are defined in the usual way; if X = R d and μ = L d , we just write L p (R d ; Y ); if, moreover, Y = R, we write L p (R d ). We use the standard notation for Sobolev spaces. We denote by [M loc (X )] m and by [M(X )] m , respectively, the space of R m -valued Radon measures and the space of R m -valued finite Radon measures. The space [M(X )] m is a Banach space with the norm μ M := |μ|(X ), where |μ| is the total variation of the measure μ. In the case m = 1, we denote the set of signed Radon measures, positive Radon measures, and finite Radon measures by M(X ), M + (X ), and M b (X ) respectively (see [11,Chapter 1]).
We say that b ∈ L 1 ( ; R m ) has bounded variation in , and we write b ∈ BV( ; R m ) if Db is representable by a R m×d -valued measure with finite total variation in . Endowed with the norm b BV( [11,Chapter 3]). We will use the notation for the continuous, jump, absolutely continuous, and Cantor parts of Db.
Given a Banach space X , by L p ([0, T ]; X ) we denote the Lebesgue-Bochner space of strongly measurable maps f : For the sake of brevity, we often write L p t X x to indicate L p ([0, T ]; X ). We add the subscript loc to denote properties which holds locally.
For a vector field b : R d+1 → R d , sometimes we also use the notation b(t) : while Db(t) denotes the space derivative of b at time t. Similar notations are used for |Db|. We write f (x±) to denote the right/left limit of f in x (when such limit exists, for example in case f ∈ BV(R), see [11]).
If A is a Borel set of positive measure, we write the average integral of f ∈ L 1 (μ) as We say that γ : If the ODE above generates a flow, we use the notation X(t, s, y) for the solution to (1.5) with initial data y at time s. The graph of X in a time interval (s, t) is denoted by X((t, s), y), and when we restrict the curve to some open set we will use the notation X(t, t − (y), y), with y ∈ ∂ and X (t − (y), t − (y), y) = y; the exit time is denoted by t + (y). For the sets (perturbed proper sets) that we are considering in this paper, all quantities (entering/exiting time t ± (y) with respect to the regular Lagrangian flow X) are well defined (cf. the discussion in [16]). If K is a compact set of initial data, we use the notation K to denote the union of its trajectories, K = y∈K X((t + (y), t − (y)), y).

Structure of the Paper
The proof of our main result is quite technical. In this section, we outline its structure and the reason of the technicalities. Moreover, we provide a sketch of the proof under the stronger assumption b ∈ L 1 t W 1,1 x (which makes the argument much easier) and show where the difficulties for the BV case lie.
In Section 3, we present some preliminary results that are needed in the proof of our main theorem. In Section 3.1, we collect some technical results on the existence of open sets ⊂ [0, T ] × R d with particularly nice properties for the vector field (1, b), the so-called proper sets, introduced in [16]. Roughly speaking, these are open sets where the problem can be meaningfully localized. Since the argument of the proof is based on the analysis of local properties of the vector field b, the tool of proper sets plays a fundamental role. The main results are Lemma 3.2, which states that there are sufficiently many of them, and Theorem 3.4, which allows us to perturb them so that there are finitely many "time-flat" boundary regions where the majority of the flow of (1, b) is entering or leaving. The motivation for this construction is that it is much easier to state the differentiability of the flow X when it is parameterized by its crossing point y on a flat surface; we acknowledge that it is also possible to avoid this, but we decided to use perturbed proper sets since this tool has already been established in the literature (see [16]). Section 3.2 deals with Smirnov's decomposition of (1, b), which is stated in Theorem 3.5: that is, thanks to the superposition principle, which has been established by Ambrosio in [8] (see also [60] in the context of a general normal 1-current and [61]), every non-negative (possibly measure-valued) solution to the PDE (1.1) can be written as a superposition of solutions obtained via propagation along the characteristics of b (such representation is also called a Lagrangian representation, see [16,Section 5]). Theorem 3.5 is used to construct L ∞ solutions ρ satisfying (1.1) by considering the curves γ a of the decomposition which start from 0 and arrive to T , and such that the Jacobian of the transformation γ a (0) → γ a (t) is uniformly bounded.
In Section 3.3, we observe that our main theorem also gives the differentiability in measure of the Smirnov decomposition of (1, b): by a countable partition of the set of curves {γ a } a used in the Smirnov decomposition, one can find countably and apply Theorem 1.1 to this set of trajectories. Finally, in this section, we also select the curves for which we address the differentiability in order to have a uniform control of the rescaled conditional probabilities (Db) y and (div b) y and to have y → X(·, y) continuous in C 0 . The precise statement is in Proposition 3.6, which is an application of Lusin's theorem.

Case
We sketch the proof of differentiability in measure for the case b ∈ L 1 t W 1,1 x . Under this assumption, we can directly estimate lim r 0ˆRdˆBd Here we make use of the fact that the rescaled conditional probabilities (Db) y are given by ∇ b(t, X(t, y))J (t, y) due to the change of variable (t, x) → (t, X(t, y)) and Fubini's theorem. We remark that, by Fubini's theorem, we also have ∇ b(t, X(t, y)) ∈ L 1 (0, T ), so that the ODE (6.1) is well-defined.

(2.2)
This estimate follows from integrating of the infinitesimal error at time t > 0 b(t, X(t, y + r z)) − b(t, X(t, y)) − ∇b(t, X(t, y)) X(t, y + r z) − X(t, y) , along the trajectory, and multiplying it by the Lipschitz constant e´T 0 |Db|(t,X(t,y)) dt of the semigroup generated by (2.1). Since we are considering trajectories {X(·, y)} y∈K such that for some fixed M (this is part of the statement of Proposition 3.6, see discussion above), we have the exponential factor in (2.2) is bounded by e M and the Jacobian is controlled by where C d is a dimensional constant and The last integral in (2.5) converges to 0 due to the continuity of translations in L 1 , and this shows that the set of trajectories starting in B d r (y) and exiting the cylinder can be made arbitrarily small and, for the remaining ones, the double integral converges to 0: more precisely, since by (2.1) and (2.3) one has |W (t, y)| e M , then, by Chebyshev's inequality, This yields the convergence in measure.

Case
The argument above also highlights what is the key difficulty of the BV case: the dependence R d y → (Db) y ∈ M(R) is only weakly continuous, and then (2.5) gives only a bound in terms of the constant |Db| y (0, T ) M, and the last integral of (2.5) does not converge to 0. The present paper deals precisely with how to remove this difficulty.
The following diagram represents a general scheme of the proof and outlines its various components as well as the relations among them: Section 5: the general argument on how to prove differentiability in measure from a local approximate version of differentiability in measure.
Section 7: the local differentiability for the singular part D sing b. The sections are almost independent from each other, and their arrangement in the paper could be altered. We first study the ODE (Section 4) to obtain some useful bounds on W (t, y), and then present the local-to-global argument (Section 5), in order to have a clear picture of the local estimates one has to prove. As one can imagine, the most complex part of the paper is the one concerning local estimates for the singular part D sing b.
In the remaining part of this introduction, we present a detailed description of these core sections. According to the notations of Section 1.2, we write (t − , t + ) for the interval of time a trajectory spends inside an open set (and (t − i (y), t + i (y)) if the trajectory is X(t, y) and the open set is i ). When we are considering a single proper set , trajectories are parameterized by their entrance point y, and are considered distinct after reentering. This is in accord with the property of proper sets that the restriction of a Lagrangian representation to a proper set is still a Lagrangian representation (see [16,Section 5]).
In Section 4, we study the ODE (1.3) for the Jacobian matrix W (t, y), that is Since this is not the classical setting, we provide a constructive proof of the wellposedness theorem (Theorem 4.1) based on the convergence of an Euler scheme. An interesting observation (Remark 4.3) is that if we require the ODE for W to be time invertible, that is that W (T − t, y) satisfies the rank-one property of the vector field is needed (see [1]). This remark could be used in the case of 2d-autonomous vector fields to have another proof of Alberti's rank-one theorem, because in this case the well-posedness does not require rankone (see [4]), although clearly there are much simpler proof of rank-one property in the literature (see, for example, [38,39,52]). The core of the proof is in the next four sections: in order, first, we present the argument to prove the differentiability in measure if there exists a partition into perturbed proper sets where suitable properties are satisfied (Section 5), then these properties are proved for the a.c. part of the derivative (Section 6) and for the singular part (Section 7), and finally the partition is constructed (Section 8).
The local-to-global argument is in Section 5: we prove that the existence of a partition into (perturbed) proper sets where approximate local differentiability assumptions are satisfied implies a global result on differentiability in measure. In the beginning (page 18), the key assumptions on the partition into perturbed proper sets are stated, which can be explained as follows: apart from the smallness of a measure μ P controlling the total error (Assumption (1)) and the fact that the trajectories considered for the differentiability are sufficiently close (Assumptions (2) and (3)), the key assumption is that there exists an approximate flowX(r, y; t, z) which approximates both the perturbation X(t, y + z) − X(t, y), when the latter quantity has R d -norm smaller than r , and also the derivative W (t, y)z (Assumptions (5) and (7)). Moreover, the approximate flowX has a controlled growth, as in Assumption (6). The reason why we need to introduce this approximate flowX is because y → (Db) y is only weakly continuous, as we explained before in Section 2.1, so we choose a flowX that solves an ODE for which the convergence ofẊ to (Db) y is in mass and not in the weak sense (or, equivalently, their difference in norm is small). This comparison works only at the initial and final time (as shown also in Assumption (7), where the comparison is directly between X(t, y + z) − X(t, y) and W (t, y)z). There are some additional technical assumptions, in particular that the estimates are valid only after removing some trajectories (Assumption (4)), which is also the reason why we obtain only differentiability in measure (instead of approximate differentiability).
The argument to pass from these local assumptions to a global differentiability result is presented in Proposition 5.1. First, we remove all trajectories which do not satisfy the previous estimates in some of the sets i of the partition: these are controlled by the measure μ P , which is assumed to be small (Step 1-3 of the proof). Second, we control the perturbations X(t, y + z) − X(t, y) which do not remain close to 0 (that is X(t, y + z) not close to X(t, y)) for all t ∈ [0, T ] (Step 5-11 of the proof): the idea here is that, in order to exit the ball B d R (X(t, y)), a trajectory has first to growth much more of the approximate flowX(R, y; t, z), and a suitable choice of the initial distance r and of R yields a control on these runaway trajectories (similar to (2.6)). For the remaining ones, a suitable comparison with the linearized flow W (t, y)z holds. This yields the differentiability in measure (Step 12-13).
Sections 6 and 7 show that it is possible to construct proper sets where the local estimates required at the beginning of Section 5 are satisfied. The analysis of the absolutely continuous part is roughly the same as the one sketched in Section 2.1 for the b ∈ L 1 t W 1,1 x case; as an additional error term, the mass of the singular part D sing b inside the proper set also appears. The analysis of the singular part is instead the core of the paper, and requires many technical estimates. The first step is to consider a small neighborhood of a Lebesgue point of the singular part of the derivative (Section 7.1). This allows us to write Db ξ ⊗η|Db| (by Alberti's rank-one theorem), and to use the latter measure to build an approximate vector field whose flow isX. The definition of the approximate vector fieldb H (r, y; t, w) is in Section 7.2, and its explicit expression is in formula (7.5), namely assuminḡ η = e 1 andξ =ξ 1 e 1 +ξ 2 e 2 , : this allows to prove that removing a small set of trajectories we still have that X 2 to the W (t, y)z; this is analyzed in Section 7.6: first, we can replace (Db) y with ξ ⊗η|Db| y with a controlled error; then, the explicit solution to the ODĖ which turns out to be close to the perturbed flowX H . This concludes the estimates, which are collected in Sections 7.7 and 7.8 . Finally, Section 8 concerns the construction of a disjoint partition of [0, T ]×R d into perturbed proper sets as required in Section 5 and is based on the analysis of the absolutely continuous part (Section 6) and the singular part (Section 7) of the derivative Db. First, we cover a large portion of the singular part D sing b with disjoint perturbed proper sets so that the required estimates holds, and then the remaining part. This is done in Theorem 8.1 and Proposition 8.2. The proof of our main theorem is thus concluded.
In Appendix A, we give a proof of the estimate (2.2) in our setting.

Preliminaries and Setting of the Problem
In this section we collect some preliminary information on proper sets and the decomposition of a BV vector field; then we present the setting of our problem.

Proper sets
The analysis of open sets such that b maintains suitable regularity properties has been carried out in [16]. In this section, we present the main definitions and results. (1) ∂ has finite H d -measure and it can be written as where N is a closed set with H d (N ) = 0 and {U i } i∈N are countably many C 1 -hypersurfaces such that the following holds: for every (t, (2) If the functions ϕ δ,± are given by where n is the outer normal to ∂ .
The condition (2) above can be seen as the requirement that the distributional flow is actually crossing the boundary, for example there are no regions where the flow is 0 but the trajectories hit the boundary: it requires indeed that the flux seen by the functions ϕ δ,± (which sees the trajectories next to ∂ ) converges in norm to the distributional flux. See [16] for a deeper discussion.
It is possible to prove that balls B r (t, x) and cylinders are proper sets for almost everywhere r > 0 (see [16,Lemma 4.10]). their difference 1 \ 2 is also a proper set. We prove the last claim in the following lemma:

Lemma 3.3. Let 1 , 2 be proper sets such that
Proof of Lemma 3.3. If is proper, so is int(R d+1 \ ) since the conditions to be proper are given on ∂ = ∂(int(R d+1 \ )). Thus, by writing the result follows from [16,Proposition 4.11].
Furthermore, it is possible consider a perturbation ε of a proper set in order to have a large part of the inflow and outflow of ρ(1, b) across ∂ ε occurring on finitely many time-constant hyperplanes, that is regions of the boundary ∂ ε such that their outer normal is n = (±1, 0). We shall call S 1 the union of the hyperplanes of inflow and S 2 the union of the hyperplanes of outflow. More precisely, the following theorem holds true (see [16,Theorem 4.18]): Here, we denote by ρ[ (1, b) · n] ± the positive/negative part of the trace ρ(1, b) · n respectively.

Decomposition of BV vector fields
The following result summarizes [16, Main Theorem 1, p. 18] applied to the PDE and w a I a > 0; (3) when w a is extended to 0 outside I a , then it is a BV function, , then f is a partition via characteristics as above also for , that is the same results as above are true replacing and μ, μ a with ν, ν a in (3.5).
A possible choice of f is to take countably many sets {t = t i } i∈N and define f(γ ) = γ (t i ). This choice is more suitable when one wants to construct a flux from the partition via characteristics. Indeed, with this choice, the function w a becomes naturally the Jacobian J (t, y), where γ (t i ) = y and (3.4) is the equation for the evolution of J .
A corollary of formula (3.3) is that, given a proper set , we can estimate the flux across its boundary as follows. Let γ −1 a ( ) = ∪ i I i,a , I i,a ⊂ I a open intervals with a → I i,a Borel. By [16,Proposition 5.11] the Lagrangian representation of ρ(1, b) is written as the sum of the Lagrangian representation of ρ(1, b) restricted to the sets i = ∪ a (I, γ a )(I i,a ). Since γ a I i,a is not crossing the boundary of , we have Summing up with respect to i and observing that the first integral in the last formula above is the divergence of ρ(1, b) inside while, being proper, the last integral becomes the trace of −ρ(1, b) on ∂ , we conclude that where n is the inner normal to . In particular, from [16, Theorem 6.8 and Proposition 6.10], we obtain that, for N ⊂ ∂ , that is the flux through N controls the measure of trajectories crossing N .

Setting of the problem
We consider the set of trajectories starting from t = 0 and arriving to t = T living inside the ball of radius R 0 and such that J (t, y) ∈ [1/C,C]. By an elementary partition argument, the partition via characteristics of (1, b) can be decomposed as a countable union of such a sets by varyingC and the initial and final time (here for definiteness we have assume them to be 0, T , respectively). We can define ρ = 1/J and obtain a solution to div t,x (ρ (1, b) We denote with K 0 a compact set made of these trajectories, that is Being y → X(·, y) a Borel function, the above sets are compact, and K 0 can be parameterized by the initial data, that is Since the values of b outside (0, T ) are not important, we assume that b(t) = 0 for t / ∈ (0, T ), and also outside the ball of radius 2R 0 . We will often write R d+1 in the estimates, even if we are working in the ball of radius 2R 0 . In the set K 0 we disintegrate according to the partition (3.7), and the uniqueness of the normalization for (Db) y is assured by requiring Db y = 1 for m ⊥ -almost everywhere y ∈ R d . Hence the measure is uniquely defined, and it corresponds to the part of Db K 0 whose image measure is not absolutely continuous, and we have Being the flow defined for L d -almost everywhere y ∈ K 0 , we can assume that (Db K 0 ) r = 0 by removing a negligible set of trajectories. Since then, for every M, then there exists a compact set K 1 ⊂ K 0 of trajectories such that We also define K 1 ⊂ K 0 as the union of the graphs of the trajectories starting in K 1 , as in (3.7). We observe that, by the monotonicity properties of measures, if K is another compact set of trajectories such that Summing up, we are in the following situation: Proposition 3.6. We can restrict to a compact set of trajectories K 1 ⊂ K 0 such that (2) X K 1 is continuous; for some constantC.

The ODE Satisfied by the Derivative of the Flow
We consider the Cauchy problem where the Jacobian J (t, y) satisfies and, by assumption, In this section the variable y is a fixed parameter.
Moreover, it is the limit of every sequence of Euler scheme solutions W δt (t, y) corresponding to a partition Proof. For the sake of brevity, we use the notation By the assumptions on the disintegration and near incompressibility, we have As a first step, we prove existence of a solution to the ODE by means of an Euler scheme (see [14]). Secondly, we prove uniqueness by a Gronwall-type argument.
Step 1. Construction of a solution. The construction of a solution is done by the Euler method: for every partition of and we have used the fact that J (0, y) = 1. With an abuse of notation, we used the apex δt (the maximal size of interval of the partition) to denote the partition with point {t i } i , later we will also denote a sequence of functions depending on the partitions {t n i } i with the apex δt n (again the maximal size of the interval of the partition).
The function W δt is piece-wise constant, right continuous, and its jump at each t i is given by We have that W δt is uniformly bounded. Indeed, (4.6) Moreover its total variation is controlled by Therefore, by Helly's Compactness Theorem (see [26,Theorem 2.3]), for every sequence of intervals such that δt → 0 there is a subsequence δt n such that By the estimate on the total variation, for every t < τ we have where we have used the estimate in the third line, and the Jacobian bound As the set of times for which W δt n (t, y) is convergent dense in [0, T ], it follows by letting t τ that the limit W δt n exists for every t and moreover t → W (t, y) is left continuous by (4.9): clearly W (0, y) = I. A similar result can be stated for (4.10) In this case, the proof is elementary. Hence, we can pass to the limit to the approximate ODE for W δt n ; its equation is where, as in the previous equation, the matrix valued measure (Db) y δt n (dt) is defined as We write the ODE (4.11) in integral form: (4.12) Here we observed that and we have to leave out the final interval for which t ∈ [t i−1 , t i ).
From the pointwise convergence, we obtain that while, from δt n → 0 and the boundedness of W (t, y)/J (t, y), Hence, for every δt → 0, there exists a subsequence converging to a solution.
Step 2. Uniqueness of the solution. The uniqueness of the solution can be proved by observing that where we have allowed the initial data to be general, and the τ i 's denote the jump set of W (·, y), a subset of the set where the jump part of (Db) y (dt) is concentrated: the first inequality follows from (4.1) for the a.c. part and log(x) x − 1, the second inequality again from (4.1) for the jump part.
Thus, we conclude that which gives the uniqueness.

Remark 4.2.
(Time reversibility of the ODE) We note that the ODE is time reversible. Being b(t) a BV function, by Alberti's rank-one theorem we can write for the singular part of (Db) y as follows: By (4.15), we have the relations In particular, we have that Remark 4.3. (Time reversibility and rank-one property) We remark that, in turn, the invertibility of the ODE does not imply that the vector field satisfies the rank-one property. The invertibility condition is that for the singular part (4.16) However, it turns out that the above condition is valid also for the matrix ⎛ which is not of rank one. In the 2 × 2 case, on the other hand, where the proof of the existence of the flow is independent from the rank one property (see [18]), condition (4.16) is a characterization of rank-one matrices (since it is equivalent to det A = 0).

Local-to-Global Argument
The key idea of our proof is to build the derivative in measure by patching together local estimates. In this section, we show how the existence of a partition into (perturbed) proper sets where an approximate differentiability in measure property is satisfied leads to a global estimate on the differentiability in measure.
We assume that there is a finite partition proper sets (up to the negligible set made of their boundaries) such that the following local estimates hold true.
(2) Removal of a small set of initial points: in each set i of the partition, let S i,1 , S i,2 be the part of the boundary where the outer normal is (∓1, 0) respectively, as in the paragraph following Lemma 3.3. Then there exists a set of initial point S i,1 ⊂ S i,1 ∩ K 0 whose co-measure is The set S i,1 is the boundary part of the (perturbed) proper set i defined in Theorem 3.4, with ε = μ P ( i ). Moreover, up to a H d -negligible set, S i,2 is contained in ∪ j S j,1 ∪ {t = 0, T } up to a H d -negligible set: this means that the trajectories exiting one (perturbed) proper set from S i,2 are entering another (perturbed) proper set trough S j,1 (or are initial-final points). An equivalent way of expressing (5.1) is to say that the measure of trajectories we remove is less than μ P ( ). (3) Cylinders where the linear approximation is constructed: there exists R i such that for every y i ∈ S i,1 the set is the time coordinates of the points on S i,1 ). In particular, y i + B d R i (0) ⊂ S i,1 , and similarly for the exit point ) Bad set of trajectories for the linear approximation: for every y i ∈ S i, 1 and r i R i there exists a set of initial points (5) Error estimate for the flow generated by an approximate vector field: for every y i ∈ S i,1 and r i R i , there exists an approximated vector field b i (r i , y i ; t, w i ) such that the flowX generated by ) Control on the approximate flow: the approximated solutionX(r i , y i ; t, z i ) satisfies for r i r î : then the remaining trajectories satisfŷ With the above assumptions, we proceed to prove the differentiability in measure of the flow. (1) there is a set K 2 ⊂ K 1 of initial points of co-measure where the factor O(1) depends only on M,C and d; (3) in the remaining set, we havê Together with Point (1) of Proposition 3.6, this gives the differentiability in measure of Theorem 1.1, under the assumptions (1)-(7) above. In the following sections, we will show how to construct the partition and obtain the estimates.
Proof. The proof is organized into several steps. The idea is that one uses the comparison with the linear flow when the perturbed trajectory X(t, y + z) is not exiting the cylinder X(t, y) + B d r (0), while the estimate using the approximated flow controls how many trajectories are exiting from X(t, y) + B d R (0), 0 < r < R. Then, a suitable choice of r, R allows to prove the claim.
(1) Removal of trajectories which are not inside S i,1 . We remove trajectories of K 0 for which X(t − i (y), y) / ∈ S i,1 (and we control also the trajectories not entering in S 1,i or leaving from S 2,i , that is the ones which cross on the lateral boundaries, because of the last part of Point (2) of the assumptions: by nearly incompressibility and formula (3.6), the measure of trajectories we remove is less than Thus we restrict to a compact set K 2 ⊂ K 1 whose co-measure in K 2 is bounded byCε P . This set is the set of Point (1) of the statement. (2) Choice of the radius of the cylinders and definition of the partition of sets crossed by a trajectory. LetR = min i R i and, for each y ∈ K 2 , let i y be the sequence of proper sets i which the trajectory X(t, y) is crossing. We will abuse the notation, writing (i − 1) y for the predecessor of i y , 1 y for the initial i y , 0 y = 0, and so on; we also note that one trajectory may cross a given i several times, however from [16, Corollary 6.9] the number of crossings is finite for L d -almost everywhere y ∈ K 2 , so there are only finitely many indexes i y . The exit time of a trajectory X(t, y) from i y will be denoted by t i y .
(3) Removal of the set of perturbations which do not behave mildly. For every y ∈ K 2 , remove all z ∈ B dR (0) ∩ (K 0 − y) such that −1) y , y)). (5.6) This means that at time t (i−1) y we remove the trajectories which do not satisfy (5.2) while in i y . Here we have used the notation The i ranges from 0 y toī y (z) corresponding to the index of the set i such that the trajectory X(t, y + z) is exiting for the first time from X(t, y) + B dR (0) within i . This new set has measure bounded by (we use the nearly incompressibility property of the map z → X(t, y + z) − X(t, y), which is the Lagrangian flow of the vector field (t, z) → b(t, X(t, y) + z) − b(t, X(t, y))) X(t i y , y)) dy (3.10) and Fubini C 2 (5.8) (4) Change of coordinate for the disintegration. The disintegration formula of (5.4), Point (7) of page 19, is computed in the coordinates y i on the surface S 1,i . When using instead the coordinates y at t = 0, we have to replace Indeed this is just the composition properties for the solution to (4.1). (5) Estimate on the growth of the perturbation. We now use the estimate of Equation (5.4) up to the last time t (ī−1) y (z) such that the trajectory remains at distanceR from X(t, y), that is when crossing (ī−1) y . We define, for Let us set the initial data as 0 y (y, z) = z, and consider the difference equation By Duhamel's formula for the difference equation, that is a n = b n a n−1 + c n , a n =   From (5.11), we obtain the estimate (5.14) (8) Measure of trajectories with large growth. Thus, using (5.4), we get In the third line, we have used that the trajectories under consideration are not exiting X(t, y) + B dR (0) in i , see the definition of E 2,i (r i , y i ) in Assumption (7). Hence, we can remove a set F y ⊂ (B d r (0) ∩ (K 0 − y))\E y such that (5.13). In other words, for r, r chosen as in (5.13), we have an estimate of the trajectories remaining inside X(t, y) + B dR (0) and such that their distance from X(t, y) is actually bounded by r <R up to the time t (ī−1) y (z) . The rest of the growth needed to exit X(t, y) + B dR (0) while in the set ī y (z) is studied in the next step. where t + i y (y, (ī y −1)(y) (y, z)) is the exit time of the trajectory X(t, y + z) from X(t, y) + B d R (0). (10) Measure of exiting trajectories. We have the estimatê (y, ī y −1 (y,z))) dz dy (5.16),Fubini Choosing (see also (5.23))  20) and the remaining trajectories lie inside X(t, y) + B dR (0). The total set of trajectories E y ∪ F y ∪ G y we remove from (B d r (0) ∩ (K 0 − y)) has measurê Hence, using i y j y =1 y W i y = W (t + i (y i ), y), again the solution formula (5.10) gives (note that in this case the initial data is 0)  Remark 5.2. We observe that the estimate gives some sort of differentiability in measure even with i y depending on y. This is not surprising since the sets S i,1 are subsets of finitely many sets {t = const}. However, the set K 2 depends on the partition: indeed, the derivative W (t, y) has discontinuities; thus, at any time τ i of discontinuity, we have, in general, for every linear map A. As an example one may consider the vector field in (t, so that at any time T the set of trajectories for which the differential cannot be computed is y 1 = −T . Thus for every T the set of trajectories which have to be removed is different. In next two sections we will show how to prove Assumptions (2)-(7) in two cases: (1) when one takes into account only the a.c. part of Db (Section 6); (2) in the Lebesgue points of the singular part of Db (Section 7).
The choice of the measure μ P will be obtained by piecing together these two cases.

Local Estimate with the Absolutely Continuous Part
We fix a perturbed proper set i and in it we consider the following vector field: In order to make the notation lighter, going forward we will neglect the index i. Define where ε P will be chosen at the end.
(1) Control of the derivative. First, for M > 0 chosen in Proposition 3.6, we have (2) Cylinders where the linear flow is constructed. By choosing R 1, we can also assume that the cylinder X(t, y) + B d R (0) has bases inside the entering and exiting flat parts of i : again we can assume that we remove a set of trajectories of measure smaller than εL d+1 ( ), where ε → 0 when R → 0. Let S 1 ⊂ S 1 be the set of initial data of the remaining trajectories: the choice of R corresponds to Point (3) of page 18. In order to satisfy Point (2) of page 18, we will choose ε < ε P /(2T L d (B d R 0 (0)). (4) Comparison with Lipschitz flow. We can compare the evolution of a trajectory with the evolution of the Lipschitz linear flow as follows: X(s, y + z)) − b(s, X(s, y)) − (Db) a.c. y (s) X(s, y)
(6.4) (6) Integral over all trajectories. The last integral can be evaluated after integrating with respect to y as follows: X(s, y))w ds dw dy where ω is the modulus of continuity in L 1 of the a.c. part of Db.

Conclusion.
We now show that Assumptions (2)-(7) hold with the choice of More precisely, (1) concerning Point (4) of page 18, we set E 1 (r, y) = ∅; (2) concerning Point (5) of page 19, by Point (3) and Point (6) above we havê This concludes the analysis of the a.c. part of Db.

Local Estimates with the Singular Part
The analysis of the singular part is more complicated and depends on the choice of several parameters: in particular, we will need the set E 1 of Point (4) of page 18, which collects the perturbed trajectories which do not behave mildly. As before, we will neglect the index i; moreover, here we assume that the perturbed proper set is in a small neighborhood of a Lebesgue point of the singular part D sing b. We will first compute our estimates in the case of "contracting" flow, that is div b < 0. Then, we will show how to deduce the general case from this.

Localization and coordinates
Letε 1 be given. For every Lebesgue point of the singular part D sing b of Db, we can choose as follows.
(1) Entering and exiting sets: the (perturbed) proper set is a proper small perturbation of a ball centered in the Lebesgue point, such that the set of trajectories N not entering from S 1 and not leaving from S 2 has η-measure η(N ) <εL d+1 ( ), (7.1) where η denotes the Lagrangian representation of ρ(1, b)L d+1 as in [16, Definition 3.1].
(2) Lebesgue point of the derivative: by the rank-one property of the singular part of the derivative of BV functions [1], there exist vectorsξ,η such that
With the above choice ofξ,η we have In particular almost all of the derivative occurs when moving along the 1direction, and the variation lies in the 1,2-directions. Hence the other components have small derivative. This implies that the flow is essentially contracting forward in time. However the nearly incompressibility (3.10) yields that the contraction is controlled, as we will see later on.
The proof of the needed estimates is divided into several subsections. Sometimes will will consider it as embedded into R d : in this case its definition refers to the coordinates (

Construction of the approximate vector field
Letb H be the approximate vector fields defined bỹ Notice that it depends only on the first component w 1 .
In order to simplify the notation, we will often assume w 1 0, mainly when we need to integrate in intervals [0, w 1 ]; the other case gives exactly the same estimates, as one can check.
We begin with a series of estimates for the vector fieldb (1) there exists R > 0 and K ⊂ S 1 compact such that (7.6) and each trajectory starting in y ∈ K satisfies X(t, y) + B d R (0) ⊂ for all t ∈ (t − (y), t + (y)), and X(t + (y), y) X(t, y) + w) − b(t, X(t, y)) In particular from the first point, since then the set The choice of R i of Point (3) of page 18 is done at this step by setting R i = √ ε R/2.

Estimate on the first component e 1
The ODE for the first component is The first observation is that by the choice (7.3), it holds for

0.
We have thus proved the following result.  X(t, y ), let ∪ i (t − i (y, y ), t + i (y, y )) be the set of time where it belongs to the cylinder X(t, y) + B d r (0), so that for every t ∈ (t − i (y, y ), t + i (y, y )) it holds s, X(s, y )) − b 1 (s, X(s, y)) s, X(s, y )) − b 1 (s, X(s, y)) Integrating over all y ∈ K we obtain Integrating over all y ∈ K and repeating the computations for (7.7) we obtain s, X(s, y) Note that since we are estimating only the first component the constant is improved. We collect the above estimate in the following Proposition: Proposition 7.3. We have the following estimate: if X(t, y ) ∈ B d R (X(t, y)) for t ∈ (t − i (y, y ), t + i (y, y )), then : has measure bounded bŷ K L d (E 1 (y, r )) dy (7.15)

Comparison with the disintegration
Aim of this section is to compareb H (r, y; t, w) with the disintegrated measure (Db 1,2 ) y w 1 . Being these measures singular, the estimate is done by considering their time integral: this reflects the fact that we want to compare the flow generated byb H and (Db) y , not the vector fields themselves.
Here we need to consider the flowX H 1 generated by the vector field (7.11). The proof of the final theorem requires several steps, which are listed below.  (Q H (r ))|Db|( ), (7.16) where C d is a constant depending only on the dimension d of the space.

First selection of initial point in order to have continuity of the flow and disintegration
Consider a compact set K ,1 ⊂ K of trajectories X(t, y), where y → X(t, y) ∈ C([0, T ], R d ) is continuous in the C 0 -topology, and such that the disintegration y → (Db) y is weakly continuous in the sense of measures and m K ,1 = L d K ,1 , which means that the singular part of m has measure 0 on K 1 .
Since we haveˆN then it follows that if L 1 (N ) = 0, In particular we can assume that the initial and end sets {t − (y)} y∈K ,1 , {t + (y)} y∈K ,1 have measure 0 with respect to (Db) y , and thus in K ,1 it holds that y → (Db) y ((t − (y), t + (y)) is continuous with continuity modulus ω dis .
We can also take a second compact set K ,2 made of Lebesgue points of K ,1 and such that the limits are uniform with continuity modulus ω dis (r ) (eventually changing ω dis of (7.22)). The total error can be taken by Egorov and Lusin Theorems. We thus have proved the following lemma: Lemma 7.6. There exist two compact sets K ,2 ⊂ K ,1 ⊂ K such that the following holds: (1) their difference in measure is small, that is (Db) y+z ((t − (y), t + (y))) − (Db) y ((t − (y), t + (y))) m(dz) ω dis (r ). (7.23c) , t + (y)) dy, (7.25) where in the last step we use the definition of disintegration. We estimate the integral (7.26) where K in ,1 (y) are the trajectories in K ,1 which remain inside ∪ t X(t, y) + [0,X H 1 (r, y; t, w 1 )] × Q H (r ). Recall that we denote with K the union of the graph of the trajectories starting in K , and the same with K ,1 .
Step 2. We write the last term of (7.25) aŝ We have (1) term K in ,1 (y): in this set the measure (D 1 b i ) z are continuous by (7.22) and the trajectories remain inside the set by the definition of K in ,1 (y), so that ω dis (Hr) (7.16) 2M min 1, where we have observed that and, by (7.23a), (2) term K \K in 1 : these trajectories satisfy |(D 1 b i ) z | M and exit, so that 2 m-measure of exiting/entering trajectories dy 2 flow on the boundary dy (7.16) M Finally, collecting all estimates, (7.27) Step 3. We thus havê , t + (y)))w 1 dw dy (7.25), (7.27 (7.28) Step 4. Observing that
√ε |Db|( ). (7.38) which in particular is equivalent tô We have used Integrating the third term with respect to y and using (7.2), we get where we used the fact thatX H 1 (r, y; s, z 1 ) w 1 in the first inequality and |w 1 | r in the last one.
− X 1 (s, y) × Q H (r ) (7.48) 1 such that, if J (t, y) is the Jacobian of the flow X(t, y) (see Proposition 3.6), In this set the solution to (7.52) is defined for all t ∈ [t − (y), t + (y)] and it holdŝ Hence integrating in K one obtainŝ K J (·, y) −J (·, y) L ∞ (t − (y),t + (y)) dy Hence by Chebyshev's inequality we can remove a set of trajectories of measure < √ε |Db|( ) and in the remaining set the estimate (7.55) holds: In particular we deduce that so that the solutionW (t, y) does exist on this set, and in the same way as in (4.14) one gets |W (t, y)| e 2C M . (7.58)

The time reverse case
To study the caseξ ·η > 0, we use the estimates we have already proved by reversing time or, equivalently, by changing variables and using as initial set the set S 2 instead of the set S 1 . In order to have more flexibility in the proof, we will choose the parameter H determining Q H (r ) later.
We proceed as follows: (1) First of all, we will consider as initial points S 2 the image of the set K ,3 , that is S 2 ∩ K ,3 : by the near incompressibility and the fact that up toCεL d+1 ( ) all trajectories start from S 1 and leave from S 2 for a perturbed proper set, we obtain that This is Point (2) of page 18 for the caseξ ·η > 0. (4) Using the bound (7.73), we can estimate the size ofX 1 (r, y; t + (y), r ): by the balance final area + lateral flow 1 C initial trajectories, one gets where we have used (7.73). The above equation gives for y ∈ K ,3 \Z 1 that X H 1 (r, y; t + (y), r ) This lower bound on X H 1 (r, y; t + (y), r ) will be used in the next section to bound E + 1 (r, y). Note that r + (r ) O(1)C −1 r by the choice of H in the next points, as one has to expect from the near incompressibility (4.3). (5) We can thus estimate the subset E + 1 (r, y) of B d r + (r ) (X(t + (y), y)) coming from trajectories crossing the boundary of (7.69): We also estimates the set X(t + (y), y) + E + 2 (r, y) ⊂ X(t + (y), y) + B d r + (r ) (0) of trajectories arriving from points which do not belongs to K ,3 : (6) The remaining trajectories in X(t + (y), y) + B d r + (r ) (0) are arriving from some set which we denote as and are not crossing the boundary of (7.69); hence these trajectories cannot arrive from E 2 ((1 + H )r, y)), being the latter defined as the set of trajectories in X(t − (y), y) + B d (1+H )r (0) which exit X(t, y) + B d r (0) before t + (y). Thus by changing the coordinates from the initial points y, z at time t − (y) to the end points at time t + (y) and using the near incompressibility we can writê K ,3ˆA (y)\E 1 ((1+H )r,y) X(t, y + z) − X(t, y) − W (t + (y), t − (y), y)z dy dz For shortness we have used the notation y(y ) inverting the function X t + (y(y )), y(y ) = y . and the estimate of (7.76) becomeŝ The image of the set E 1 ((1 + H )r, y) is controlled by (7.63): hence using the nearly incompressibility its image has are controlled bŷ E 1 (r, y))) dy r + (r ) (X(t + (y), y)))|Db|( ).

(7.84)
Up to pushing the measure L d (dy) to the end points X(t + (y), t − (y), y)) (thus multiplying the right-hand side byC when integrating in L d (dy )), the estimate (7.84) corresponds to Point (3) of page 18, as well as the evaluation of the measure of E 2 (r, y) of Point (7) for the expanding case.
We note in particular that the fraction of E 2 (r, y) can be made small around a large set of initial points: this is what is proved here for the final points, but the argument can be repeated also in the contractive case.
Since the trajectories not in E(r + , y) are not exiting, we can just use Point (7) of page 19 for the previous two points: (5.3) follows from the properties of the disintegration applied to the linear flow W (·, y).
This concludes the proof that the assumptions of Section 5 hold around Lebesgue points of the singular part of the derivative.

Construction of a Suitable Partition into Proper Sets
The differentiability in measure follows from the estimates in the previous sections if we can find a suitable partition into perturbed proper sets such that the assumptions of Section 5 hold. Without loss of generality, we can assume that is a proper set. The construction of a disjoint covering is done as follows: (1) Consider a Lebesgue negligible set S where |D sing b| is concentrated. By Besicovitch's Theorem (see [11,Theorem 2.17]), we can cover S with countably many disjoint closed proper balls such that the estimates of Section 7.7 and Section 7.8 hold: these are collected in Point (2) of page 49 in the proof that the partition satisfies the assumptions of Section 5. (2) Hence we can consider finitely many closed proper balls {B d+1 (3) Being these balls at positive distance from one another and from R d+1 \ , we can perturb them into disjoint proper balls { In order to show that the sets { sing i , rem } satisfy the statement, we just need to prove that rem is a perturbed proper set: by Lemma 3.3, it is a proper set, being the difference of two proper sets whose boundaries have empty intersection. It remains to show that the flow occurs mostly on the time-flat parts of the boundary. To this end, we need to estimate the trajectories in K which cross ∂ rem outside {t = 0, T }∪∪ i S sing i ∪∪ j S a.c. j : indeed these are the non flat parts of the boundary of rem from which a trajectory in K may enter. We observe that these trajectories are leaving one of the sing i , a.c.
j not from some flat parts, so that their total estimate is bounded by (8.1) byεL d+1 ( rem ).
We conclude this section by proving that the partition constructed in Theorem 8.1 satisfies the assumptions of Section 5 with a suitable measure μ P . This will conclude the proof of Theorem 1.1.
Proof. We consider separately sing i and rem .
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Appendix A. Bressan's Lemma on the Approximation of Lipschitz Continuous Flows
A key tool in the proof of our main result is the following lemma (see [25,Lemma 4] or [26, Theorem 2.9]): given an absolutely continuous curve γ and a Lipschitz continuous semigroup S t , we can estimate the distance between γ and the trajectory of the semigroup starting at γ (0). Proof. Let us consider the curve t → X (t) = S T −t γ (t).

Lemma A.1. Let t → γ (t) be an a.c. curve, and S t is a L-
We have Using the assumption on S t , we have By the nearly incompressibility and Fubini theorem, for L d -almost everywhere trajectory the above limit is equal for L 1 -almost everywhere t to |b(t, X(t, y)) −b(t, X(t, y))|. We thus proved the claim.
In particular, we remark that assumption (A.1) holds in the following two cases (which are relevant to Sections 6 and 7 respectively): (1) the linear flow generated by a matrix A(t) ∈ L 1 ((0, T )); (2) the solution to the differential inclusioṅ x ∈ −A(t, x), with A(t) a quasi monotone operator defined in R d and such that |A(t, 0)| ∈ L 1 . The first case is elementary; the second one is analyzed in [19].