Positive solutions of transport equations and classical nonuniqueness of characteristic curves

The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio's superposition principle we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Sz\'ekelyhidi in the recent groundbreaking work [Ann. PDE, 4, 2018]. On the opposite side, we introduce a new class of asymmetric Lusin-Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna-Lions theory.


Introduction
In this paper we study positive solutions of the continuity equation ∂ t ρ + div (uρ) = 0 (1) and the related system of ordinary differential equationsγ(t) = u(t, γ(t)). To avoid technicalities we restrict our attention to periodic vector fields, i.e. u : I × T d → R d , where T d is the d-dimensional torus and I ⊂ R. In the sequel we omit the superscript d and simply write T and use the notation L d for the Lebesgue measure on the whole space R d and on T. Note that in Definition 1.1 it matters how u is defined at every point: different pointwise representatives for u might have different integral curves starting at the same x. When u is smooth (Lipschitz) the trajectories are unique and, after "bundling them" into a flow map X : (0, T ) × T → T, solutions of (1) can be recovered via Liouville's classical theorem. This fact can be elegantly encoded using measure theory in the formula (X(t, ·)) ♯ (ρ(0, ·)L d ) = ρ(t, ·)L d . For less regular vector fields it is customary, after the seminal paper [3,19], to introduce the notion of regular Lagrangian flows. The latter consists, following one of its equivalent formulations given in [3], of a measurable selection X of integral curves of the ODE for which X(t, ·) # L d ≤ CL d . 1 Such uniqueness and stability result is sometimes inappropriately regarded as "almost everywhere uniqueness of integral curves", even though it is well known among the experts that the DiPerna-Lions theory does not imply the statement "for a.e. x there is a unique integral curve of u starting at x". In fact whether such "classical" uniqueness theorem holds for Sobolev vector fields is a long-standing open question, see [19, p. 546], [3, p.231], [2,Section 2.3], [4,Open problems,section 4]. This question has had a positive answer for specific vector fields, such as suitable weak solutions of the Navier-Stokes system [26,27], based on estimates of the dimension of the singular set originally due to [11]. Recently, in [12] the authors use a suggestion of Jabin to prove almost everywhere uniqueness of the trajectories when u ∈ C([0, T ], W 1,r (T, R d )) for some r > d. One aim of this paper is to show that in general, under the assumptions of the DiPerna-Lions theory, the answer is negative. Theorem 1.3. For every d ≥ 2, r < d, s < ∞ and every T > 0 there is a divergence-free vector field u ∈ C([0, T ], W 1,r (T, R d ) ∩ L s ) such that the following holds for every Borel map v with u = v L d+1 -a.e.: (NU) There is a measurable A ⊆ T with positive Lebesgue measure such that for every x ∈ A there are at least two integral curves of v starting at x.
Given [12,Theorem 5.2], the above statement covers the optimal range, except for the endpoint r = d. In fact an improvement of the argument in [12,Theorem 5.2] allows to prove almost everywhere uniqueness of trajectories for a function space which shares the same scaling properties of W 1,d , namely when Du belongs to the Lorentz space L d,1 , see Corollary 10.1 below. We also expect Theorem 1.3 to hold for some Sobolev vector fields that are continuous in space-time [1], thereby answering the question of [2, Section 2.3] and providing an example under the classical assumptions of Peano's theorem.
The theorem above is a consequence of Ambrosio's superposition principle (see [4,Theorem 3.2]) and of the following nonuniqueness result at the PDE level, which in turn will be proved using "convex integration type" techniques borrowed from a groundbreaking work of Modena and Székelyhidi [24,25], improved later by Modena and Sattig [23](we refer to [9,10,15,17,18,21] and the references therein for the birth of this and related lines of research).
Compared to the results in [24] and [23] the addition (crucial for our application) is the positivity of the solution ρ. While it is relatively simple to modify the approach of Modena and Székelyhidi in [24] in order to achieve Theorem 1.4 when 1 p + 1 r > 1 + 1 d−1 , we have not been able to do the same with the one in [23] to cover the range 1 + 1 d−1 ≥ 1 p + 1 r > 1 + 1 d . Our proof is therefore relatively different from the one of [23] and in fact less complicated and shorter. At the technical level we introduce suitable space-time flows which compared to the basic building blocks of [23] are more similar to Mikado flows: in a nutshell our flows are a perturbation of point masses traveling on a space-time line. This approach makes a part of our argument more similar to [24], but it has the technical drawback that we need to introduce a suitable partition of unity to discretize the time velocities of the moving particles (a similar idea was used first in [18]). One subtle part of our proof is a combinatorial argument to ensure that the supports of the flows are disjoint in 2 space dimensions. Since in 3 space dimensions and higher the latter can be completely omitted and the proof is simpler we have decided to first present the full arguments for Theorem 1.4 when d ≥ 3 and then show in Section 7 which modifications are necessary in the case d = 2.
Our interest in Theorem 1.4 was triggered by the gap between the DiPerna-Lions theory, which guarantees uniqueness for 1 p + 1 r ≤ 1, and the nonuniqueness results of [23][24][25]. In particular we are able to show that in some intermediate range of exponents (strictly containing the DiPerna-Lions range, but not reaching the full complement of the Modena-Sattig-Székelyhidi range) positive solutions are in fact unique.
where X denotes the unique regular Lagrangian flow of Definition 1.2.
Remark 1.6. Observe that, under the above assumptions [12,Corollary 5.4], in which the case r > d has been settled as a consequence of the L d -a.e. uniqueness result for trajectories mentioned above. The proofs of the latter and of Theorem 1.5 employ all some suitable Lusin-Lipschitz type estimates for u, an idea pioneered in [5] and [14] and which has proved quite fruitful in different contexts (see for instance [6-8, 13, 16]). As it is well known, for sufficiently regular domains Ω ⊂ R d and when p ∈ (1, ∞], a Borel map u belongs to W 1,p (Ω, R) if and only if there is a function g ∈ L p (Ω) such that |u(x) − u(y)| ≤ (g(x) + g(y))|x − y| for a.e. x, y.
In fact g can be taken to be the classical Hardy-Littlewood maximal function of |Du|. It seems less known (but anyway classical) that for p > d the symmetry in (3) can be broken to show Theorem 1.5 is based on the idea that an appropriate symmetry-breaking is still possible for smaller exponents p. More precisely we have the following proposition, which has its own independent interest.
Proposition 1.7. Let 1 < r ≤ d be fixed. For any u ∈ W 1,r (T) and any α ∈ (0, r d ) there exist a negligible set N ⊂ T and a nonnegative function g ∈ L r (T) satisfying the inequalities Moreover, we can assume N = ∅ provided we choose an appropriate representative of u ∈ W 1,r (T) and there is a continuous selection W 1,r ∋ u → g ∈ L r .
A simple corollary of the latter statement is an inequality of the form |u(x) − u(y)| ≤ (a(x) + b(y))|x − y| where one function, say b, can be taken more integrable at the prize of giving up some integrability for the other. Theorem 1.5 follows from the extreme case where the integrability of b is maximized at the expense of reducing the integrability of a to the bare minimum, namely L 1 , cf. Corollary 9.1. We moreover show that in this case the range of exponents for b obtained in the latter is in fact optimal.
Clearly, it is tempting to advance the conjecture that, for positive solutions of the continuity equations, well-posedness holds in the range 1 + 1 d > 1 p + 1 r , namely the complement of the the closure of the range of Theorem 1.4. An even more daring conjecture is that the latter statement holds for any solution. However nothing is known without assuming a one-sided bound or, as is the case of [12], some technical property of trajectories of the ODEs.

Iteration and continuity-Reynolds system
As in [24] we consider the following system of equations We then fix three parameters a 0 , b > 0 and β > 0, to be chosen later only in terms of d, p, r, and for any choice of a > a 0 we define The following proposition builds a converging sequence of functions with the inductive estimates where α is yet another positive parameter which will be specified later.
Proposition 2.1. There exist α, b, a 0 , M > 5, 0 < β < (2b) −1 such that the following holds. For every a ≥ a 0 , if (ρ q , u q , R q ) solves (6) and enjoys the estimates (7), (8), then there exist (ρ q+1 , u q+1 , R q+1 ) which solves (6), enjoys the estimates (7), (8) with q replaced by q + 1 and also the following properties: Compared to [24] we are using a slightly different notation and a more specific choice of the parameters. None of that is however substantial: the really relevant differences are in estimate (b) and in the range of exponents, which is the same as the one in [23]. In the same range of exponents of [24] the positivity could be achieved by a slight tweak in the approach of [24]. However we have not been able to find a similar modification of the arguments of [23]. For this reason our proof of Proposition 2.1 differs from both that of [24] and that of [23]. However we still make use of some crucial discoveries in [24] and we will refer to that paper for the proofs of some relevant lemmas. From now on, in order to simplify our notation, for any function space X and any map f which depends on t and x, we will write f X meaning max t f (t, ·) X .

Preliminary lemmas
3.1. Geometric lemma. We start with an elementary geometric fact, namely that every vector in R d can be written as a "positive" linear combination of elements in a suitably chosen finite subset Λ of Q d ∩ ∂B 1 . This is reminiscent of the geometric lemma in [18]. In both [24] and [23] the positivity of the coefficients is not needed and hence the authors can choose Λ as the standard basis of R d .
For each fixed j each vector R ∈ R d can be written in a unique way as linear combination of the vectors ξ j,1 , . . . ξ j,d . If we denote by b j,i (R) the corresponding coefficients (which obviously depend linearly on R), then the latter are all strictly positive if R belongs to Σ(v j ). We consider a partition of unity χ j on the unit sphere ∂B 1 associated to the cover {Σ(v j )} and for every ξ j,i = ξ ∈ Λ we set The coefficients a ξ are then smooth nonnegative functions of R.
Remark 3.2. With Lemma 3.1 at hand, it is easy to generate a finite number of disjoint families Λ (1) , ..., Λ (k) where each one enjoys the property of Lemma 3.1: it is enough to take suitable rational rotations of one fixed set Λ.

3.2.
Antidivergences. We recall that the operator ∇∆ −1 is an antidivergence when applied to smooth vector fields of 0 mean. As shown in [24, Lemma 2.3] and [23, Lemma 3.5], however, the following lemma introduces an improved antidivergence operator, for functions with a particular structure. .
we have that div R(f g λ ) = f g λ − f g λ and for some C := C(k, p) Proof. It is enough to combine [23, Lemma 3.5] and the remark in [23, page 12].
3.3. Slow and fast variables. Finally we recall the following improved Hölder inequality, stated as in [24,Lemma 2.6] (see also [10,Lemma 3.7]). If λ ∈ N and f, g : T → R are smooth functions, then we have and

Building blocks
Given µ ≪ 1 we define the 1-periodic functions Given Λ as in Lemma 3.1, for any ξ ∈ Λ we chose ξ ′ ∈ ∂B 1 such that ξ · ξ ′ = 0 and we define . Notice thatW ξ,µ,σ is divergence free since it is also the divergence of the skew-symmetric matrix Ω µ ξψ µ . By construction we haveW , hence the following properties are easily verified.
For any k ∈ N and any s ∈ [1, ∞] one has In our construction ξ will take values in a finite set of ξ's, which will be fixed throughout the iteration, i.e. it is independent of the step q in Proposition 2.1. The parameter σ will also vary in a finite set, but the cardinality of the latter will depend (and in fact diverge to infinity) on the iteration step q. In dimension d ≥ 3 we consider suitable translations ofW ξ,µ,σ andΘ ξ,µ,σ which guarantee that, as ξ varies in these fixed set of directions, pairs of (W ,Θ) with distinct ξ's have disjoint supports. The precise statement is given in the following lemma.
This claim follows from a simple induction argument along with the observation: orthogonal to ξ and ξ ′ , has this property. Indeed, if we assume by contradiction the existence of s, t, k, α, β, γ, ξ ′′ as above such that αξ By the previous lemma and by (17) we notice that, if we consider the translations of for ξ in a suitable finite set of directions, these functions satisfy the same properties as in Lemma 4.1 (with the exception of the description of the support, which is now translated) and moreover they have disjoint support for µ sufficiently large and for every σ. Notice finally that in fact both µ and σ could vary for different ξ and the supports would still remain disjoint, as long as µ(ξ) is larger than µ 0 for every ξ.
The latter approach is clearly not feasible in dimension d = 2. In that case we will need to take advantage of the discreteness in the parameter σ as well. As already mentioned this is more delicate, since the set of values taken by σ depends on the step q. At each step q we need to choose rather carefully the set of parameters σ which enter the construction: for distinct values of σ we need to ensure that their ratio is not too close to 1, compared to the size of µ −1 . The relevant statements depend thus on how the building blocks enter in the definition of the maps (u q+1 , ρ q+1 , R q+1 ). For this reason we detail next the definition of the maps when d ≥ 3 and show first how to prove Proposition 2.1 in that case. We then give a detailed description on how to modify the arguments to handle the case d = 2.

Iteration scheme
5.1. Choice of the parameters. We define first the constant where we have used crucially Notice that, up to enlarging r, we can assume that the quantity in the previous line is less than 1/2, namely that γ > 2. Hence we set α : Finally, we choose a 0 and M sufficiently large (possibly depending on all previously fixed parameters) to absorb numerical constants in the inequalities. We set 5.2. Convolution. We first perform a convolution of ρ q and u q to have estimates on more than one derivative of these objects and of the corresponding error. Let φ ∈ C ∞ c (B 1 ) be a standard convolution kernel in space-time, ℓ as in (20) and define We observe that (ρ ℓ , u ℓ , R ℓ +(ρ q u q ) ℓ −ρ ℓ u ℓ ) solves system (6) and by (7), (19) enjoys the following estimates By Young's inequality we estimate the higher derivatives of R ℓ in terms of R q L 1 to get for every N ∈ N. Finally, for the last part of the error we show below that (where we have assumed that a is sufficiently large). The claim follows a well-known bilinear trick used often and originating (at least in the context of fluid dynamics) from the proof of Constantin, E and Titi of the positive part of the Onsager conjecture. We include a proof for the reader's convenience.
Lemma 5.1. Consider a mollification kernel φ compactly supported in time and space. Then there is a constant C = C(φ) such that for every smooth functions u and ρ depending on time and space Proof. To simplify our notation we introduce the variable z = (x, t) and set Σ : We then use Combining these last estimates with (25) we infer the desired conclusion.

5.3.
Definition of the perturbations. Let µ q+1 > 0 be as in (21) and 3 4 ]. Notice that n∈Zχ (τ − n) ∈ [1, 2] and χ ·χ = χ. Fix a parameter κ = 20 δ q+2 and consider two disjoint sets Λ 1 , Λ 2 as in Lemma 3.1. Next, define [i] to be 1 or 2 depending on the conrguence class of i. Finally, consider the building blocks introduced in Section 4 in such a way that, for ξ ∈ ∪ 2 i=1 Λ i , their spatial supports are disjoint. We define the new density and vector field by adding to ρ ℓ and u ℓ a principal term and a smaller corrector, namely we set The principal perturbations are given, respectively, by where we understand that the terms in the second sum are all 0 at points where R ℓ vanishes. In the definition of w (p) and θ (p) the first sum runs for n in the range Indeed (23) we obtain an upper bound for n. The aim of the corrector term for the density is to ensure that the overall addition has zero average: The aim of the corrector term for the vector field is to ensure that the overall perturbation has zero divergence. Thanks to (13), we can apply Lemma 3.3 to define Moreover, since W ξ,µ q+1 ,n/κ is divergence-free, the argument inside R has 0 average for every t ≥ 0.
Notice finally that the perturbation equals 0 on every time slice where R ℓ vanishes identically.
6. Proof of the Proposition 2.1 in the case d ≥ 3 Before coming to the main arguments, we record some straightforward estimates for the "slowly varying coefficients".

6.2.
Estimate on w q+1 L p ′ and on Dw q+1 L r . Exactly with the same computation as in (27), replacing p with p ′ , we have that Concerning the corrector term w Computing the gradient of w (p) q+1 and combining Lemma 6.1 with (16) we have Concerning the corrector, by Lemma 3.3 and similar computations as above, In the second equality above we have used that (ρ ℓ , u ℓ , R ℓ + (ρ q u q ) ℓ − ρ ℓ u ℓ ) solves (6), that div u ℓ = div w q+1 = 0, and that θ by using that Θ ξ,µ q+1 ,n/κ and W ξ,µ q+1 ,n/κ solve the transport equation (12) and Lemma 3.1 we get the cancellation of the error R ℓ up to lower order terms div (θ We have We can now define R q+1 which satisfies (29) as Notice that R quadr is well defined since by (14) the function (Θ ξ,µ q+1 ,n/κ W ξ,µ q+1 ,n/κ )(λ q+1 t, λ q+1 x) − n κ ξ has 0 mean. We now estimate in L 1 each term in the definition of R q+1 . From the second equality in (29) and since the average of (∂ t θ (p) q+1 ) 2 is m by integration by parts, we deduce that (∂ t θ q+1 + m has 0 mean, so that R time is well defined. Recall that the estimate on (ρ q u q ) ℓ − ρ ℓ u ℓ L 1 has been already established in (24).
By the property (9) of the antidivergence operator R, Lemma 6.1 and (26) we have To estimate the terms which are linear with respect to the fast variables, we take advantage of the concentration parameter µ q+1 . First of all, by Calderon-Zygmund estimates we get From (32), (12) and (11) we get Similarly, we have that In the last inequality we used 2βb 2 ≤ 1, the definition of γ, and b(1 + 1/p) ≥ 2(1 + α)(d + 2) + 1. Finally, from (27) and (28) 6.4. Estimates on higher derivatives. By the choice of α, since in particular α ≥ 2 + γ(d + 1), we have that An entirely similar estimate is valid for ∂ t ρ q+1 C 0 and the one for u ℓ +w From (16) and Lemma 6.1 A similar computation is valid for ∂ t w (c) q+1 L 1 .

The building blocks and the iterative scheme in dimension d = 2
We describe in this section how the proof of Proposition 2.1, concluded above in dimension d ≥ 3, should be modified to cover the case d = 2. The main obstruction in this regard is that the building blocks in Section 4 cannot be translated, as we did in dimension d ≥ 3 in Lemma 4.2, to make sure that their support is disjoint in space. In dimension d = 2, we need to make them disjoint in space-time, observing that they are described by a small ball which translates at constant speed according to a translating vector field which is supported on a slightly larger ball. 7.1. Iteration scheme and definition of the perturbations. We choose the parameters as in Section 5.1 and we perform the convolution step as in Section 5.2. Similarly, the cutoffs χ ∈ C ∞ c (− 3 4 , 3 4 ) and χ ∈ C ∞ c (− 4 5 , 4 5 ) are chosen as in Section 5.3, as well as κ = 20 δ q+2 and the sets Λ 1 , Λ 2 . Starting from the building blocks introduced in Section 4 we will choose positive real numbers v n (which will satisfy |v n − n| ≤ 1) and real numbers a ξ,n and define for any n ≥ 1 and ξ ∈ Λ [n] . The difficult part will be to choose v n and a ξ,n so that whenever ξ = ξ ′ , |n − m| ≤ 1 and n, m ≤ Cλ d(1+α)+1 q . Assuming for the moment that this can be done, we define the new density and vector field as we did in Section 5.3 up to replacing all functions Θ ξ,µ q+1 ,n/κ and W ξ,µ q+1 ,n/κ with Θ ξ,µ q+1 ,vn/κ and W ξ,µ q+1 ,vn/κ . The proof of the proposition would then follow the same arguments: we only need to modify sligthly the definition of R q+1 . Most of this section will be devoted to choose v n and a ξ,n so that (33) holds. Once we have achieved the latter, we will then show how to change the definition of R q+1 . 7.2. Geometric arrangement. The main geometric construction is given by the following proposition. Λ 1 ∪ Λ 2 is the set of possible space directions for the building blocks, while the sequence {w n } is in fact the set of values µ d/p ′ q+1 n κ 1/p ′ . Observe that ,when wn w n−1 is a rational number, the relative position of the space supports of the building blocks is time-periodic. If each space support were merely a point we could easily make them always disjoint and in fact we could identify their minimum distance. If we write wn w n−1 = 1 + A(n) N (n) with A(n) and N(n), intuitively such minimum distance should be made comparable to 1 N (n) .
Then there exists a constant c 0 := c 0 (C, Λ 1 , Λ 2 ) > 0 with the following property. For every ξ ∈ Λ k and n ∈ N there exists a ξ,n ∈ [0, 1] such that the family of curves The proof of the proposition is based on the following elementary lemma.
Lemma 7.2. Fix two different vectors ξ, ξ ′ ∈ S 1 ∩ Q 2 and a number w = 1 + A N < 10, with A and N positive integers and coprime. Then there exists C = C(ξ, ξ ′ ) such that Proof. Set T int := {(t, t ′ ) : ξt = ξ ′ t ′ on T 2 } and observe that T int ⊂ Q 2 since the matrix with columns ξ and ξ ′ is invertible with rational coefficients. Moreover T int is an additive discrete subgroup of R 2 , hence it is a free group of rank k ∈ {0, 1, 2}. Denoting by T and T ′ the period of, respectively, t → ξt and t → ξ ′ t one has that (T, 0), (0, T ′ ) ∈ T int . This implies that the rank of T int is two, hence we can find two generators (t 1 , t ′ 1 ), (t 2 , t ′ 2 ) ∈ T int . Let us finally introduce A := {ξt ∈ T 2 : (t, s) ∈ T int for some s ∈ R} to denote the set of points in T 2 where the supports of the curves t → tξ and t → tξ ′ intersect.
The key ingredient is Lemma 7.2 and indeed the constant c 0 is chosen such that where C is the constant appearing in Lemma 7.2(i)&(ii). Notice that we are interested in pairs (ξ m,k , n) such that k ≡ n mod 2. Without loss of generality we can thus assume that k is a function of n and takes the values 1 or 2 depending on the congruence class of n modulo 2. In particular we will use the shorthand notation a m,n for the point a ξ m,k ,n . We will find a m,n inductively, after endowing the set {1, . . . , m 0 } × N \ {0} with the lexicographic order. More precisely we write (m, n) ≤ (m ′ , n ′ ) if • either n < n ′ • or n = n ′ and m < m ′ . At the starting point of the induction we set a 1,1 = 0. For the inductive step, we fix (m ′ , n ′ ) and assume that a m,n has been already defined for any (m, n) ≤ (m ′ , n ′ ). If m ′ < m 0 we need to define a m ′ +1,n ′ , otherwise we have m ′ = m 0 and we need to define a 1,n ′ +1 . We explain how to proceed just in the case m ′ < m 0 , since the other case follows from the same argument. To fix ideas, let us assume that the congruence class of n ′ is 1, so that the congruence class of n ′ − 1 is 2. We look for a m ′ +1,n ′ ∈ [0, 1] such that d T 2 ((tw n ′ + a m ′ +1,n ′ )ξ m ′ +1,1 , x n ′ ξ m,1 (t)) ≥ c 0 ≥ c 0 n ′ for every t > 0, when 1 ≤ m < m ′ + 1 and d T 2 ((tw n ′ + a m ′ +1,n ′ )ξ m ′ +1,1 , x n ′ −1 ξ m,2 (t)) ≥ c 0 n ′ for every t > 0, when 1 ≤ m ≤ m 0 (we interpret the latter condition as empty when n ′ = 1). Define the sets G 1 m , G 2 m ⊆ [0, 1] as G 1 m := {a : d T 2 (tw n ′ ξ m ′ +1,1 + aξ m ′ +1,n ′ , x n ′ ξ m,1 (t)) ≥ c 0 for every t > 0} Note that the existence of a m ′ +1,n ′ is equivalent to the fact that G is not empty. According to Lemma 7.2(i) L 1 ([0, 1] \ G 1 m ) ≤ Cc 0 for every m ∈ {1, . . . , m ′ }, while according to Lemma 7.2(ii) L 1 ([0, 1] \ G 2 m ) ≤ CCc 0 for every m ∈ {1, . . . , m 0 }. In particular, by our choice of c 0 in (34) we have L 1 ([0, 1] \ G) ≤ C(Cm 0 + m ′ )c 0 ≤ 2CCm 0 c 0 < 1, which in turn implies that G is not empty and completes the proof. 7.3. Suitable discretized speeds. Clearly we cannt apply the geometric arrangement of the previous section if we choose v n = n for the values of the parameter σ since the rations of µ d/p ′ q+1 n κ 1/p ′ are not even guaranteed to be rational. The aim of the next lemma is to show that it suffices to perturb {n} n∈N\{0} slightly to a new sequence {v n } n∈N\{0} in order to achieve that the w n+1 wn are rational numbers with a denominator which is not too large (in fact comparable to n). (i) |v n − n| ≤ 1; Proof. We prove the statement by induction on n. For n = 1 and n = 2 we set v 1 = v 2 = 1, A 2 = 0, N 2 = 1, and the statement is satisfied. Suppose now that the claim is verified forn. If vn ≥n, we set vn +1 = vn, An +1 = 0, Nn +1 = Nn and the claim is verified. Hence we can assume that vn <n. We claim that we can choose An +1 =N, vn +1 and Nn +1 with vn +1 ∈ [n + 1,n + 2] and Indeed, consider the continuous, decreasing function it is enough to show that to find t ∈ [n + 1,n + 2] such that f (t) ∈ N. Since the function g(n, vn) is increasing with respect to the variable vn,N g(n, vn) ≥N g(n,n − 1) =N 1 + 2 n − 1 We finally chooseN :=N (p ′ ) in such a way that infn ≥2 g(n,n − 1) ≥N −1 ; we notice that this infimum is positive since the function g(n,n − 1) is positive for everyn ≥ 2 and, by a simple Taylor expansion, it grows linearly asn → ∞. This proves the claim (35).
7.5. Proof of the Proposition 2.1 in the case d=2. The estimates up to Section 6.2 are done in the same exact way, up to observing that v n /κ is comparable to n/κ up to a factor 2. In Section 6.3, we compute in (30) the product of θ as a consequence of (33), (26), the fact that χ ·χ = χ and χ(κ|R ℓ | − n) · χ(κ|R ℓ | − n) = 0 when |n − m| > 1.
Since the average of Θ ξ,µ q+1 ,n/κ W ξ,µ q+1 ,n/κ which appears from the forth line of formula (30), in the definition of R quadr and in m is now v n /κξ rather than n/κξ, the definition ofR ℓ should now be replaced byR .
Clearly ρ and u solve the continuity equation and ρ is nonnegative on T by Moreover, ρ does not coincide with the solution which is constantly 1, because Finally, since ρ 0 (t, ·) ≡ 1 for t ∈ [0, 1/3], point (c) in Proposition 2.1 ensures that ρ(t, ·) ≡ 1 for every t sufficiently close to 0. to the map e 0 , which is ρ 0 -a.e. well defined; since by assumption for a.e. x ∈ T, the integral curve starting from x is unique (and hence coincides with the regular Lagrangian flow), we deduce that η x is a Dirac delta on the curve t → X(t, x) and consequently ρ(t, ·)L d = X(t, ·) # (ρ 0 L d ). This concludes the proof of the first claim.
Let u be the vector field given by Theorem 1.4 and observe that the Cauchy problem for the continuity equation (1) from the initial datum ρ 0 ≡ 1 has two different nonnegative solutions in [0, T ]: ρ (1) ≡ 1 and the nonconstant solution ρ (2) given by Theorem 1.4. Hence, by the previous observation we conclude that there exists a set of initial data of positive measure such that the corresponding integral curves are nonunique.
Since the fact that the two functions are distinct solutions of the continuity equation is independent of the pointwise representative chosen for the vector field, this completes the proof of Theorem 1.3.

9.
Asymmetric Lusin-Lipschitz estimates 9.1. Proof of Proposition 1.7. We will prove the inequality up to constants and we assume α ≥ 1/d, since for any α ∈ (0, 1/d) a simple application of the Young inequality gives We next introduce the following localized Hardy Littlewood maximal function: regarding any integrable function f : T → R as a periodic function on R 3 , we set We will show below that the conclusion of the proposition hold for g(x) := (M|Du| q ) 1/q where q = αd. In particular the map L r ∋ Du → g ∈ L r is continuous. Note first that it suffices to prove the estimate for x, y ∈ {g < ∞} ⊂ {M|Du| < ∞}. On {g = ∞} we can arbitrarily define u to be 0: this will not matter for our purposes because when one of the two points x, y belong to {g = ∞} the right hand side of (5) is infinite, making the inequality trivial. On {g < ∞} we wish instead to define u everywhere in a sensible way. We fix thus a smooth convolution kernel ϕ supported in the ball of radius 1, assume x ∈ {g < ∞} and consider u k := u * ϕ 2 −k . Recalling the Poincaré inequality 1 2 kd (where the constant depends on ϕ) we infer |u k+1 (x) − u k (x)| ≤ C2 −k g(x). This implies that {u k (x)} k is a Cauchy sequence and has a limit: we define then u(x) to be such limit. We next fix x, y ∈ {g < ∞}, regard u as a periodic function defined on the whole R d and set R := |x−y|. W.l.o.g. R ≤ 1. Moreover we recall the classical inequality When u ∈ C 1 we refer the reader to [22,Lemma 3.1] for a proof. Otherwise, the inequality can be validated passing to the limit on the respective ones for the approximating functions u k 's (using that lim k u k (x) = u(x), lim k u k (y) = u(y) and standard facts about convolutions). A classical telescoping argument gives next Recall that α ∈ (1/d, r/d) and fix ε ∈ (0, 1] to be chosen later. We write Let us study I and II separately. Using the Hölder inequality we get For what concerns II, we argue as in (38) getting Putting the two estimates together and choosing that along with (38) gives the sought conclusion.

9.2.
A second version of the asymmetric Lusin-Lipschiz. A simple application of the Young inequality gives the following linear version of (5).
Corollary 9.1. Let u ∈ W 1,r (T) for some 1 < r ≤ d. Then, for any q ∈ [r, r d−1 d−r ) there exist a positive constant C := C(r, d, q) and nonnegative functions a ∈ L 1 and b ∈ L q satisfying together with |u(x) − u(y)| ≤ |x − y|(a(x) + b(y)) for any x, y ∈ R d \ N.
Moreover, we can take N = ∅ provided we choose a suitable representative of u, a and b and the latter can be selected so that the respective map W 1,r ∋ u → (a, b) ∈ L 1 × L q is continuous.
Let us now fix 1 ≤ r < d. For any β < d/r − 1 consider the function u(x) := |x| −β ∈ W 1,r loc (R d ) and cut it off with a smooth cut-off function so that it is compactly supported in (−1/2, 1/2) d . Extend then the function by periodicity and regard it as a function in W 1,r (T). Proposition 9.2 ensures that the exponent q in Corollary 9.1, associated to u, must satisfy q ≤ d−1 β and therefore q ≤ r d−1 d−r + ε with ε → 0 when β → d/r − 1.
9.3. The critical case p = d. We discuss here possible improvements of (5) in the critical case p = d. First of all observe that, in general, we cannot expect (4) to hold for u ∈ W 1,d (T) since it would in turn imply u ∈ L ∞ . However, following closely the proof of Proposition 1.7 one can show: where N ⊂ R d is negligible. We do not give the details since (42) does not play any role in the sequel. We instead show a generalization of (5) for maps with Du in the Lorentz space L d,1 (see e.g. [20] for the relevant definition): as a corollary we get the L d -a.e. uniqueness of trajectories of vector fields enjoying such regularity. We recall, in passing, that the assumptions Du ∈ L d,1 implies the continuity of u.
Proposition 9.3. Assume u ∈ W 1,1 (T) satisfy Du ∈ L d,1 . Then there exists g ∈ L d,∞ such that |u(x) − u(y)| ≤ (C(d)M|Du|(x) + g(x))|x − y| for any x, y ∈ R d \ N for some negligible set N. The latter can be assumed to be empty if u is appropriately defined pointwise and moreover there is a continuous selection map Du ∋ L d,1 → g ∈ L d,∞ .
Proof. Fix x, y ∈ R d and R = 2|x − y| and argue as in the proof of Proposition 1.7. Our conclusion will follow from (37) and (38) provided we show for some g ∈ L d,1 (R d ) satisfying (43).
The Hölder inequality for Lorentz spaces gives Observe that |Du|1 B R (y) L d,1 ≤ |Du|1 B 3R (x) L d,1 ≤ 3R sup 0<t<3 t −1 |Du|1 Bt(x) L d,1 , let us set g(x) := sup 0<t<3 t −1 |Du|1 Bt(x) L d,1 and check (43). First notice that where in the latter estimates we regard |Du| as a function on the torus. Now we argue by duality. Fix h ∈ L d d−1 ,1 . Recall that g d L 1,∞ = g d L d,∞ for any nonnegative g ∈ L d,∞ . Hence, using the weak (1, 1) estimate for the maximal function, we get Since h ∈ L

Well posedness theorems
First of all we observe that, arguing as in [12,Corollary 5.4], Proposition 9.3 implies the following result.
For L d -a.e. x ∈ T there exists a unique trajectory of u starting at x at time t = 0.