Non-renormalized solutions to the continuity equation

We show that there are continuous, $W^{1,p}$ ($p<d-1$), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.

in a d-dimensional periodic domain, d ≥ 3, for a time-dependent incompressible vector field u : [0, 1] × T d → R d and an unknown density ρ : [0, 1] × T d → R. Here and in the sequel T d = R d /Z d is the d-dimensional flat torus. We will also always assume, without loss of generality, that the time interval is [0, 1]. We prove in these notes the following theorem.  By weak solution we mean solution in the sense of distributions.
In particular, for Lipschitz vector fields u the well-posedness theory for (1) follows from the Cauchy-Lipschitz theory for ordinary differential equations applied to (3). It is in general of great interest to investigate the existence and uniqueness of weak solutions to the Cauchy problem for (1) in the case of non-smooth vector fields, and the connection to the Lagrangian problem (3)- (4). The general question can be formulated as follows. Fix an exponent r ∈ [1, ∞], denote by r ′ its dual Hölder, 1/r + 1/r ′ = 1, and assume that a vector field (5) u ∈ L 1 0, 1; L r (T d ) is given. What can be said about existence and uniqueness of weak solutions in the class of densities (6) ρ ∈ L ∞ 0, 1; L r ′ (T d ) ?
The choice of class (6) for ρ is motivated by the fact that for classical solutions to (1) every spatial L r ′ norm is preserved in time. Once (6) is fixed, the choice of the class (5) for u is natural, since in this way ρu ∈ L 1 ((0, 1) × T d ) and thus the notion of distributional solution to (1) is well defined. While existence of weak solutions can be easily shown under the assumptions (5)- (6), the uniqueness question is much harder. In 1989 R. DiPerna and P.L. Lions [9] proved that uniqueness holds in the class (6) if (7) Du ∈ L 1 0, 1; L r (T d ) i.e. if u enjoys Sobolev regularity with exponent r. Moreover, in this case, the incompressibility assumption can be relaxed to div u ∈ L ∞ . In the class of bounded densities the uniqueness result was later extended by L. Ambrosio [1] in 2004, for fields u ∈ L 1 (0, 1; BV ) with div u ∈ L ∞ and very recently by S. Bianchini and P. Bonicatto [3] in 2017 in the case of BV nearly incompressible vector fields. In all of these results an important additional feature is the connection to a suitable extension to (3), i.e. the link between the Eulerian and the Lagrangian picture. More precisely, under assumption (7), there exists a unique distributional solution to (3) for a.e. x, such that x → X(t, x) is measure preserving for all t (assuming div u = 0): such flow map is called regular Lagrangian flow (see [2] for a general discussion). Then the unique solution to the continuity or the transport equation is given by (4), as in the smooth case.
On the other side, several non-uniqueness counterexamples are known, but they mainly concern the case when the field is "very far" from being incompressible (e.g. div u / ∈ L ∞ , see [9]) or the case when no bounds on one full derivative of u are available (see, for instance, the counterexample in [9] for a field u ∈ L 1 (0, 1; W s,1 ) for every s < 1, but u / ∈ L 1 (0, 1; W 1,1 ) or the counterexample in [8] for a field u ∈ L 1 (ε, 1; BV ) for every ε > 0, but u / ∈ L 1 (0, 1; BV )). In all these counterexamples, however, non-uniqueness for the PDE (1) is a consequence of a Lagrangian non-uniqueness for the associated ODE (3). We refer to [12] and to [2] for a more detailed discussion.
Very recently, we proved in [12] the analog of Theorem 1.1, for fields and densities in the class The result in [12] shows that uniqueness can fail even for incompressible, Sobolev vector fields (i.e. fields for which the Lagrangian problem (3) is well posed, in the sense of the regular Lagrangian flow), if the integrability exponent p of Du is much lower than the one provided in (7) by DiPerna and Lions' theory, as specified in (9). The end-point r = ∞, corresponding in (9) to p < d − 1, is excluded in [12]. The main result of this notes, namely Theorem 1.1, shows that such end-point case can indeed be reached and, in addition, quite surprisingly, the vector field produced by Theorem 1.1 is continuous in time and space, not only bounded.
We postpone to Section 2 a technical discussion about why the case r = ∞ was out of reach in [12] and which new ideas are introduced in these notes to deal with such problem.
We would like now to briefly comment about the continuity of the vector field u produced by Theorem 1.1. It was observed by L. Caravenna and G. Crippa [7] that the boundedness or the continuity of the vector field (in addition to some Sobolev regularity) could play a key role in the uniqueness problem in the class of integrable densities ρ ∈ L 1 ((0, 1) × T d ). It thus turns out to be a very interesting question to ask if, in fact, boundedness or continuity plus Sobolev regularity are enough to guarantee uniqueness. Theorem 1.1 shows that this is not the case, if the integrability of Du is lower than a dimensional threshold (precisely, d − 1).
The idea that the boundedness or the continuity of u can play a crucial role in the uniqueness problem is confirmed by the fact that the majority of the result concerning existence and uniqueness of the regular Lagrangian flow associated to a Sobolev or BV vector field u assume that u ∈ L ∞ (see, for instance, the recent survey [2]).
On a different point of view, it is a classical result (see, for instance, [10]) that the boundedness of u, even without any further Sobolev regularity, is enough to have uniqueness, if a small viscosity is added to the continuity equation: (10) ∂ t ρ + div (ρu) = ν∆ρ, ν > 0, while in [12] we showed that uniqueness for (10) can drastically fail is u is Sobolev, but not bounded. The result in Theorem 1.1 is quite surprisingly, even in comparison with our previous result in [12]. Indeed, for vector fields produced by Theorem 1.1 the Lagrangian picture is very well behaved: first, the Sobolev regularity implies the existence and uniqueness of the regular Lagrangian flow. Second, the continuity of the field implies that the trajectories provided by the regular Lagrangian flow are C 1 in time (and this was not the case for the fields produced in [12]). Third, the bound (2) means that the length of each trajectory is at most ε > 0, i.e. particles almost don't move (and, again, this was not the case for the fields produced in [12]). Observe also that ε in (2) depends neither on the length of the time interval [0, 1] nor on the L 1 distance between the initial and the final datum ρ(1)− ρ(0) L 1 (T d ) = ρ L 1 (T d ) . Nevertheless uniqueness in the Eulerian world gets completely lost.
We conclude this introduction observing that Theorem 1.1 is an immediate application of the following theorem, whose proof is the topic of all next sections.
Then there exist ρ : (a) ρ, u have the following regularity: Condition (d) can be substituted by the following: Acknowledgement. This research was supported by the ERC Grant Agreement No. 724298.

Comments on the proof
We describe in this section what problems arise when one tries to extend the proof provided in [12] to Theorem 1.1, i.e. to the end-point case r = ∞ and which new ideas are introduced to solve such problems.
2.1. Sketch of the paper [12]. We first briefly sketch the proof provided in [12] for the analog of Theorem 1.1 under the conditions (8), (9) . The proof is based on a convex integration scheme, with both oscillations and concentration playing a key role. More precisely, the density ρ and the field u are defined as limit of a sequence (ρ q ) q , (u q ) q of smooth approximate solutions to the continuity equation where R q is a smooth vector field converging strongly to zero In this way ρ, u are a weak solution to (1) and, moreover, they have the desired regularity. The sequence (ρ q , u q , R q ) is constructed recursively. Assuming (ρ q−1 , u q−1 , R q−1 ) are given, one defines where λ q is an oscillation parameter and µ q is a concentration parameter, suitably chosen at each step of iteration, F, G are nonlinear functions and {Θ µ } µ>0 (resp. {W µ } µ>0 ) is a family of Mikado densities (resp. Mikado fields) (see Proposition 5.1 below and in particular estimates (47)). It is proven in [12] that ϑ q , w q satisfy the following estimates: Notice that γ 1 > 0 because r < ∞ (and thus r ′ > 1), γ 2 > 0 because r > 1 and γ 3 > 0 because of (9). Estimates (19a)-(19b) together with the inductive assumption (14) applied to R q−1 guarantee the convergences in (15). Estimate (19e) guarantees the convergence in (16), provided at each step µ q ≫ λ q . A computation then shows that, in order for (13) to be satisfied, R q must be defined as quadratic term In order to prove (14), one first use the oscillation parameter λ q to make the (antidivergence of the) quadratic term small. Then, in order to estimate the linear term, one can use concentration. For instance, for the term ϑ q u q−1 , we can use (19c) provided µ q is chosen large enough. A similar estimate holds for ∂ t ϑ q , again using (19c), while for ρ q−1 w q one must use (19d). This shows that R q can be suitably defined in order to satisfy (20), thus concluding the proof in [12] for the analog of Theorem 1.1 under the assumptions (8), (9). Let us now discuss why the above proof does not apply to Theorem 1.1, i.e. to the case r = ∞, r ′ = 1.

First Issue.
If r = ∞, then estimate (19b) becomes w q Ctx 1 and this is not enough to prove the convergence in (15b). This issue is solved, modifying the definition of ρ q , u q in (17) as In this way, using (19a), we get so that the convergences in (15) still holds, and, moreover, the limit vector field u = lim u q is continuous, being the uniform limit of smooth fields. See Section 4 and, in particular, estimates (43), (44). (20) and in particular estimate (21) and the companion estimate for ∂ t ϑ q . Indeed, if r = ∞ and r ′ = 1, then γ 1 = 0 and thus the concentration paramter µ q can not be used in (21) to make the linear term smaller than δ q . This issue is solved using the inverse flow map associated to u q−1 , an idea used in [4] in the framework of the Euler equation, see also [11], [5]. Precisely, one separately considers

Second Issue. The second issue concerns the analysis of the linear term in
While for the Nash term an estimate similar to (21) still holds, since γ 2 = d − 1 > 0, in order to treat the transport term, one modifies the definition of ϑ q and w q as follows. The time interval [0, 1] is split into N small intervals {I i } i of size 1/N . Denoting by t i the middle point of each I i , one considers the inverse flow map Φ i associated to u q−1 and a partition of unity {α i } subordinated to the partition {I i } i of [0, 1]. The definition in (18) is then modified as follows: With this new definition, the transport term in (22) assumes the form The oscillation parameter λ q can now be used to show that See Section 6.3.

Third Issue.
The third issue appears because of the new definition (23) of ϑ q , w q . Indeed if at some time t ∈ [0, 1] two cutoffs α i (t) = 0, α i+1 (t) = 0 are active, then in the quadratic term in (20) a term of the form appears, i.e. a non-trivial interaction between a Mikado density and a Mikado field. In general there is no reason why one should be able to find a small antidivergence of such term. The problem can be solved, using, at each step q of the construction, two different oscillation parameters λ ′ q , λ ′′ q and two different concentration parameters and modifying one more time the definition of ϑ q , w q as follows: With this new definition, the main term in the non-trivial interaction in (24) becomes of the form x) , i.e. the product of a fast oscillating function (with frequency λ ′ q ) with a very fast oscillating function (with frequency λ ′′ q ), where one of the two factors (namely W µ ′ q or W µ ′′ q ) is small in L 1 (T d ) because of the concentration mechanism (compare with estimate (19d)). One can then use an improved Hölder inequality (see Lemma 3.4) to show that the terms in (25) are small in L 1 and thus conclude the proof of Theorem 1.1. See Section 6.2 and in particular Lemma 6.1.

Technical tools
In this section we provide some technical tools which will be frequently used in the following. We start by fixing some notation: we denote by f C k the sup norm of f together with the sup norm of all its derivatives in time and space up to order k; -f (t) C k (T d ) the sup norm of f together with the sup norm of all its spatial derivatives up to order k at fixed time t; -f (t) L p (T d ) the L p norm of f in the spatial derivatives, at fixed time t.
Since we will take always L p norms in the spatial variable (and never in the time variable), we will also use the shorter notation f (t) L p to denote the L p norm of f in the spatial variable.
is the set of smooth functions on the torus with zero mean value.
• We will use the notation C(A 1 , . . . , A n ) to denote a constant which depends only on the numbers A 1 , . . . , A n .
3.1. Diffeomorphisms of the flat torus. We discuss in this section standard properties of diffeomorphisms of the flat torus. Let Φ : R d → R d be a smooth diffeomorphism. We say that Φ is a diffeomorphism of T d , and we write Φ : We say that a diffeomorphism Φ : Given a diffeomorphism Φ, we will often consider Recall also that for a given invertible matrix A, Lemma 3.1. Let Φ : T d → T d be a measure-preserving smooth diffeomorphism. Then, for every k ∈ N, where C k is a constant depending only on k (and on the dimension d).
Proof. For any fixed x ∈ T d it holds The conclusion now follows from the definition of cofactor matrix.
Proof. We show that for every ϕ ∈ C ∞ (T d ) it holds thus concluding the proof of the lemma.
The proof is an easy application of the chain rule and thus it is omitted.

3.2.
Properties of fast oscillations. We discuss now some properties of fast oscillating periodic functions. For a given g : T d → R and λ ∈ N * , we set g λ (x) := g(λx).
Observe that for every p ∈ [1, ∞] and k ∈ N, 3.2.1. Improved Hölder inequality. In the same spirit as in [12] and [6], we now prove an improved Hölder inequality for the product of a slow oscillating function with a fast oscillating functions composed with a diffeomorphism.
Since Φ is a measure-preserving diffeomorphism, it holds Therefore we can apply (30) to get 3.2.2. Antidivergence operators. In this section we introduce two antidivergence operators, a standard and an improved one, in the same spirit as in [12].
is thus well defined. We define the standard antidivergence operator as ∇∆ −1 : It clearly satisfies div (∇∆ −1 f ) = f . Lemma 3.5. For every k ∈ N and p ∈ [1, ∞], the standard antidivergence operator satisfies the bounds For the proof, see [12,Lemma 2.2] . We now use introduce an improved antidivergence operator.
Lemma 3.6. Let f, g : T d → R be smooth function with Let λ ∈ N * and Φ : T d → T d be a smooth, measure-preserving diffeomorphism. Then there exists a smooth vector field u : and for every k ∈ N, p ∈ [1, ∞], We will use the notation where · denotes the scalar product.
Proof. Since g has zero mean value, we can define Let us first check that u satisfies (33). It holds We prove now that (34) holds. We can write Let us estimate A: Similarly, for B: Remark 3.8. In Lemma 3.6, if f, g, Φ are smooth functions of (t, x), t ∈ [0, 1], x ∈ T d and at each time t ∈ [0, 1], they satisfy the assumptions of Lemma 3.6, then we can apply R at each time and define a time-dependent vector field u(t, ·) satisfying (33) and (34). Moreover u turns out to be a smooth function of (t, x).

Mean value and fast oscillations.
In this section we prodide an estimate on the mean value of the product of a slow oscillating function with a fast oscillating function composed with a diffeomorphism.

Statement of the main proposition and proof of Theorem 1.2
We assume without loss of generality T d is the periodic extension of the unit cube [0, 1] d . The following proposition contains the key facts used to prove Theorem 1.2. Let us first introduce the continuity-defect equation: (38) ∂ t ρ + div (ρu) = −div R, div u = 0.
We will call R the defect field.
and, moreover, if at some time t ∈ [0, 1], R 0 (t) = 0, then Proof of Theorem 1.2 assuming Proposition 4.1. For ρ 0 , u 0 in the statement of Theorem By (11), R 0 is well defined, it is smooth and (ρ 0 , u 0 , R 0 ) solve the continuity-defect equation. Let (p q ) q∈N be a fixed increasing sequence of real numbers such that p q → d − 1 as q → ∞. Let also (η q ) q∈N , (δ q ) q∈N be two sequence of positive real numbers, which will be fixed later. Starting from (ρ 0 , u 0 , R 0 ), we can recursively apply Proposition 4.1 to obtain a sequence (ρ q , u q , R q ) q∈N of smooth solutions to the continuity-defect equation such that for all times t ∈ [0, 1] and ρ q+1 (t) = ρ q (t), u q+1 (t) = u q (t), R q+1 (t) = 0, for all times t such that R q (t) = 0. Therefore, by induction, we get from (40a) and (40d) that for all t ∈ [0, 1] and all q ∈ N, where we set δ −1 := max t∈[0,1] R 0 (t) L 1 and, moreover, where E was defined in (12). We now choose (δ q ) q∈N so that . There is q * so that p q > p for every q > q * . We now have, for all t ∈ [0, 1], It follows from (42) that ρ(t) = ρ 0 (t) and u(t) = u 0 (t), whenever t ∈ E, and thus part (c) is also proven. To prove (d), we observe that, from (43), for all t ∈ [0, 1], and thus (d) follows choosing Alternatively, to achieve (d'), we observe that, from (44), for all t ∈ [0, 1],

The perturbations
In this and the next two sections we prove Proposition 4.1. In particular in this section we fix the constant M in the statement of the proposition, we define the functions ρ 1 and u 1 and we estimate them. In Section 6 we define R 1 and we estimate it. In Section 7 we conclude the proof of Proposition 4.1.

5.1.
Mikado fields and Mikado densities. We recall the following proposition from [12].
For every µ > 2d and j = 1, . . . , d there exist a Mikado density Θ j µ : T d → R and a Mikado field W j µ : T d → R d with the following properties. (a) It holds in Proposition 5.1. In this way for each direction j = 1, . . . , d, we obtain a family of Mikado densities {Θ j µ } µ>2d and fields {W j µ } µ>2d , obeying the following estimates:

Definition of the perturbations.
We are now in a position to define ρ 1 , u 1 .
The constant M has already been fixed in (48). Let thus p ∈ [1, d − 1), η, δ > 0 and (ρ 0 , u 0 , R 0 ) be a smooth solution to the continuity-defect equation (38). Let be parameters, which will be fixed later. Set Notice that for every time t ∈ [0, 1] there is at most one odd index i 1 and one even index i.e. the inverse flow map associated to the vector field u 0 , starting at time t i . Notice that, for fixed t, Φ i (t) : T d → T d is a measure-preserving diffeomorphism. We denote by R 0,j the components of R 0 , i.e.
Let also ψ : [0, 1] → R be a smooth function such that ψ(t) ∈ [0, 1] for every t ∈ [0, 1] and (54) We set where ϑ, ϑ c , w are defined as follows. First of all, let Θ j µ , W j µ , j = 1, . . . , d, be the family (depending on µ) of Mikado densities and fields provided by Proposition 5.1, with a, b chosen as in (49). We set (55) The factor (DΦ i (t, x)) −1 is the inverse matrix of DΦ i (t, x). Observe that for fixed t 0 ∈ [0, 1], there are at most one odd index i 1 and one even index i 2 so that α i (t) = 0 if i = i 1 , i 2 and t is close enough to t 0 (say, |t − t 0 | ≤ 2τ /3). Therefore for such times t we can write (56) Notice that ϑ 0 and w are smooth functions. Notice also that ϑ + ϑ c has zero mean value in T d at each time t. Finally observe that w is a sum of terms of the form (DΦ) −1 (G•Φ), with Since div (W µ ) λ = 0 for every µ, λ (see Proposition 5.1), we get from Lemma 3.2 that each one of these terms is divergence free and thus div w = 0. Therefore div u 1 = div u 0 + div w = 0.
Remark 5.2. Observe that, thanks to the cutoff in time ψ, if R 0 (t) ≡ 0, then

5.3.
Estimates on the perturbation. In this section we estimate ϑ, ϑ c , w.
Proof. Since we have to estimate ϑ(t) L 1 (T d ) for every fixed time t, we can assume that ϑ(t) has the form (56). In (56) each term in the summation over j has the form f (g λ • Φ), with Therefore we can apply the improved Hölder inequality, Lemma 3.4, to get Proof. As in the proof of Lemma 5.3, we can use for ϑ(t) the form (56) and we observe that each term in the summation over j has the form f (g λ • Φ), with f, Φ, g, λ as in (57). We can thus apply Lemma 3.9 to get: Proof. As in the proof of Lemma 5.3 we can use for w(t) the form (56). Therefore Lemma 5.6 (W 1,p norm of w). For every time t ∈ [0, 1], Proof. As in the proof Lemma 5.5 we can use for w(t) the form (56). Taking one partial derivative ∂ k , we get We now apply the classical Hölder inequality to estimate ∂ k w(t) L p : A similar (and even easier) computation holds for w(t) L p , thus concluding the proof of the lemma.

The new defect field
In this section we continue the proof of Proposition 4.1, defining the new defect field R 1 and estimating it.
6.2. Definition and estimates for R interaction , R flow , R ψ , R quadr . In this section we define and estimate the vector fields R quadr , R interaction , R ψ and R flow so that (61a) holds. First of all, we want to compute more explicitly div (ϑ(t)w(t) − R 0 (t)), for every fixed time t. We can use the form (56) for ϑ(t) and w(t). Exploiting the fact that for j = k, Θ j µ and W k µ have disjoint support (see Proposition 5.1), we have where we set On the other side, using the fact that N i=1 where we set with Id being the identity matrix, and where in the last equality we used the fact that div ((Θ j µ ) λ (W j µ ) λ − e j ) = 0 for every µ, λ, j (see Proposition 5.1) and Lemma 3.2. We now observe that each term in the two summations over j has zero mean value (being a divergence) and it has the form We can therefore apply Lemma 3.6 and define so that (61a) holds. We now separately estimate R interaction , R flow , R ψ , R quadr . We start with R interaction .
Lemma 6.4. For every t ∈ [0, 1], Proof. R quadr (t) is defined in (66) using Lemma 3.6. Applying the bounds provided by such proposition, with k = 0 and p = 1, we get (by (50)) We now observe that each term in the last line in (69) has the form f Since Θ j µ has zero mean value (see Proposition 5.1), we can apply Lemma 3.6 and define Proof. First of all, we observe that We defined R transport in (70) using the antidivergence operator provided by Lemma 3.6. We can thus apply the bounds provided by such proposition, with k = 0 and p = 1, to get where in the last line we used (50).
6.4. Estimates for R Nash and R corr . In this section we estimate R Nash and R corr .