UNIQUENESS OF L^{∞} SOLUTIONS

FOR A CLASS OF CONORMAL BV VECTOR FIELDS

Ferruccio Colombini, Nicolas Lerner

Università di Pisa, Université de Rennes 1

February 14, 2003

Abstract. Let X be a bounded vector field with bounded divergence defined in an open set Ω of ℝ^{d}, transverse to a hypersurface S. Let Ω_{0} be an open subset of Ω such that the Hausdorff measure H^{d−1}(Ω\Ω_{0}) = 0. We suppose that the vector field X belongs to BV_{loc}(Ω_{0}) “conormally”, an assumption made precise in the text, which is satisfied whenever the gradients of the coefficients of X have locally only a single component which is actually a Radon measure. This class can be invariantly defined and contains the so-called piecewise W^{1,1} functions studied in [Li]. We prove the uniqueness of L^{∞} solutions for the Cauchy problem related to X across the hypersurface S. We use for the proof some simple arguments of geometric measure theory to get rid of closed sets with codimension > 1. Next, we need an anisotropic regularization argument analogous to the one used in [Bo].

Contents

1. Introduction Our general framework. The one and two dimensional cases. The DiPerna-Lions’ theorem on W^{1,1} vector fields with bounded divergence. The Colombini-Lerner’s result on BV vector fields with bounded divergence.

2. A new result Definition 2.1. The class conormal BV_{loc}. Theorem 2.4. Uniqueness of L^{∞} solutions for a class of BV vector fields.

3. First part of the proof Step 1: non-negative solutions are unique. Lemma 3.1. Step 2: Leibnizian vector fields. Lemma 3.2.

4. Getting rid of subsets whose H^{d−1} Hausdorff measure is zero.

Lemma 4.1. A solution on a set whose codimension is > 1 is a global solution.

Lemma 4.2. The same occurs for an inequality.

5. Commutation arguments Preliminary remarks. Tailoring the mollifiers on the geometry of the vector field. Beginning of the proof of the estimate. Lemma 5.1: translations in L^{∞} and in BV. The key argument. Straightening the singular set. Choosing the different speeds. Concluding remarks.

6. Appendix

2000 Mathematics Subject Classification 35F05, 34A12, 26A45. Key words and phrases. Vector fields, Transport equation, Weak solutions, BV.

Typeset by AMS-TEX

1. Introduction

Let us give first the general framework of our study: we shall consider a real vector field X defined in an open set Ω of ℝ^{d} and a Lipschitz oriented hypersurface S so that

(1.1)  X ∈ L_{loc}^{∞}, div X ∈ L_{loc}^{∞}, X is positively transverse to S.

Note that the first condition is simply the requirement of finite speed of propagation so that a local problem makes sense for the equation Xu = f. The second condition is essentially necessary to get a uniqueness result: in our appendix A1, we give an example of a two-dimensional W^{1,1} ∩ L^{∞} vector field whose divergence is a positive unbounded L^{1} function so that no uniqueness property is satisfied (see also section IV.1 of [DL]). Let us clarify the third condition. Let ν be a unit vector field conormal to the oriented Lipschitz hypersurface S. The vector field X is said to be positively tranverse to S if for all x_{0} ∈ S, there exists a neighborhood V_{0} of x_{0} such that

(1.2)  essinf_{V_{0}} X(x) · ν(x) > 0.

When the dimension is ≤ 2, the conditions (1.1) essentially ensure uniqueness of L^{∞} solutions. In one dimension, the autonomous ODE

(1.3)  ẋ = f(x), x(0) = x_{0},

has a unique solution, provided f is merely continuous and f(x_{0}) ≠ 0. The existence is given by Peano’s theorem whereas the uniqueness follows from the direct integration of \frac{dx}{f(x)} = dt. In fact, setting G(x) = ∫_{x_{0}}^{x} \frac{dy}{f(y)}, we find a neighborhood of x_{0} in which G ∈ C^{1}, G′ ≠ 0 so that G has an inverse function g ∈ C^{1}. Then for a C^{1} solution x(t) defined near 0 of (1.3), we get

\frac{d}{dt}(G(x(t)) = \frac{ẋ(t)}{f(x(t))} = 1

which implies G(x(t)) = t and thus x(t) = g(t) and the uniqueness. In two dimensions, let us examine a divergence-free L^{∞} vector field X. It is then a Hamiltonian vector field H_{σ}

X = \frac{∂σ}{∂y} \frac{∂}{∂x} − \frac{∂σ}{∂x} \frac{∂}{∂y} ,

where σ is a Lipschitz function. Denoting by ρ the Lipschitz equation of the hypersurface, one can assume from (1.2) that {σ, ρ} ≥ 1 near 0. It means that the Jacobian determinant of the mapping κ given by (ρ, σ) = κ(x, y) is

det κ′ = | \begin{matrix}∂_{x}ρ & ∂_{y}ρ \\ ∂_{x}σ & ∂_{y}σ \end{matrix} | = X(ρ) ≥ 1,

and κ is thus a bi-Lipschitz local homeomorphism. Setting ν = κ^{−1}, and for a function F ∈ C_{c}^{1}, we get for u ∈ L^{∞} such that Xu = 0,

0 = ⟨Xu, F (ρ, σ)⟩ = − ∫∫ u(x, y)\frac{∂F}{∂ρ} (ρ(x, y), σ(x, y))X(ρ)(x, y)dxdy

= − ∫∫ (u ∘ ν)(ρ, σ)\frac{∂F}{∂ρ} (ρ, σ) det κ′(ν(ρ, σ)) det ν′(ρ, σ)dρdσ

= − ∫∫ (u ∘ ν)(ρ, σ)\frac{∂F}{∂ρ} (ρ, σ)dρdσ.

It means

\frac{∂(u ∘ ν)}{∂ρ} = 0

and since u ∘ ν_{|ρ<0} = 0, we obtain u = 0.

In dimension ≥ 3, the Cauchy uniqueness under the sole conditions (1.1) does not appear to be true. In fact a three-dimensional counterexample, due to M.Aizenman [Ai], shows that the existence of a flow is not guaranteed for a divergence-free L^{∞} vector field. It is then natural to require some additional regularity for the coefficients of the vector field. A standard result in this direction is the Eulerian version of the classical Cauchy-Lipschitz theorem, ensuring the uniqueness of L_{loc}^{1} solutions for Lipschitz vector fields satisfying (1.1). In 1989, an important step forward was accomplished by R.DiPerna and P.-L.Lions, who proved in [DL] a uniqueness result for W^{1,1} vector fields. Let us give a local version of their theorem.

Theorem 1.1. Let X be a vector field and S be an hypersurface satisfying (1.1) on an open set of ℝ^{d}. Assume moreover that X ∈ W_{loc}^{1,1} and let c be a L_{loc}^{1} function. Let u be a L_{loc}^{∞} function such that

Xu = cu, supp u ⊂ S_{+},

where S_{+} is the half-space above the oriented S. Then if c_{+} belongs to L_{loc}^{∞}, the function u vanishes in a neighborhood of S. The same conclusion holds if we replace in (1.1) the condition div X ∈ L_{loc}^{∞} by (div X)_{+} ∈ L_{loc}^{∞}.

A natural question raised in [PP], [PR], [Li], with important implications in fluid mechanics, is to know if the same result holds, replacing W^{1,1} by BV. In [CoL], the authors proved the following theorem.

Theorem 1.2. Let X be a vector field and S be an hypersurface satisfying (1.1) on an open set Ω of ℝ^{d}. Assume moreover that X ∈ BV_{loc} and let c be a Radon measure on Ω. Let u be a continuous function on Ω such that

Xu = cu, supp u ⊂ S_{+},

where S_{+} is the half-space above the oriented S. Then, if c_{+} belongs to L_{loc}^{∞}, the function u vanishes in a neighborhood of S. The same conclusion holds if we replace in (1.1) the condition div X ∈ L_{loc}^{∞} by (div X)_{+} ∈ L_{loc}^{∞}.

Remarks. If X ∈ L_{loc}^{1}, div X ∈ L_{loc}^{1} and u ∈ L_{loc}^{∞}, we define

Xu = ∑_{1≤j≤n} \frac{∂}{∂x_{j}} (a_{j}u) − udiv X.

The same formula can be used if X ∈ L_{loc}^{1}(Ω), div X ∈ D′^{(0)}(Ω) (the Radon measures M(Ω)) and u ∈ C^{0}(Ω). These definitions are of course consistent with the usual definition of Xu whenever u is smooth and with the weak definition

⟨Xu, ϕ⟩ = − ∫ u(X(ϕ) + ϕ div X)dm, ∀ϕ ∈ C_{c}^{1}(Ω),

where dm stands for the Lebesgue measure. In fact, if (M, ω) is a smooth oriented manifold, and X a locally bounded measurable vector field on M, the divergence of X can be defined by the equality

(1.4)  div X = −^{t}X − X.

For φ, ψ, C_{c}^{1} test functions, we define

⟨^{t}Xφ, ψ⟩ = ⟨φ, Xψ⟩ = ∫ φ(Xψ)ω.

In both theorems above, the one-sided condition can be replaced by the more elegant (c + div X)_{+} ∈ L_{loc}^{∞}. On may remark that an unbounded divergence makes a real vector field an irreversible equation, since the divergence acts as a diffusion term. Theorem 1.2 gives uniqueness of continuous solutions, which are indeed weak solutions, but whose existence is not guaranteed. We are seeking uniqueness of L^{∞} solutions, whose existence is known for vector fields satisfying (1.1) (see prop.II.1. in [DL]).

2. A new result

We shall now describe a new result, giving uniqueness of L^{∞} solutions for a class of BV vector fields going beyond the so-called piecewise W^{1,1} vector fields introduced in [Li]. We refer the reader to the appendix A2 for the trivial verification that the assumptions (1.1) and X ∈ BV_{loc} are indeed invariant by C^{1,1} diffeomorphism.

Definition 2.1. Let Ω_{0} be an open set of ℝ^{d}. The space conormalBV_{loc}(Ω_{0}) is defined as a subspace of BV_{loc}(Ω_{0}).

(i) A function a ∈ BV_{loc}(Ω_{0}) belongs to conormalBV_{loc}(Ω_{0}) if each x ∈ Ω_{0} has a neighborhood V ⊂ Ω_{0} such that, on V, there exist C^{1,1} coordinates x_{1}, …, x_{d} so that

\frac{∂a}{∂x_{1}} ∈ M(V), \frac{∂a}{∂x_{k}} ∈ L_{loc}^{1}(V), for k ≥ 2.

(ii) A vector field X ∈ BV_{loc}(Ω_{0}) belongs to conormalBV_{loc}(Ω_{0}) if each x ∈ Ω_{0} has a neighborhood V ⊂ Ω_{0} such that, on V, there exist C^{1,1} coordinates x_{1}, …, x_{d} so that, whenever X = ∑_{1≤j≤d} a_{j}∂_{x_{j}},

∀j ∈ \{1, …, d\}, \frac{∂a_{j}}{∂x_{1}} ∈ M(V), \frac{∂a_{j}}{∂x_{k}} ∈ L_{loc}^{1}(V), for k ≥ 2.

We shall denote the space conormalBV_{loc}(Ω_{0}) by CBV_{loc}(Ω_{0}).

Definition 2.1 is equivalent to the more intrinsic

Definition 2.1’. Let Ω_{0} be C^{1,1} oriented manifold, equipped with a C^{1,1} local foliation of codimension 1. A vector field X ∈ BV_{loc}(Ω_{0}) belongs to CBV_{loc}(Ω_{0}) if, for all Lipschitz continuous vector fields Y tangent to the foliation, the bracket [X, Y] is in L_{loc}^{1}(Ω_{0}). A function a ∈ BV_{loc}(Ω_{0}) belongs to CBV_{loc}(Ω_{0}) if, for all Lipschitz continuous vector fields Y tangent to the foliation, the function Y (a) is in L_{loc}^{1}(Ω_{0}).

Note that in definition 2.1, the foliation is simply given by the hypersurfaces {x_{1} = c}, which can always be assumed locally. As said in the abstract, we shall need the BV assumption only on a “small” open set Ω_{0}, included in our reference open set Ω. This is the reason for introducing the following

Definition 2.2. Let Ω be an open set of ℝ^{d}. We define the class B(Ω) as

(2.1) B(Ω) = {a ∈ L_{loc}^{∞}(Ω)| ∃Ω_{0} open ⊂ Ω with H^{d−1}(Ω\Ω_{0}) = 0 and a ∈ CBV_{loc}(Ω_{0})},

where H^{d−1} stands for the d − 1 dimensional Hausdorff measure. A vector field X ∈ L_{loc}^{∞}(Ω) belongs to B(Ω) if there exists an open set Ω_{0} ⊂ Ω such that H^{d−1}(Ω\Ω_{0}) = 0 and X ∈ CBV_{loc}(Ω_{0}).

In the appendix A2, it is shown that the assumptions (1.1) and X ∈ B(Ω) are invariant by a C^{1,1} diffeomorphism. It might be helpful for the reader to get a couple of examples of CBV functions and vector fields.

Examples 2.3.

(a) The class B(Ω) contains the so-called piecewise W^{1,1} class accepting jumps across C^{1,1} hypersurfaces (see proposition 6.4 of the appendix A3 for a proof of this statement). In fact, an important new feature of our result is that we can get rid of subsets whose (d − 1) dimensional Hausdorff measure is zero, so that our BV assumption is made only on a “small” open set Ω_{0} ⊂ Ω, such that

(2.2)  H^{d−1}(Ω\Ω_{0}) = 0,

disregarding the geometric complexity coming from singular subsets of codimension > 1.

(b) The first simple example of a CBV function in ℝ^{d} = ℝ_{x_{1}} × ℝ_{x_{2}}^{d−1} is the tensor product

a(x_{1}, x_{2}) = b_{1}(x_{1})b_{2}(x_{2}), where b_{1} ∈ BV (ℝ), b_{2} ∈ W^{1,1}(ℝ^{d−1}).

Note that this example is not in general piecewise W^{1,1} since there is no restriction on the singularity with respect to the variable x_{1}, beyond the BV assumption.

(c) The class B(Ω) is not included in BV_{loc}(Ω), because our regularity assumption is made only on an open subset Ω_{0} of Ω such that (2.2) holds. In particular, we are able to handle vector fields which are not locally BV, but only L^{∞} and conormal BV on a “small” open set Ω_{0} such that (2.2) holds. For instance, we provide an example of a function in our class B(ℝ^{2}), which is not in BV_{loc}(ℝ^{2}). Let us consider for x = (x_{1}, x_{2}) ∈ ℝ^{2}, r = \sqrt{x_{1}^{2} + x_{2}^{2}}, the function

a(x_{1}, x_{2}) = cos(r^{−2}).

The function a belongs to L^{∞}(ℝ^{2}) but is not in BV_{loc}(ℝ^{2}) since, on {x ≠ 0},

\frac{∂a}{∂x_{1}} = 2r^{−4} sin(r^{−2})x_{1}

and testing^{1} this distribution against the C_{c}^{0}(ℝ^{2}) function 1_{[0,π^{−1/2}]}(r)H(x_{1})x_{1}sin(r^{−2}), where H is the characteristic function of ℝ_{+}, we get

∫_{0}^{π^{−1/2}} 2r^{−1} sin^{2}(r^{−2})dr ∫_{−π/2}^{π/2} cos^{2} θdθ = \frac{π}{2} ∫_{π}^{+∞} \frac{sin^{2} s}{s}ds = +∞.

Nevertheless, since the function a belongs to C^{∞}(ℝ^{2}\{0}), it is indeed in CBV_{loc}(ℝ^{2}\{0}) and since H^{1}({0}) = 0, the function a belongs to B(ℝ^{2}). Although we have the inclusion L_{loc}^{∞}(ℝ^{2}) ∩ C^{∞}(ℝ^{2}\{0}) ⊂ B(ℝ^{2}) we should also notice that, for a line Δ,

L_{loc}^{∞}(ℝ^{2}) ∩ C^{∞}(ℝ^{2}\Δ) ⊄ B(ℝ^{2}), (e.g. x_{1} sin(1/x_{1}) ∉ B(ℝ^{2}), see the ftnt of lemma 6.5).

1We should use a sequence of smooth test functions χ_{k}(x)x_{1}sin(r^{−2}) where χ_{k} ∈ C_{c}^{∞}(ℝ^{2}; [0, 1])

is supported in {x ∈ ℝ^{2}, 0 < |x| < π^{−1/2} and x_{1} ≥ 0}, and such that, for almost all x, lim_{k} χ_{k}(x) = 1_{[0,π^{−1/2}]}(|x|)H(x_{1}).

(d) One can go much beyond the second example (which is already not piecewise W^{1,1}) as shown by the following example. If a(x_{1}, x_{2}) = b_{1}(x_{1})b_{2}(x_{2}) with b_{1} ∈ BV (ℝ), b_{2} ∈ BV (ℝ) such that

b′_{1} = ∑_{k∈ℕ} α_{k}δ_{y_{k}} + L_{loc}^{1}, b′_{2} = ∑_{l∈ℕ} β_{l}δ_{z_{l}} + L_{loc}^{1}, (α_{k})_{k∈ℕ}, (β_{l})_{l∈ℕ} ∈ ℓ^{1},

so that the sequences of real numbers (y_{k})_{k∈ℕ} and (z_{l})_{l∈ℕ} have both limit 0 then the set

Ω_{0}^{c} = {(0, 0)} ∪_{(k,l)∈ℕ^{2}} {(y_{k}, z_{l})} ∪_{k∈ℕ} {(y_{k}, 0)} ∪_{l∈ℕ} {(0, z_{l})}

is closed in ℝ^{2}, and H^{1}(ℝ^{2}\Ω_{0}) = 0. Consequently, to check that the function a belongs to B(ℝ^{2}), it is enough to verify that a ∈ CBV (Ω_{0}). In fact, for x ∈ Ω_{0}, there exists a neighborhood V of x such that V ⊂ Ω_{0} and a_{|V} is as in example (b) above (or a(x_{2}, x_{1}) is as in example (b)). This example is not piecewise W^{1,1} because of the accumulation of singular lines at {0} × ℝ and ℝ × {0}. Note however that the worst point (0, 0) belongs to Ω_{0}^{c} as well as the simple intersections (y_{k}, z_{l}). On the other hand, there is no difficulty to handle the points (0, x_{2}) provided that x_{2} does not belong to the closure of the set {z_{l}}_{l∈ℕ} since there is then a neighborhood of (0, x_{2}) in which a is of the type of example (b). In the appendix A4, we provide a picture of the singular support of the function a which illustrate the importance and flexibility of our assumption of local foliation on Ω_{0} in the definition 2.1’.

(e) Let a be a function in L_{loc}^{∞}(ℝ^{d}) such that for all j ∈ {1, …, d}

\frac{∂a}{∂x_{j}} = ∑_{k∈ℕ} α_{k,j}δ_{S_{k}^{j}} + L_{loc}^{1}, (α_{k,j})_{k∈ℕ} ∈ ℓ^{1},

where for each j, (S_{k}^{j})_{k∈ℕ} is a countable family of parallel hyperplanes, δ_{S_{k}^{j}} is the simple layer at S_{k}^{j}. We assume that for j ≠ j′, the hyperplanes S_{k}^{j} and S_{l}^{j′} are transverse and such that, for each j, the sequence (S_{k}^{j})_{k∈ℕ} converges. Argumenting as in example (d), one finds with

Ω_{0}^{c} = \overline{∪_{1≤j≠j′≤d} ∪_{k,l∈ℕ} (S_{k}^{j} ∩ S_{l}^{j′})}

that H^{d−1}(ℝ^{d}\Ω_{0}) = 0 and a ∈ B(ℝ^{d}).

(f) Let us give a vector field example in ℝ^{d} of the previous type. Assume that, for all j ∈ {1, …, d}, there exists a countable family (S_{k}^{j})_{k∈ℕ} of parallel hyperplanes so that, for j ≠ j′, the hyperplanes S_{k}^{j} and S_{l}^{j′} are transverse and such that, for each j, the sequence (S_{k}^{j})_{k∈ℕ} converges. Let X = ∑_{1≤i≤d} a_{i}∂_{x_{i}} be a bounded measurable vector field such that, for all i, j ∈ {1, …, d}

\frac{∂a_{i}}{∂x_{j}} = ∑_{k∈ℕ} α_{k,j}^{i}δ_{S_{k}^{j}} + L_{loc}^{1}, (α_{k,j}^{i})_{k∈ℕ} ∈ ℓ^{1}.

Then X belongs to B(ℝ^{d}).

Since the notion of conormal BV makes sense on a C^{1,1} manifold, all the previous examples can be pushed-forward by a C^{1,1} diffeomorphism.

The main theorem of this paper is the following

Theorem 2.4. Let Ω be an open subset of ℝ^{d}, X be a vector field and S be an hypersurface satisfying (1.1) on Ω. Assume moreover that X belongs to B(Ω) (cf.def.2.2) and let c be a function in L_{loc}^{1}(Ω). Let u be a function in L_{loc}^{∞}(Ω) such that

Xu = cu, supp u ⊂ S_{+},

where S_{+} is the half-space above the oriented S. Then if c_{+} belongs to L_{loc}^{∞}(Ω), the function u vanishes in a neighborhood of S. The same conclusion holds if we replace in (1.1) the condition div X ∈ L_{loc}^{∞}(Ω) by (div X)_{+} ∈ L_{loc}^{∞}(Ω).

3. First part of the proof

We collect in this section the standard facts related to the proof, postponing the introduction of the new ingredients to the next section.

Step 1: non-negative solutions are unique. The following lemma is proved in [CoL]. For the convenience of the reader, we recall its statement here and the proof in the appendix A5. Let Ω be an open set of ℝ^{d} and X be a L_{loc}^{∞} vector field on Ω with divergence in L_{loc}^{1}, c ∈ L_{loc}^{1} and w ∈ L_{loc}^{∞}. We shall say that

Xw ≤ cw

if for all non-negative test functions θ ∈ C_{c}^{1}(Ω),

− ∫ w(Xθ + θ div X)dm = ⟨Xw, θ⟩ ≤ ∫ cwθdm.

Lemma 3.1. Let Ω be an open set of ℝ^{d}, X be a vector field on Ω, and S a Lipschitz oriented hypersurface of Ω such that X ∈ L_{loc}^{∞}, div X ∈ L_{loc}^{1} and X is positively transverse to S. Let w be a L_{loc}^{∞} function such that, for some function c ∈ L_{loc}^{1},

Xw ≤ cw, supp w ⊂ S_{+} and w ≥ 0.

Then if (c + div X)_{+} ∈ L_{loc}^{∞}, the function w vanishes in a neighborhood of S. Note that the assumptions of this lemma are satisfied whenever c_{+} ∈ L_{loc}^{∞} and (1.1) is fulfilled.

Remark. This result does not require any regularity for X, besides finite speed, bounded divergence and transversality to S.

Step 2: prove that Xu = 0 implies X(u^{2}) = 0. Then use Step 1 to get u = 0. More pedantically, one could say that Leibnizian vector fields satisfying (1.1) have unique L^{∞} solutions across transverse hypersurface, where the following property would stand as a definition of a Leibnizian vector field:

(3.1)  u, v ∈ L^{∞}, Xu, Xv ∈ L^{1} ⟹ X(uv) = X(u)v + uX(v).

Let us now assume that u ∈ L_{loc}^{∞} satisfies Xu = cu where c is an L_{loc}^{1} function such that c_{+} ∈ L_{loc}^{∞}, X is a vector field and S a hypersurface satisfying (1.1). We compute Xu^{2} using the property (3.1) and we get, since Xu = cu ∈ L_{loc}^{1}

Xu^{2} = 2uXu = 2cu^{2}.

We can now use step 1 to get the answer since 0 ≤ u^{2}, c_{+} ∈ L_{loc}^{∞} and X, S satisfy (1.1) with supp u^{2} = supp u ⊂ S_{+}. In fact, the following lemma asserts that an even weaker statement than (3.1) will be sufficient to get our uniqueness result.

Lemma 3.2. Let Ω be an open set of ℝ^{d}, X be a vector field on Ω, and S a Lipschitz oriented hypersurface of Ω such that X ∈ L_{loc}^{∞}, div X ∈ L_{loc}^{1}, (div X)_{+} ∈ L_{loc}^{∞} and X is positively transverse to S. Let u be a L_{loc}^{∞} function such that, for some function c ∈ L_{loc}^{1} with c_{+} ∈ L_{loc}^{∞},

(3.2)  Xu = cu, supp u ⊂ S_{+}.

Assume also that there exist a L_{loc}^{∞} function C such that for all non-negative test functions θ ∈ C_{c}^{1}(Ω) there exists a sequence (u_{ϵ}) of Lipschitz continuous functions, bounded in L^{∞}(supp θ) by ∥u∥_{L^{∞}(supp θ)} and converging a.e. in supp θ to u such that

(3.3)  lim sup_{ϵ} {∫ X(u_{ϵ})uθdm} ≤ ∫ Cu^{2}θdm.

Then the function u vanishes in a neighborhood of S.

Proof. We shall prove that there exists a non-negative L_{loc}^{∞} function c̃ such that

(3.4)  Xu^{2} ≤ c̃u^{2}.

To get the conclusion, we shall use lemma 3.1. It means that we need to check that, for all non-negative test functions θ ∈ C_{c}^{1}(Ω), we have, with some non-negative L_{loc}^{∞} function c̃,

− ∫ u^{2}(Xθ + θ div X)dm = ⟨Xu^{2}, θ⟩ ≤ ∫ c̃u^{2}θdm,

i.e.

(3.5)  0 ≤ ∫ u^{2}(Xθ + θ(div X + c̃))dm.

Let u_{ϵ} be a sequence of Lipschitz continuous functions, bounded in L^{∞}(supp θ) by ∥u∥_{L^{∞}(supp θ)}, converging a.e. to u. Then we have, since u_{ϵ} is Lipschitz continuous, for all bounded measurable c̃,

∫ u^{2}(Xθ + θ(div X + c̃))dm = lim_{ϵ} ∫ uu_{ϵ}(Xθ + θ(div X + c̃))dm

= lim_{ϵ} {−⟨X(u_{ϵ}u), θ⟩ + ∫ uu_{ϵ}θc̃dm}

= lim_{ϵ} {−⟨X(u_{ϵ})u, θ⟩ − ⟨u_{ϵ}X(u), θ⟩ + ∫ uu_{ϵ}θc̃dm}

= lim_{ϵ} {− ∫ X(u_{ϵ})uθdm} + ∫ u^{2}θ(c̃− c)dm

≥ lim_{ϵ} {− ∫ X(u_{ϵ})uθdm} + ∫ u^{2}θ(c̃− c)dm

≥ ∫ u^{2}θ(c̃− c − C)dm. (3.6)

We can take c̃ = (c + C)_{+} to infer (3.5) from (3.6). □

The proof of theorem 2.3 is thus reduced to proving the estimate (3.3), which will be done in section 5.

4. Getting rid of subsets whose H^{d−1} measure is zero

We focus our attention in this section on the first new feature of our proof. We shall show first that we need only to prove this estimate for non-negative test functions θ in C_{c}^{1}(Ω_{0}) where Ω_{0} is an open subset of Ω such that H^{d−1}(Ω\Ω_{0}) = 0. It is an important aspect of our argument to reduce checking (3.3) in an open set Ω_{0} such that H^{d−1}(Ω\Ω_{0}) = 0, somehow getting rid a priori of subsets whose (d−1) Hausdorff measure is zero.

Lemma 4.1. Let Ω_{0} ⊂ Ω be open subsets of ℝ^{d} such that H^{d−1}(Ω\Ω_{0}) = 0. Let X be a vector field in L_{loc}^{∞}(Ω) such that div X ∈ L_{loc}^{1}(Ω), and let v ∈ L_{loc}^{∞}(Ω) be a (weak) solution on Ω_{0} of the equation Xv = f where f belongs to L_{loc}^{1}(Ω). It means that for all φ ∈ C_{c}^{1}(Ω_{0}),

(4.1)  ∫ fφdm = − ∫ v(X(φ) + φ div X)dm.

Then the equation Xv = f is satisfied weakly on Ω, i.e. (4.1) is true for all φ ∈ C_{c}^{1}(Ω).

Proof. We first make the more restrictive assumption

M^{d−1}(Ω\Ω_{0}) = 0

where M^{d−1} stands for the (d − 1) Minkowski content i.e., setting F = Ω\Ω_{0},

L^{d}(F + ϵB_{1}) = ϵα(ϵ), with lim_{ϵ→0_{+}} α(ϵ) = 0,

where L^{d} stands for the d-dimensional Lebesgue measure and B_{1} is the closed unit ball of ℝ^{d}. Let φ be a test function ∈ C_{c}^{1}(Ω). Using the notation A^{c} for the complement of A in ℝ^{d}, we note that the set

K = F ∩ supp φ = Ω_{0}^{c} ∩ supp φ

is a compact subset of Ω satisfying M^{d−1}(K) = 0. We define the Lipschitz continuous function

(4.2)  σ(t) = min((2t − 1)_{+}, 1), so that \{ \begin{matrix}σ(t) = 0 & if t ≤ 1/2, \\ σ(t) = 1 & if t ≥ 1, \\ 0 ≤ σ′ ≤ 2. \end{matrix}

We note that Xφ belongs to L_{comp}^{∞}(Ω), supp Xφ ⊂ supp φ and σ(ϵ^{−1}|x − K|) tends to 1 on the complement of K when ϵ goes to 0_{+}. Since the Lebesgue measure of K is zero, we have^{2}

(4.3) − ∫ v(X(φ) + φ div X)dm

= − lim_{ϵ→0_{+}} ∫ v(x)((Xφ)(x) + φ(x)(div X)(x))σ(ϵ^{−1}|x − K|)dx.

Setting for ϵ > 0, ω_{ϵ}(x) = σ(ϵ^{−1}|x − K|), we note that the function ω_{ϵ} is Lipschitz continuous and

supp ω_{ϵ} ⊂ {|x − K| ≥ ϵ/2} ⊂ K^{c} = F^{c} ∪ (supp φ)^{c} = Ω^{c} ∪ Ω_{0} ∪ (supp φ)^{c}

and thus

(4.4)  supp ω_{ϵ} ∩ supp φ ⊂ Ω_{0}.

2In the sequel, we shall use the notation dx for the Lebesgue measure when the variable x appears in the integrand.

We obtain from (4.3), with T = ∇_{x}(|x − K|) (note that ∥T ∥_{L^{∞}} ≤ 1), integrating by parts as we may on the open set Ω_{0} (that is using the assumption of the lemma),

(4.5) − ∫ v(X(φ) + φ div X)dm

= lim_{ϵ→0_{+}} {∫ σ(ϵ^{−1}|x − K|)fφdx + ∫ φvσ′(ϵ^{−1}|x − K|)⟨T (x) · X(x)⟩ϵ^{−1}dx} .

We have

lim sup_{ϵ→0_{+}} ∫ |φvσ′(ϵ^{−1}|x − K|)⟨T (x) · X(x)⟩| ϵ^{−1}dm

≤ 2 ∥vφX∥_{L^{∞}} lim sup_{ϵ→0_{+}} ϵ^{−1}L^{d}(K + ϵB_{1}) = 0,

since our assumption is precisely ϵ^{−1}L^{d}(K + ϵB_{1}) = α(ϵ) → 0 with ϵ. Thus (4.5) implies

− ∫ v(X(φ) + φ div X)dm = ∫ fφdm, q.e.d.

Let us now make the less stringent^{3} assumption H^{d−1}(F) = 0. Let φ be a test function ∈ C_{c}^{1}(Ω) and consider as before the compact set K = F ∩ supp φ ⊂ Ω. The fact that H^{d−1}(K) = 0 means that for any ϵ > 0, there exist δ_{ϵ} > 0 such that, for any δ ∈]0, δ_{ϵ}], there exist a sequence of

(4.6)  open sets S_{j} with diameter ≤ δ such that S_{j} ∩ K ≠ ∅

and

(4.7)  K ⊂ ∪_{j∈ℕ}S_{j}, ∑_{j∈ℕ} (diamS_{j})^{d−1} ≤ ϵ.

In particular, one can assume that δ ∈]0, ϵ]. Let us choose from now on some δ ∈]0, ϵ]. Since K is compact, we can assume that K ⊂ ∪_{j∈J}S_{j} with a finite set of indices J. Setting δ_{j} = diamS_{j} (note that 0 < δ_{j} ≤ δ since S_{j} is a non-empty open set), we consider

ϕ_{j} ∈ C_{c}^{∞}(S_{j} + 2δ_{j}B_{1}; [0, 1])

a smooth function equal to 1 on \overline{S_{j}} +δ_{j}B_{1} with ∥∇ϕ_{j}∥_{L^{∞}} ≤ δ_{j}^{−1}C(d), where C(d) depends only on the dimension. We consider now the Lipschitz continuous function

χ = σ(∑_{j∈J} ϕ_{j}), where σ is defined in (4.2).

3We are indebted to Giovanni Alberti for this improvement.

For x ∈ K + (min_{j∈J} δ_{j})B_{1}, there exists at least an index j ∈ J such that x ∈ S_{j} + δ_{j}B_{1} and thus ϕ_{j}(x) = 1. Consequently, the function χ is equal to 1 on K + (min_{j} δ_{j})B_{1} and is supported in the set

{∑_{j∈J} ϕ_{j} ≥ 1/2} ⊂ ∪_{j∈J} supp ϕ_{j} ⊂ ∪_{j∈J}(S_{j} + 2δ_{j}B_{1}) ⊂ K + 3δB_{1},

where the last inclusion is due to the assumption (4.6). Moreover, the gradient of χ satisfies

|∇χ| = |σ′(∑_{j∈J} ϕ_{j}) ∑_{j∈J} ∇ϕ_{j}| ≤ 2C(d) ∑_{j∈J} δ_{j}^{−1}1_{S_{j}+2δ_{j}B_{1}}

which implies the following estimate for its integral

∫_{ℝ^{d}} |∇χ(x)|dx ≤ 2^{1−d}C(d)L^{d}(B_{1}) ∑_{j∈J} δ_{j}^{−1}(diamS_{j} + 4δ_{j})^{d}

= C_{1}(d) ∑_{j∈J} δ_{j}^{d−1} ≤ C_{1}(d)ϵ,

where C_{1}(d) depends only on the dimension. Eventually, for any ϵ > 0, we were able to construct a Lipschitz continuous function χ_{ϵ}, valued in [0,1], supported in K + 3ϵB_{1}, equal to 1 in K + ρ_{ϵ}B_{1} with some ρ_{ϵ} ∈]0, ϵ] such that

(4.8)  ∫ |∇χ_{ϵ}(x)|dx ≤ C_{1}(d)ϵ.

In particular, since L^{d}(K) = 0, we obtain, with

(4.9)  ω_{ϵ} = 1 − χ_{ϵ},

(4.10)  lim_{ϵ→0_{+}} ω_{ϵ}(x) = 1, L^{d} − a.e. and ∥ω_{ϵ}∥_{L^{∞}} ≤ 1.

It is then easy to start over the computations in (4.2–5). We check, using (4.10) and vX(φ), φv ∈ L_{comp}^{∞}(Ω), vφ div X, fφ ∈ L^{1},

− ∫ v(X(φ) + φ div X)dm = − lim_{ϵ→0_{+}} ∫ ω_{ϵ}v(X(φ) + φ div X)dm

= lim_{ϵ→0_{+}} {∫ ω_{ϵ}fφdm + ∫ φv⟨∇ω_{ϵ}(x) · X(x)⟩dx}

= ∫ fφdm from (4.10) and the following consequence of (4.8)

(4.11)  lim sup_{ϵ→0_{+}} ∫ |φv⟨∇ω_{ϵ}(x) · X(x)⟩| dx ≤ ∥φvX∥_{L^{∞}} lim sup_{ϵ→0_{+}} ∫ |∇χ_{ϵ}(x)|dx = 0.

The proof of lemma 4.1 is complete.

Remark. This proof shows also that it would be possible to weaken the assumption H^{d−1}(Ω\Ω_{0}) = 0. As a matter of fact, a sufficient requirement on F concerns its socalled W^{1,1} capacity. We need only to assume that for all ϵ > 0, and all K compact subset of F, there exists a W^{1,1} function χ_{ϵ}, supported in Ω, valued in [0,1], such that

(4.12)  K ⊂ int({χ_{ϵ} = 1}), lim_{ϵ→0_{+}} χ_{ϵ}(x) = 0 L^{d} − a.e., lim_{ϵ→0_{+}} ∫ |∇χ_{ϵ}|dm = 0.

This assumption amounts essentially to require that the W^{1,1} capacity of Ω\Ω_{0} is 0. However, we shall stick on our hypothesis involving the (d − 1) Hausdorff measure since we believe that this condition is easier to understand and more explicit than (4.12). We can view the previous arguments as proofs of the implications^{4}

M^{d−1}(F) = 0 ⟹ H^{d−1}(F) = 0 ⟹ cap_{W^{1,1}}(F) = 0.

In fact, we shall use the following lemma, dealing with an inequality, whose proof is identical to lemma 4.1’s.

Lemma 4.2. Let Ω_{0} ⊂ Ω be open subsets of ℝ^{d} such that H^{d−1}(Ω\Ω_{0}) = 0. Let X be a vector field in L_{loc}^{∞}(Ω) such that div X ∈ L_{loc}^{1}(Ω), and let v ∈ L_{loc}^{∞}(Ω) be a (weak) solution on Ω_{0} of the inequality Xv ≤ f where f belongs to L_{loc}^{1}(Ω). It means that for all non-negative φ ∈ C_{c}^{1}(Ω_{0}),

(4.13)  ∫ fφdm + ∫ v(X(φ) + φ div X)dm ≥ 0.

Then the inequality Xv ≤ f is satisfied weakly on Ω, i.e. (4.13) is true for all φ ∈ C_{c}^{1}(Ω).

Proof. Let φ be a non-negative function ∈ C_{c}^{1}(Ω), and let ω_{ϵ} satisfying (4.9 − 10) with K = supp φ ∩ Ω_{0}^{c}. Following the same lines as in the previous proofs, we get

∫ fφdm + ∫ v(X(φ) + φ div X)dm = ∫ fφdm + lim_{ϵ→0_{+}} ∫ ω_{ϵ}v(X(φ) + φ div X)dm

= ∫ fφdm − lim_{ϵ→0_{+}} ∫ (X(ω_{ϵ})v + ω_{ϵ}X(v)) φdm

= lim_{ϵ→0_{+}} {∫ fω_{ϵ}φdm + ∫ v(X(ω_{ϵ}φ) + ω_{ϵ}φ div X)dm} ≥ 0. □

4We can note also that, for analytic sets (Suslin sets) we have the equivalence H^{d−1}(F) = 0 ⟺ cap_{W^{1,1}}(F) = 0 (see e.g. lemma 5.12.3 in [Zi]).

5. Commutation arguments

Preliminary remarks. When X ∈ W_{loc}^{1,1}(Ω), one can prove (see [DL]) a much stronger statement than (3.3). Let Ω be an open set of ℝ^{d}, X be a vector field on Ω, and S a C^{1} oriented hypersurface of Ω such that X ∈ L_{loc}^{∞}, div X ∈ L_{loc}^{1}, (div X)_{+} ∈ L_{loc}^{∞} and X is positively transverse to S. Let u be a L_{loc}^{∞} function such that, for some function c ∈ L_{loc}^{1} with c_{+} ∈ L_{loc}^{∞},

Xu = cu, supp u ⊂ S_{+}.

If X ∈ W_{loc}^{1,1}(Ω), then one can prove the strong convergence in L^{1} of θX(χu)_{ϵ} to θcu, where χ, θ are smooth compactly supported functions, χ = 1 on the support of θ, (χu)_{ϵ} is a regularization of χu by any standard mollifier. Here we took any smooth compactly supported function ρ with integral 1 and set

v_{ϵ}(x) = ∫ ϵ^{−n}ρ(\frac{x − y}{ϵ} )v(y)dy.

The proof amounts to the computation of the commutator

[X, ρ̂(ϵD)].

It fails even in the piecewise W^{1,1} case if a jump occurs on a curved hypersurface. In the latter case, one should use a pseudo-differential mollifier and not only a convolution operator, or equivalently, straighten first the jump hypersurface and after this use a convolution. However, one should be careful at choosing the various speeds: if the jump occurs on the hypersurface {x_{1} = 0} then it is natural to choose ϵ_{1} ≪ ϵ_{2}, …, ϵ_{d} if ϵ is a diagonal matrix. It is still not enough to handle the simplest BV example: we must pay more attention at choosing the ϵ′s.

Beginning of the proof of the estimate (3.3). In order to obtain the result in theorem 2.4, we need only to prove (3.3) on Ω_{0}, since we shall then get that

X(u^{2}) ≤ (C + c_{+})u^{2}

on Ω_{0} and thus, from lemma 4.2, the same inequality in Ω. Then u^{2} will satisfy the assumptions in lemma 3.1, which will give the result. Let us then consider a vector field on V open subset of Ω_{0}

X = ∑_{1≤j≤d} a_{j}(x)\frac{∂}{∂x_{j}}

and assume for 1 ≤ j ≤ d

(5.1)  \frac{∂a_{j}}{∂x_{1}} ∈ M(V) = D^{′(0)}(V), and for k ≥ 2, \frac{∂a_{j}}{∂x_{k}} ∈ L_{loc}^{1}(V), div X ∈ L_{loc}^{1}(V).

Let u be a L_{loc}^{∞}(V), c ∈ L_{loc}^{1}(V) such that Xu = cu. We consider a non-negative function θ ∈ C_{c}^{1}(V) and χ ∈ C_{c}^{1}(V; [0, 1]) such that supp θ ⊂ {χ = 1}. Let ρ be a smooth compactly supported function with integral 1, ϵ be a positive diagonal d × d matrix with ϵ_{2} = ⋯ = ϵ_{d}. We need to check (3.3), setting v = χu and v_{ϵ} = v ∗ ρ_{ϵ}

⟨X(u^{2}), θ⟩ = − ∫ u^{2}(X(θ) + θ div X)dm = − ∫ χu^{2}(X(θ) + θ div X)dm

= − lim_{ϵ→0_{+}} ∫ v_{ϵ} u(X(θ) + θ div X)dm = lim_{ϵ→0_{+}} ∫ uθXv_{ϵ}dm + ∫ cu^{2}θdm

≤ lim_{ϵ→0_{+}} ∫ uθXv_{ϵ}dm + ∫ u^{2}θc_{+}dm.

We define, (for x ∈ supp χ and ϵ small enough so that supp χ + ϵ supp ρ ⊂ V),

(5.2)  (R_{ϵ}v)(x) = ∑_{1≤j≤d} ∫ (v(x − ϵz) − v(x))ϵ_{j}^{−1}(a_{j}(x) − a_{j}(x − ϵz))(∂_{j}ρ)(z)dz

and we check easily that

(5.3) (X(v ∗ ρ_{ϵ}) − (Xv) ∗ ρ_{ϵ})(x) =

(R_{ϵ}v)(x) + \overbrace{∫ (div X)(x − ϵz)ρ(z)(v(x − ϵz) − v(x))dz}^{(T_{ϵ}v)(x)} .

Consequently, we have,

(5.4)  X(v_{ϵ}) = (Xv) ∗ ρ_{ϵ} + T_{ϵ}v + R_{ϵ}v.

From the equation Xv = cv + uX(χ), we get that Xv belongs to L^{1} and the strong convergence in L^{1} of θ(Xv ∗ ρ_{ϵ}) to cθu. The term T_{ϵ}v is also easy^{5} to handle since, using the notation τ_{t}w(x) = w(x − t), we have

(5.5) ∥θT_{ϵ}v∥_{L^{1}} ≤ ∫ ∥θ(τ_{ϵz} − Id)(div X)∥_{L^{1}} 2 ∥v∥_{L^{∞}} |ρ(z)|dz

+ ∫∫ |ρ(z)||v(x − ϵz) − v(x)||(div X)(x)|θ(x)dxdz.

Since div X belongs to L_{loc}^{1} and ρ is compactly supported, the first term in the right-hand side of the above inequality goes to 0 with ϵ. For the second term, we can use the assertion (5.6) of the following simple lemma.

5It is even trivial to see that T_{ϵ}v goes to zero strongly in L^{1} if we assume that div X is bounded since v belongs to L^{1}. However, we want to show that our weaker assumption (div X)_{+} ∈ L_{loc}^{∞} and div X ∈ L_{loc}^{1} is enough.

F.Colombini, N.Lerner 17

Lemma 5.1. Let b ∈ L^{1}, a ∈ BV and v ∈ L^{∞}. Then, we have

(5.6) lim_{t→0} ∫ |b(x)||v(x + t)−v(x)|dx = 0 and
(5.7)  ∫ |a(x + t) − a(x)||v(x)|dx ≤ ∥∇a∥_{M_{b}} ∥v∥_{L^{∞}} |t|.

Proof: see the appendix A6. We get then

⟨X(u^{2}), θ⟩ ≤ lim_{ϵ→0_{+}} ∫ uθR_{ϵ}vdm + 2 ∫ u^{2}θc_{+}dm.

The key argument. We need to check the quantity

(5.8)  ∫ uθR_{ϵ}vdm.

Looking at (5.2), we calculate, with a sequence u_{ν} of continuous functions converging a.e. on supp χ to u with ∥u_{ν}∥_{L^{∞}(supp χ)} ≤ ∥u∥_{L^{∞}(supp χ)} and setting v_{ν} = χu_{ν} (5.9) ∫ u(x)θ(x)(R_{ϵ}v)(x)dx

= lim_{ν} ∑_{1≤j≤d} ∫∫ u_{ν}(x)θ(x)(v_{ν}(x − ϵz) − v_{ν}(x))ϵ_{j}^{−1}(a_{j}(x) − a_{j}(x − ϵz))(∂_{j}ρ)(z)dzdx.

Setting (when \frac{∂a_{j}}{∂x_{k}} is a Radon measure, the integral stands for a bracket of duality)

(5.10) (R_{ϵ}v_{ν})(x) = ∑_{1≤j,k≤d} ∫_{0}^{1}∫ (v_{ν}(x − ϵz) − v_{ν}(x))ϵ_{j}^{−1}ϵ_{k} \frac{∂a_{j}}{∂x_{k}} (x − sϵz)z_{k}(∂_{j}ρ)(z)dsdz,

and using the fact that ϵ is a positive diagonal matrix with

(5.11)  0 < ϵ_{1} ≤ ϵ_{2} = ⋯ = ϵ_{d},

we get

(R_{ϵ}v_{ν})(x) =

∫_{0}^{1}∫ (v_{ν}(x − ϵz) − v_{ν}(x))\frac{∂a_{1}}{∂x_{1}} (x − sϵz)z_{1}(∂_{1}ρ)(z)dsdz (5.12)

+ ∑_{2≤k≤d} ∫_{0}^{1}∫ (v_{ν}(x − ϵz) − v_{ν}(x))ϵ_{1}^{−1}ϵ_{2} \frac{∂a_{1}}{∂x_{k}} (x − sϵz)z_{k}(∂_{1}ρ)(z)dsdz (5.13)

+ ∑_{2≤j≤d} ∫_{0}^{1}∫ (v_{ν}(x − ϵz) − v_{ν}(x))ϵ_{2}^{−1}ϵ_{1} \frac{∂a_{j}}{∂x_{1}} (x − sϵz)z_{1}(∂_{j}ρ)(z)dsdz (5.14)

+ ∑_{2≤j,k≤d} ∫_{0}^{1}∫ (v_{ν}(x − ϵz) − v_{ν}(x))\frac{∂a_{j}}{∂x_{k}} (x − sϵz)z_{k}(∂_{j}ρ)(z)dsdz. (5.15)

Note that from div X ∈ L_{loc}^{1}, and (5.1) we get \frac{∂a_{1}}{∂x_{1}} ∈ L_{loc}^{1} so that

(5.16)  \frac{∂a_{1}}{∂x_{1}} , \frac{∂a_{1}}{∂x_{k}} for k ≥ 2 and \frac{∂a_{j}}{∂x_{k}} for k ≥ 2 are in L_{loc}^{1}.

This implies that the limit with ν → +∞ of the terms (5.12), (5.13) and (5.15) are respectively (a.e. in x)

∫_{0}^{1}∫ (v(x − ϵz) − v(x))\frac{∂a_{1}}{∂x_{1}} (x − sϵz)z_{1}(∂_{1}ρ)(z)dsdz,

∑_{2≤k≤d} ∫_{0}^{1}∫ (v(x − ϵz) − v(x))ϵ_{1}^{−1}ϵ_{2} \frac{∂a_{1}}{∂x_{k}} (x − sϵz)z_{k}(∂_{1}ρ)(z)dsdz,

∑_{2≤j,k≤d} ∫_{0}^{1}∫ (v(x − ϵz) − v(x))\frac{∂a_{j}}{∂x_{k}} (x − sϵz)z_{k}(∂_{j}ρ)(z)dsdz.

with domination by L^{1} functions of the variable x independent of the index ν. Consequently, we obtain from (5.9),

∫ u(x)θ(x)(R_{ϵ}v)(x)dx =

∫_{0}^{1}∫∫ u(x)θ(x)(v(x − ϵz) − v(x))\frac{∂a_{1}}{∂x_{1}} (x − sϵz)z_{1}(∂_{1}ρ)(z)dsdzdx,

+ ∑_{2≤k≤d} ∫_{0}^{1}∫∫ u(x)θ(x)(v(x − ϵz) − v(x))ϵ_{1}^{−1}ϵ_{2} \frac{∂a_{1}}{∂x_{k}} (x − sϵz)z_{k}(∂_{1}ρ)(z)dsdzdx,

+ lim_{ν} ∑_{2≤j≤d} ∫_{0}^{1}∫∫ u_{ν}(x)θ(x)(v_{ν}(x − ϵz) − v_{ν}(x))ϵ_{2}^{−1}ϵ_{1} \frac{∂a_{j}}{∂x_{1}} (x − sϵz)z_{1}(∂_{j}ρ)(z)dsdzdx,

+ ∑_{2≤j,k≤d} ∫_{0}^{1}∫∫ u(x)θ(x)(v(x − ϵz) − v(x))\frac{∂a_{j}}{∂x_{k}} (x − sϵz)z_{k}(∂_{j}ρ)(z)dsdzdx.

Thus, using lemma 5.1, we get

(5.17) |∫ u(x)θ(x)(R_{ϵ}v)(x)dx| ≤ σ_{11}(ϵ_{1}, ϵ_{2}) + ∑_{2≤k≤d} σ_{1k}(ϵ_{1}, ϵ_{2})\frac{ϵ_{2}}{ϵ_{1}} + ∑_{2≤j,k≤d} σ_{jk}(ϵ_{1}, ϵ_{2})

+ \frac{ϵ_{1}}{ϵ_{2}} 2 ∥u^{2}θ∥_{L^{∞}} ∑_{2≤j≤d} ∥\frac{∂a_{j}}{∂x_{1}} ∥_{M_{b}(supp χ)} ∫ |z_{1}(∂_{j}ρ)(z)|dz,

where the functions σ_{11}, σ_{1k}, σ_{jk} tend to 0 with ϵ. We infer from (5.17) that there exists a constant C_{1} and a function σ such that

|∫ u(x)θ(x)(R_{ϵ}v)(x)dx| ≤ σ(ϵ_{1}, ϵ_{2}) + σ(ϵ_{1}, ϵ_{2})\frac{ϵ_{2}}{ϵ_{1}} + C_{1} \frac{ϵ_{1}}{ϵ_{2}} , (5.18)

(5.19) where 0 = lim_{ϵ→0} σ(ϵ_{1}, ϵ_{2}).

We set

(5.20)  β(ϵ_{2}) = sup_{0<ϵ_{1}≤ϵ_{2}} σ(ϵ_{1}, ϵ_{2}),

we note that lim_{ϵ_{2}→0} β(ϵ_{2}) = 0 and we choose in (5.18)

(5.21)  ϵ_{1} = β(ϵ_{2})^{1/2}ϵ_{2},

which is ≤ ϵ_{2}, for ϵ_{2} small enough, obtaining

(5.22)  |∫ u(x)θ(x)(R_{ϵ}v)(x)dx| ≤ β(ϵ_{2}) + β(ϵ_{2})^{1/2} + C_{1}β(ϵ_{2})^{1/2} ⟶_{ϵ_{2}→0} 0.

We obtain

⟨X(u^{2}), θ⟩ ≤ 2 ∫ u^{2}θc_{+}dm,

that is the inequality

X(u^{2}) ≤ 2c_{+}u^{2}

is satisfied on Ω_{0}. Following the previous remarks, this completes the proof of theorem 2.2.

Concluding remarks. Note that the choice of ϵ_{1}, ϵ_{2} depends on the geometry: the condition (5.21) implies ϵ_{1} ≪ ϵ_{2}, a condition forced by the term ϵ_{1}/ϵ_{2} in the right-hand-side of (5.18). But this natural geometric condition is not enough to handle that matter: the choice of ϵ_{1}, ϵ_{2} depends also on the function u under scope, since in (5.20-21), the functions σ and β depend on u.

6. Appendix

A1. Three simple examples with pictures. We give in this section three simple examples, demonstrating that the condition (div X)_{+} ∈ L_{loc}^{∞} is necessary for the forward uniqueness property.

Example 6.1. Let T_{1} = sign x\frac{∂}{∂x} + \frac{∂}{∂y} be a vector field on ℝ^{2}. The vector field T_{1} belongs to L^{∞} ∩ BV, is tranverse to the hypersurface S = {y = 0} since T_{1}(y) = 1, and fails to have the uniqueness property across S since

T_{1}((y − |x|)_{+})= 0, and (y − |x|)_{+}_{|y<0} = 0.

Note that div T_{1} = 2δ(x) which is non-negative and not in L^{∞}. The next example shows that the negative part of the divergence is unimportant for uniqueness.

Example 6.2. Let T_{2} = − sign x\frac{∂}{∂x} + \frac{∂}{∂y} be a vector field on ℝ^{2}. The vector field T_{2} belongs to L^{∞} ∩ BV, is tranverse to the hypersurface S = {y = 0} since T_{2}(y) = 1, and has the uniqueness property across S from theorem 2.4 since (div T_{2})_{+} = 0.

Example 6.3. Let T_{3} = x ln^{2} |x|\frac{∂}{∂x} + \frac{∂}{∂y} be a vector field on ℝ^{2}. The vector field T_{3} belongs to ∩_{1≤p<∞}W_{loc}^{1,p}, is tranverse to the hypersurface S = {y = 0} since T_{3}(y) = 1, and fails to have the uniqueness property across S since, on the open set {|x| < 1},

T_{3}((y − \frac{1}{| ln |x||})_{+}) = 0 and (y − \frac{1}{| ln |x||})_{+}_{|y<0} = 0.

Note that div T_{3} = ln^{2} |x| + 2 ln |x| is non-negative near the origin and not bounded. The behaviour of the T_{j} is also apparent on the following pictures.

Integral curves of T_{1}, T_{2}, T_{3}

The integral curves of T_{1} (resp.T_{3}) starting from {y < 0} cannot penetrate the shaded region where {y > |x|} (resp. {y > 1/| ln |x||}). On the other hand, the integral curves of T_{2} starting from {y < 0} fill a neighborhood of the origin.

A2. Invariance properties. Let us first check that the assumptions (1.1) and X ∈ BV_{loc} are invariant by a C^{1,1} diffeomorphism. The boundedness and transversality properties in (1.1) are already invariantly stated. Moreover the BV regularity for a vector field makes sense on a C^{1,1} manifold since with x = κ(y), where κ is a local C^{1,1} diffeomorphism,

X = ∑_{j} a_{j} \frac{∂}{∂x_{j}} = ∑_{k,j} a_{j} \frac{∂y_{k}}{∂x_{j}} \frac{∂}{∂y_{k}}

and

\frac{∂}{∂y_{l}} (a_{j}(κ(y))\frac{∂y_{k}}{∂x_{j}} (κ(y))) = ∑_{i} \overbrace{\frac{∂a_{j}}{∂x_{i}}}^{∈M} \overbrace{\frac{∂x_{i}}{∂y_{l}} \frac{∂y_{k}}{∂x_{j}}}^{∈C^{0,1}} + \overbrace{a_{j}}^{∈L^{1}} \overbrace{\frac{∂^{2}y_{k}}{∂x_{j}∂x_{i}} \frac{∂x_{i}}{∂y_{l}}}^{∈L^{∞}}

is indeed a Radon measure. The divergence is invariantly expressed in (1.4), but we can note directly that for ω a non vanishing Lipschitz continuous function, the divergence of the vector field X with respect to the n-form ω(x)dx_{1} ∧ ⋯ ∧ dx_{n} is

div X = ω^{−1} ∑_{j} \frac{∂(a_{j}ω)}{∂x_{j}} = ∑_{j} \frac{∂a_{j}}{∂x_{j}} + X(ln |ω|).

Since X is also in L_{loc}^{∞}, the divergence of X is a priori a Radon measure and the condition div X ∈ L_{loc}^{∞} (resp. (div X)_{+} ∈ L_{loc}^{∞}) is simply ∑_{j} ∂_{x_{j}}(a_{j}) ∈ L_{loc}^{∞} (resp. (∑_{j})_{+} ∈ L_{loc}^{∞}), a condition that can be easily checked as above in another chart of C^{1,1} coordinates. Note that, on a C^{1,1} manifold, the regularity of the tangent bundle and of the bundle of n-forms is Lipschitz continuity.

The condition X ∈ B(Ω) is also invariant by change of C^{1,1} coordinates from the previous discussion and the fact that it is also the case for the condition on the Hausdorff dimension.

Remark. It is tempting to formulate analogous conditions on a Lipschitz manifold, where, for a vector field X, the conditions X ∈ L_{loc}^{∞}, X positively transverse to a Lipschitz hypersurface S make sense. The vanishing of the Hausdorff measure is also invariant by bi-Lipschitz homeomorphism. However, the divergence condition does not have a simple expression (if any) in that framework, since the regularity of the function ω above could only be boundedness and measurability (for instance it could be the determinant of the Jacobian of a bi-Lipschitz homeomorphism). Moreover the notion of BV_{loc} regularity for a vector field on a Lipschitz manifold should be handled with caution since the change of coordinates formulas written above do not obviously make sense, since it is not possible to make the product of an L^{∞} function with a Radon measure. Nevertheless, the Lipschitz framework would be certainly better, and in particular would allow a good approximation of the jump set of BV functions.

A3. The class B contains the piecewise W^{1,1} functions. Let Ω be an open set of ℝ^{d}. Let us recall^{6} the definition of the class P(Ω) = L_{loc}^{∞}(Ω)∩ piecewise W^{1,1} (see page 836 of [Li]). A function a ∈ L_{loc}^{∞}(Ω) is said to belong to P(Ω) if there exists a partition of Ω, Σ_{d}, Σ_{d−1}, …, Σ_{0}, Σ_{−1} such that

6In fact the points (i)—(iv) are consequences of the definition in [Li]. Anyhow our purpose is to prove that piecewise W^{1,1} functions are indeed in our class B.

(i) for all 0 ≤ k ≤ d, ∪_{d−k≤i≤d}Σ_{i} are open sets and a ∈ W^{1,1}(Σ_{d}), (ii) for all 0 ≤ k ≤ d − 2, for all x ∈ Σ_{k}, there exists a neighborhood U of x and a

bi-Lipschitz continuous homeomorphism

\begin{aligned}κ & : & U & → & B_{1} \\ & & y & → & (κ_{1}(y), …, κ_{d}(y)) \end{aligned}

such that Σ_{k} ∩ U = {y ∈ U| ∀j, 1 ≤ j ≤ d − k, κ_{j}(y) = 0}, (iii) for all x ∈ Σ_{d−1}, there exists a neighborhood U of x and a C^{1,1} diffeomorphism

\begin{aligned}κ & : & U & → & B_{1} \\ & & y & ↦ & (κ_{1}(y), …, κ_{d}(y)) \end{aligned}

such that

Σ_{d−1} ∩ U = {y ∈ U| κ_{1}(y) = 0}.

(iv) The points of Σ_{−1} are isolated.

Proposition 6.4. Let Ω be an open set of ℝ^{d} and P(Ω) be the set of piecewise W^{1,1} functions described above. Then P(Ω) ⊂ B(Ω). In particular for all a ∈ P(Ω), there exists an open subset Ω_{0} of Ω such that

H^{d−1}(Ω\Ω_{0}) = 0, and a ∈ CBV_{loc}(Ω_{0}).

Note that the set Ω_{0} may depend on the function a. Proof. Note that if κ : U ⟶ V is a bi-Lipschitz homeomorphism of open sets of ℝ^{d}, setting ν = κ^{−1}, we obtain by regularization

det κ′(x) det ν′(κ(x)) = 1,

so that since essup | det κ′(x)| is finite, we get also that essinf | det ν′(κ(x))| is positive. We remark first that M^{d−1}(Σ_{d−2}) = 0. In fact L^{d}(Σ_{d−2} + rB_{1}) = O(r^{2}) since for any point in Σ_{d−2} there exists a neighborhood V and local Lipschitz coordinates z such that

V ∩ (Σ_{d−2} + rB_{1}) ⊂ {z = (z_{1}, z_{2}, z′′) ∈ ℝ × ℝ × ℝ^{d−2}, |z_{1}| ≤ r, |z_{2}| ≤ r, |z′′| ≤ M_{0}}. The same argument can be applied for k ∈ {2, …, d} and with (iv), we get

∀k ∈ {2, …, d, d + 1}, M^{d−1}(Σ_{d−k}) = 0.

Consequently, we have indeed

Ω_{0} = \overbrace{Σ_{d} ∪ Σ_{d−1}}^{open set} ⊂ Ω, with H^{d−1}(Ω\Ω_{0}) = 0,

and Σ_{d−1} is a C^{1,1} hypersurface of Ω_{0}. Since a ∈ W^{1,1}(Σ_{d}), we get that a belongs to conormalBV_{loc}(Ω_{0}) from the following simple lemma.

Lemma 6.5. Let R_{0} > r_{0} > 0 be given positive numbers and d be an integer ≥ 2. We define

V_{+} = {x = (x_{1}, x′) ∈ ℝ × ℝ^{d−1}, 0 < x_{1} < r_{0}, |x′| < R_{0}},
V_{−} = {x = (x_{1}, x′) ∈ ℝ × ℝ^{d−1}, −r_{0} < x_{1} < 0, |x′| < R_{0}},
V = {x = (x_{1}, x′) ∈ ℝ × ℝ^{d−1}, −r_{0} < x_{1} < r_{0}, |x′| < R_{0}}.

Let a be an L^{∞}(V) function such that a ∈ W^{1,1}(V_{+}) ∩ W^{1,1}(V_{−}). Then a belongs to BV (V) and more precisely^{7}

\frac{∂a}{∂x_{1}} ∈ M_{b}(V), ∇_{x′}a ∈ L^{1}(V).

Proof. Let φ be a test function in C_{c}^{1}(V). With brackets of duality, ω standing for a C^{1} function of one real variable equal to 1 outside a neighborhood of the origin, setting

b_{±} = \frac{∂a}{∂x_{1}} |_{V_{±}} ∈ L^{1}(V_{±}),

we have,

⟨\frac{∂a}{∂x_{1}} , φ⟩ = − ∫∫ \frac{∂φ}{∂x_{1}} (x_{1}, x′)a(x_{1}, x′)dx_{1}dx′

= − lim_{ϵ→0} ∫∫ \frac{∂φ}{∂x_{1}} (x_{1}, x′)a(x_{1}, x′)ω(x_{1}ϵ^{−1})dx_{1}dx′

= lim_{ϵ→0} \{∫∫ φ(x_{1}, x′)[\frac{∂a}{∂x_{1}} (x_{1}, x′)ω(x_{1}ϵ^{−1}) + a(x_{1}, x′)ϵ^{−1}ω′(x_{1}ϵ^{−1})]dx_{1}dx′\}

= ∫∫ φ(x_{1}, x′)[b_{+}(x_{1}, x′) + b_{−}(x_{1}, x′)]dx_{1}dx′

+ lim_{ϵ→0} ∫∫ φ(ϵx_{1}, x′)a(ϵx_{1}, x′)ω′(x_{1})dx_{1}dx′.

= ⟨b_{+} + b_{−}, φ⟩ + lim_{ϵ→0} ∫ α(ϵx_{1})ω′(x_{1})dx_{1},

where α is the L_{comp}^{∞}(ℝ) function α(t) = ∫ φ(t, x′)a(t, x′)dx′. From the inequality

∥α∥_{L^{∞}(ℝ)} ≤ ∥φ∥_{L^{∞}(V)} sup_{|t|<r_{0}} ∫_{|x′|<R_{0}} |a(t, x′)|dx′,

7Note that the assumptions are “up to the boundary” and that the hypothesis a ∈ W_{loc}^{1,1}(V_{+}) ∩ W_{loc}^{1,1}(V_{−}) is not sufficient to get the conclusion as shown by the following function u ∈ C^{0}(ℝ) ∩ C^{∞}(ℝ^{∗})

given by u(x) = x sin(1/x) whose distribution derivative is not a Radon measure.

we get that

|lim_{ϵ→0} ∫ α(ϵx_{1})ω′(x_{1})dx_{1}| ≤ C ∫ |ω′(x_{1})|dx_{1} ∥φ∥_{L^{∞}(V)} ,

which implies that ∂a/∂x_{1} is a Radon measure on V. Finally, we need to check the x′ derivatives ; setting

c_{±} = ∇_{x′}a|_{V_{±}} ∈ L^{1}(V_{±}),

we write with the same notations as previously,

⟨∇_{x′}a, φ⟩ = − ∫∫ a∇_{x′}φdx_{1}dx′

= − lim_{ϵ→0} ∫∫ a(x_{1}, , x′)(∇_{x′}φ)(x_{1}, , x′)ω(x_{1}ϵ^{−1})dx_{1}dx′

= lim_{ϵ→0} ∫∫ φ(x_{1}, x′)∇_{x′}a(x_{1}, x′)ω(x_{1}ϵ^{−1})dx_{1}dx′

= ∫∫ (c_{+} + c_{−})φdx_{1}dx′

concluding the proof of the lemma. □

A4. A picture. We first describe the singular set of the two-dimensional example 2.3.(d)

Singular set of a in example 2.3.d

The points (1) are regular points (say W^{1,1} points). The points (2) are the jump points: in the picture above, the foliation is horizontal at (2′), vertical at (2′′). The points (3’) (resp. (3”)) are accumulation points, but in Ω_{0} with horizontal (resp. vertical) foliation. The open set Ω_{0} is the reunion of points (1),(2),(3). The other points (4),(5),(6) are the compact set Ω\Ω_{0} whose H^{1} measure is zero.

A vector field X in ℝ_{x,y}^{2} of the type defined in example (f) is transverse to the hypersurface {y = 0}, and has jumps across the hypersurfaces

Σ_{k} = {x = x_{k}}, S_{l} = {y = y_{l}}

where (x_{k}) and (y_{l}) are sequences with limit 0. The divergence condition forces the jump to occur in the tangential part to the foliation. The foliation has not to be defined at the intersection of the singular hypersurfaces, is vertical near the vertical component and horizontal near the horizontal component. That type of example is not piecewise W^{1,1}.

A5. Proof of lemma 3.1. Let us consider a point x_{0} ∈ S and φ a defining function for S in a neighborhood of x_{0} (i.e. S ∩ V_{0} = {x ∈ V_{0}, φ(x) = 0}). We know that, on an open neighborhood V_{0} of x_{0}, with w_{0} = w_{|V_{0}} ≥ 0, w_{0} ∈ L^{∞}(V_{0}), we have

(6.1)  Xw_{0} ≤ cw_{0}, supp w_{0} ⊂ {φ ≥ 0}, Xφ ≥ ρ_{0} > 0.

Let us consider the following Lipschitz continuous function defined on V_{0}

(6.2)  ψ(x) = φ(x) + |x − x_{0}|^{2}, θ(ψ(x)) = \frac{1}{2}((α^{2} − ψ(x))_{+})^{2},

where α is a positive parameter such that the closed ball B(x_{0}, α) with center x_{0} and radius α is included in V_{0}. We have

supp (θ(ψ)) ⊂ {ψ ≤ α^{2}}

and

supp(w_{0}θ(ψ)) ⊂ {φ ≥ 0} ∩ {ψ ≤ α^{2}} = K_{α} ∋ x_{0}

which is a compact subset of V_{0} (as a closed subset of B(x_{0}, α)). Let χ ∈ C_{c}^{∞}(V_{0}; [0, 1]), χ = 1 on a neighborhood of K_{α}. Since ψ and θ(ψ) are Lipschitz continuous functions, and X(χ) = 0 on a neighborhood of supp w_{0}θ(ψ), we have

∑_{1≤j≤n} a_{j}w_{0}∂_{j}(θ(ψ)χ) = w_{0}χθ′(ψ)X(ψ) + \overbrace{w_{0}θ(ψ)X(χ)}^{=0} .

We calculate, dm standing for the Lebesgue measure,

∫ cw_{0}θ(ψ)χdm ≥ ⟨Xw_{0}, \underbrace{θ(ψ)χ}_{≥0}⟩_{D^{′(1)}(V_{0}),C_{c}^{1}(V_{0})}

= − ∑_{1≤j≤n} ∫ a_{j}w_{0}∂_{j}(θ(ψ)χ)dm − ∫ w_{0}θ(ψ)χdivXdm

= ∑_{1≤j≤n} ∫ χw_{0}a_{j}∂_{j}(ψ)(α^{2} − ψ)_{+}dm − ∫ w_{0}θ(ψ)χdivXdm.

We obtain

(6.3)  0 ≥ ∫ χw_{0}(α^{2} − ψ)_{+}[X(ψ) − \frac{1}{2}(α^{2} − ψ)_{+}(divX + c)]dm.

Now on the set

{x ∈ V_{0}, φ(x) + |x − x_{0}|^{2} = ψ(x) ≤ α^{2}} ∩ {x, φ(x) ≥ 0} ⊂ B(x_{0}, α)

we have, using now (6.1–2) and the assumption of the lemma, that

X(ψ) − \frac{1}{2}(α^{2} − ψ)_{+}(div X + c) ≥

ρ_{0} − 2 ∥X∥_{L^{∞}(B(x_{0},α))} α − \frac{1}{2}α^{2} ∥(divX + c)_{+}∥_{L^{∞}(B(x_{0},α))} ≥ ρ_{0}/2

if α is chosen small enough with respect to ρ_{0} and ∥X∥_{L^{∞}(V_{0})}. On the other hand, the term

∫ χw_{0}((α^{2} − ψ)_{+})^{2}(div X + c)_{−}dm

makes sense and is non-negative. This yields

0 ≥ ∫ χw_{0}(α^{2} − ψ)_{+}dm,

and since the integrand is non-negative we get χw_{0}(α^{2}−ψ)_{+} = 0. Since on a neighborhood of x_{0}, we have χ = 1 and α^{2} − ψ > 0 , we indeed obtain that w_{0} vanishes near x_{0}. The proof of lemma 3.1 is complete. A6. Proof of lemma 5.1. To prove (5.6), we note that for all κ > 0, R > 0 we have

lim sup_{t→0} ∫ |b(x)||v(x + t) − v(x)|dx

= lim sup_{t→0} {∫_{|v(x+t)−v(x)|≤κ} |b(x)||v(x + t) − v(x)|dx + ∫_{|v(x+t)−v(x)|>κ} |b(x)||v(x + t) − v(x)|dx}

≤ κ ∥b∥_{L^{1}} + 2 ∥v∥_{L^{∞}} lim sup_{t→0} ∫_{{|v(x+t)−v(x)|>κ \atop |x|≤R}}|b(x)|dx + 2 ∥v∥_{L^{∞}} ∫_{|x|>R} |b(x)|dx

= κ ∥b∥_{L^{1}} + 2 ∥v∥_{L^{∞}} ∫_{|x|>R} |b(x)|dx

since L^{d}(A = {x, |x| ≤ R, and |v(x + t) − v(x)| > κ}) → 0 with t: in fact we have the estimates

L^{d}(A) ≤ κ^{−1} ∫_{|x|≤R,|x+t|≤R} |v(x + t) − v(x)|dx + L^{d}({x, |x| ≤ R, |x + t| > R})

≤ κ^{−1} ∥τ_{−t}v_{R} − v_{R}∥_{L^{1}} + |t|R^{d−1}|S^{d−1}|,

with v_{R}(x) = v(x)1(|x| ≤ R) which is an L^{1} function. The assertion (5.7) is an immediate consequence of a.e. convergence of C_{c}^{0} functions to v with L^{∞} bound ∥v∥_{L^{∞}}. □

References

[Ai] M.Aizenman, On vector fields as generators of flows: a counterexample to Nelson’s conjecture, Ann. Math. 107 (1978), 287–296.

[BC] H.Bahouri, J.-Y.Chemin, Équations de transport relatives à des champs de vecteurs non-lip-schitziens et mécanique des fluides., Arch. Rational Mech. Anal. 127 (1994), 159–181.

[Bo] F.Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal. 157 (2001), 75-90.

[BD] F.Bouchut, L.Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients, Diff. and Int. Eq. 14 (2001), 1015-1024.

[BJ] F.Bouchut, F.James, One dimensional transport equations with discontinuous coefficients, Non linear analysis 32 (1998), 891–933.

[ChL] J.-Y.Chemin, N.Lerner, Flot de champ de vecteurs non lipschitziens et équations de Navier-Stokes, J.Differ. Eq. 121 (1995), 314–328.

[CoL] F.Colombini, N.Lerner, Uniqueness of continuous solutions for BV vector fields, Duke Math.J. 111 (2002), 357–384.

[De1] B.Desjardins, A few remarks on ordinary differential equations, Comm.PDE 21 (1996), 1667– 1703.

[De2] B.Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Diff. and Int. Eq. 10 (1997), 577–586.

[De3] B.Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space, Diff. and Int. Eq. 10 (1997), 587–598.

[DL] R.J. DiPerna, P.-L.Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511–547.

[Fe] H.Federer, Geometric measure theory, Grund. der math. Wiss., vol. 153, Springer-Verlag, 1969.

[Fl] T.M.Flett, Differential analysis, Cambridge Univ. Press, 1980. [Hö] L.Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques et appli-

cations, vol. 26, Springer-Verlag, 1996.

[Li] P.-L.Lions, Sur les équations différentielles ordinaires et les équations de transport, C.R. Acad.Sc. Paris, Série I, 326 (1998), 833–838.

[PP] G.Petrova, B.Popov, Linear transport equations with discontinuous coefficients, Comm.PDE 24 (1999), 1849–1873.

[PR] F.Poupaud, M.Rascle, Measure solutions to the linear multidimensional transport equation with non-smooth coefficients, Comm.PDE 22 (1997), 337–358.

[Sa] X.Saint Raymond, L’unicité pour les problèmes de Cauchy linéaires du premier ordre, Enseign. Math. (2), 32 (1986), 1–55.

[Tr] F.Treves, Topological vector spaces, distributions and kernels, Pure & Appl. Math.Ser., Academic Press, 1967.

[Vo] A.I.Vol’pert, The space BV and quasi-linear equations, Math.USSR Sbornik 2 (1967), 225–267.

[Zi] W.P.Ziemer, Weakly differentiable functions, Graduate texts in mathematics, vol. 120, Springer-Verlag, 1989.

Dipartimento di Matematica, Università di Pisa, Via F.Buonarroti 2, 56127 Pisa, Italia

   E-mail address: colombin@dm.unipi.it

Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France

   E-mail address: lerner@univ-rennes1.fr
   Web-page: http://www.maths.univ-rennes1.fr/~lerner