The Chain Rule for the Divergence of BV –like Vector Fields Camillo De Lellis Abstract. We illustrate an approach to a conjecture of Bressan concerning compactness of solutions to ordinary differential equa- tions. This approach combines classical results in the theory of BV functions with recent developments in the theory of renormalized solutions to transport equations. Moreover it leads naturally to some new questions in the theory of BV functions, which turn out to be connected with regularity issues for solutions to Hamilton– Jacobi PDEs and hyperbolic systems of conservation laws. 1. Introduction In [9] the author advanced the following conjecture on compactness of ODEs d d Conjecture 1.1 (Bressan’s compactness conjecture). Let bn : Rt £ Rx ! R be smooth maps and denote by Φn the solution of the ODEs: 8 d < dt Φn(t; x) = bn(t; Φn(t; x)); (1) : Φn(0; x) = x : Assume that the fluxes Φn are nearly incompressible, i.e. that for some constant C we have ¡1 (2) C · det(rxΦn(t; x)) · C; and that kbnk1 + krbnkL1 is uniformly bounded. Then the sequence fΦng is 1 strongly precompact in Lloc. Thanks to the compactness of the space of BV functions, after possibly extract- ing a subsequence (not relabeled) we can assume the existence of a vector field 1 1 b 2 BVloc \ L such that bn ! b strongly in Lloc. Therefore a positive answer to Conjecture 1.1 would provide the existence of a map Φ which solves (in a suitable 2 CAMILLO DE LELLIS weak sense) the ODE 8 d < dt Φ(t; x) = b(t; Φ(t; x)); (3) : Φ(0; x) = x : The ODEs (1) are naturally connected with the transport equations 8 < @tun + bn ¢ rxun = 0; (4) : u(0; x) = u(x) : Indeed classical solutions of (4) are constant along the trajectories of (1). Next, denote by Jn(t; ¢) the Jacobian determinant of rxΦn(t; ¢) and by Ψn(t; ¢) the inverse of the map Φn(t; ¢). If we set ½n(t; x) := Jn(t; Ψn(t; x)), then a classical theorem of Liouville states that ½n solves 8 < @t½n + divx(½nbn) = 0; (5) : ½n(0; x) = 1 : Therefore the un solving (4) satisfy @t(½nun) + divx(½nunbn) = 0 : The bounds (2) imply that, up to subsequences, we can assume the existence of a 1 ¡1 ¤ 1 ½ 2 L with ½ ¸ C such that ½n * ½ in L . This ½ solves 8 < @t½ + divx(½b) = 0; (6) : ½(0; x) = 1 ; in the sense of distributions. In [3] the authors suggested Conjecture 1.2 below (compare with Conjecture 4.1 of [3]). Using the DiPerna–Lions theory of renormalized solutions to transport equations (see [10]) they proved that a positive answer to Conjecture 1.2 would imply Bressan’s Compactness Conjecture (see Proposition 4.4 of [3]). 1 m m 1 m Conjecture 1.2. Let f 2 L \ BVloc(R £ R ; R ) and ½ 2 L (R £ R ) be such that ½ is positive and bounded away from zero and @t½ + divx(½f) = 0 in the sense 1 1 of distribution. If u 2 L satisfies @t(½u)+divx (½uf) = 0 and ¯ 2 C (R; R), then @t(½¯(u)) + divx(½¯(u)f) = 0. Actually a positive answer to this conjecture would imply even more: In partic- ular it would show that the limit Φ of the fluxes Φn depends only on the limit b of the coefficients bn. Therefore this flux would be a sort of “canonical” flow for the ODE (3). In this paper we want to illustrate some recent partial results on Conjecture 1.2, and related questions. THE CHAIN RULE FOR THE DIVERGENCE OF BV –LIKE VECTOR FIELDS 3 2. Chain Rule First of all let us recast Conjecture 1.2 in a more general formulation. Consider the vector function v = (v1; v2) := (½; ½u), the vector field B := (1; b): R £ Rm ! R £ Rm ; and the coordinates z := (t; x). Then we have divz(v ­ B) = 0. To simplify the notation, from now on we will write D ¢ (vB) instead of div (v ­ B). If we define ¯˜ : R+ £ R ! R ˜ as ¯(x1; x2) := ¯(x2=x1), the goal of Conjecture 1.2 can be rewritten as D ¢ ˜ 1 (¯(v)B) = 0. Since ½ = v1 is bounded away from zero, we can find a C func- tion h which coincides with ¯˜ on the range of v. Hence we might restate our problem in the following way: Problem 2.1. Let Ω ½ Rn be an open set, B 2 L1 \ BV (Ω; RN ) and v 2 L1(Ω; Rk). Assume that D ¢ (vB) = 0. Can we right a formula which allows to compute D ¢ (h(v)B) for h 2 C1(Rk; R)? This time we cannot simply hope that D ¢ (h(v)B) = 0. Indeed, let us look at the following example. Consider an open set A ½ RN with smooth boundary and two maps B 2 L1 \ BV (RN ; RN ) and v 2 L1(RN ) such that 1 ² BjA, ujA, BjRN nA, and BjRN nA are C and continuous up to the boundary @A; ² D ¢ (vB) = 0 on RN . Denote by r¢B the bounded function which on RN n@A coincides with the pointwise divergence of B. Then it is easy to show that " # X @h D ¢ (h(v)B) = h(v) ¡ vi (v) r ¢ B @yi n i o (7) + h(v+)hº; B+i ¡ h(v¡)hº; B¡i HN¡1 @A where ² º is the exterior unit normal to @A; ² B§, v§ denote the interior and exterior traces of B and v on @A; ² Hn¡1 @A denotes the nonnegative Radon measure which to any Borel set E assigns the n ¡ 1–dimensional Hausdorff measure (that is the n ¡ 1– dimensional volume) of E \ @A. One can easily show examples where none of the terms in the right hand side of (7) vanish. However, when h takes the special form induced by the hypotheses of Conjecture 1.2, these terms do vanish (cp. with Lemma 7.6 of [5]). 4 CAMILLO DE LELLIS 3. Decomposition of Measures The BV Structure Theorem states that a BV function is approximately continuous outside a rectifiable set J of codimension 1, and on this set it undergoes jump discontinuities (in a suitable measure theoretic sense). For the precise statement of this theorem we refer to Section 2.4 of [3] and to the book [7] (Section 3.7). Moreover, the total variation measure jDBj of the (matrix–valued) Radon measure DB can be decomposed as the sum of three mutually orthogonal measures jDBja, jDBjc, and jDBjj, where ² jDBja is absolutely continuous with respect to the Lebesgue measure; ² jDBjj is absolutely continuous with respect to HN¡1 J; ² jDBjc is a singular measure such that jDBjc(A) = 0 for every Borel set A with HN¡1(A) < 1. Recall that DB = MjDBj for some bounded Borel matrix–valued function M, and that D ¢ B = trM jDBj. Therefore we can write DB = DaB + DcB + DjB := MjDaBj + MjDcBj + MjDjBj D ¢ B = Da ¢ B + Dc ¢ B + Dj ¢ B := trMjDaBj + trMjDcBj + trMjDjBj (see Sections 2.1 and 2.4 of [5] and Section 3.9 of [7]). The measures of these decompositions are called absolutely continuous part, Cantor part, and jump part. In the recent ground–breaking paper [2] the author extended the DiPerna–Lions theory of renormalized solutions to BV vector fields with bounded divergence. One consequence of his analysis is the following Theorem 3.1. Let B, v, and h be as in Problem 2.1. Then " # X @h (8) D ¢ (h(v)B) = h(v) ¡ v (v) Da ¢ B + ¹ ; i @y i i where j¹j · CjDc ¢ Bj + CjDj ¢ Bj, for some constant C. A more general statement, proved with the techniques introduced in [2], can be found in Section 3 of [5]. 4. Ambrosio–Crippa–Maniglia Trace Theorem Consider a bounded vector field C such that D ¢ C is a Radon measure. For every bounded open set Ω with C1 boundary we can define the normal trace of C on Ω as the distribution Tr C given by @ZΩ Z 1 N (9) hTr@ΩC; 'i := r' ¢ C + 'd [D ¢ C] for every ' 2 Cc (R ). Ω Ω It was proved in [8] that there exists a unique g 2 L1 (@Ω) such that Z loc hTr@ΩC; 'i = g' : @Ω The normal trace is local in the following sense: THE CHAIN RULE FOR THE DIVERGENCE OF BV –LIKE VECTOR FIELDS 5 (L) Let E = @Ω \ @Ω0 and the exterior unit normals to Ω and Ω0 coincide on N¡1 0 E. Then, up to a set of zero H measure, Tr@ΩC = Tr@ΩC on E . § This locality property allows to define the left and right traces TrAC on any recti- fiable set A of codimension 1, once we fix , a Borel unit vector field º normal to A (see for instance Section 2.2 of [5]). In [4], the authors proved the following Chain–Rule formula (see Theorem 4.2 therein): Theorem 4.1. Assume that C = wB, where w is a bounded function and B is a BV vector field. If ¯ is a smooth function and J is a rectifiable set, then µ § ¶ § TrJ (wB) § (10) TrJ (¯(w)B) = ¯ § TrJ B; TrJ B § where we use the convention that the right hand side is zero whenever TrJ C = 0. As a corollary of this chain–rule we have (cp. with Theorem 4.1 of [5]): Theorem 4.2. Let B, v, and h be as in Problem 2.1. Then ½ µ ¶ µ ¶ ¾ Tr+(vB) Tr¡(vB) D ¢ (h(v)B) = h J Tr+B ¡ h J Tr¡B¯ HN¡1 J Tr+B J Tr¡B J " J # J X @h (11) + h(v) ¡ v (v) Da ¢ B + º ; i @y i i where jºj · CjDc ¢ Bj.