A uniqueness result for the decomposition of vector fields in \(\mathbb {R}^{{d}}\)

  • Stefano BianchiniEmail author
  • Paolo Bonicatto


Given a vector field \(\rho (1,\mathbf {b}) \in L^1_\mathrm{loc}(\mathbb {R}^+\times \mathbb {R}^{d},\mathbb {R}^{d+1})\) such that \({{\,\mathrm{div}\,}}_{t,x} (\rho (1,\mathbf {b}))\) is a measure, we consider the problem of uniqueness of the representation \(\eta \) of \(\rho (1,\mathbf {b}) {\mathcal {L}}^{d+1}\) as a superposition of characteristics \(\gamma : (t^-_\gamma ,t^+_\gamma ) \rightarrow \mathbb {R}^d\), \(\dot{\gamma } (t)= \mathbf {b}(t,\gamma (t))\). We give conditions in terms of a local structure of the representation \(\eta \) on suitable sets in order to prove that there is a partition of \(\mathbb {R}^{d+1}\) into disjoint trajectories \(\wp _\mathfrak {a}\), \(\mathfrak {a}\in \mathfrak {A}\), such that the PDE
$$\begin{aligned} {{\,\mathrm{div}\,}}_{t,x} \big ( u \rho (1,\mathbf {b}) \big ) \in {\mathcal {M}}(\mathbb {R}^{d+1}), \quad u \in L^\infty (\mathbb {R}^+\times \mathbb {R}^{d}), \end{aligned}$$
can be disintegrated into a family of ODEs along \(\wp _\mathfrak {a}\) with measure r.h.s. The decomposition \(\wp _\mathfrak {a}\) is essentially unique. We finally show that \(\mathbf {b}\in L^1_t({{\,\mathrm{BV}\,}}_x)_\mathrm{loc}\) satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible \({{\,\mathrm{BV}\,}}\) vector fields.

Mathematics Subject Classification

35F10 35L03 28A50 35D30 



The authors would like to thank the Center of Mathematical Sciences and Applications (CMSA) of Harvard University and the Institut des Hautes Études Scientifiques (IHES) where part of this work has been done. They are also grateful to Guido de Philippis for useful discussions. During the revision of the paper, the second author was supported by ERC Starting Grant 676675 FLIRT.


  1. 1.
    Alberti, G., Bianchini, S., Crippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(4), 863–902 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alberti, G., Bianchini, S., Crippa, G.: A uniqueness result for the continuity equation in two dimensions. JEMS 16(2), 201–234 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Bouchut, F., De Lellis, C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Commun. Partial Differ. Equ. 29(9–10), 1635–1651 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws, volume 5 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, pp. 3–57 (2008)Google Scholar
  5. 5.
    Ambrosio, L., Crippa, G., Maniglia, S.: Traces and fine properties of a \(\text{ BD }\) class of vector fields and applications. Ann. Fac. Sci. Toulouse Math. (6) 14(4), 527–561 (2005)Google Scholar
  6. 6.
    Ambrosio, L., De Lellis, C.: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. Int. Math. Res. Not. 41, 2205–2220 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ambrosio, L., De Lellis, C., Malý, J.: On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems. In: Perspectives in Nonlinear Partial Differential Equations, volume 446 of Contemporary Mathematics. American Mathematical Society, Providence, RI, pp. 31–67 (2007)Google Scholar
  8. 8.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Oxford (2000)zbMATHGoogle Scholar
  9. 9.
    Alberti, G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinb. Sect. A 123(2), 239–274 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. math. 158(2), 227–260 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Anzellotti, G.: Traces of bounded vectorfields and the divergence theorem (1983) Google Scholar
  12. 12.
    Babadjian, J.-F.: Traces of functions of bounded deformation. Indiana Univ. Math. J. 64, 1271–1290 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bianchini, S., Bonicatto, P., Gusev, N.A.: Renormalization for autonomous nearly incompressible BV vector fields in two dimensions. SIAM J. Math. Anal. 48(1), 1–33 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bianchini, S., Gloyer, M.: An estimate on the flow generated by monotone operators. Commun. Partial Differ. Equ. 36(5), 777–796 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bianchini, S., Gusev, N.A.: Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization. Arch. Ration. Mech. Anal. 222(2), 451–505 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bonicatto, P.: Untangling of trajectories for non-smooth vector fields and Bressans Compactness Conjecture. PhD thesis, SISSA (2017)Google Scholar
  17. 17.
    Bressan, A.: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova 110, 103–117 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bressan, A.: A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110, 97–102 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    De Lellis, C.: Notes on hyperbolic systems of conservation laws and transport equations. In: Handbook of Differential Equations: Evolutionary Equations. Vol. III, Handbook of Differential Equations. Elsevier, Amsterdam, pp. 277–382 (2007)Google Scholar
  20. 20.
    De Lellis, C.: A note on Alberti’s rank-one theorem. In: Transport Equations and Multi-D Hyperbolic Conservation Laws, volume 5 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, pp. 61–74 (2008)Google Scholar
  21. 21.
    Depauw, N.: Non-unicité du transport par un champ de vecteurs presque BV. In: Seminaire: Équations aux Dérivées Partielles, 2002–2003, Sémin. Équ. Dériv. Partielles, pages Exp. No. XIX, 9. École Polytech., Palaiseau (2003)Google Scholar
  22. 22.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fremlin, D.H.: Measure Theory. Vol. 4. Torres Fremlin, Colchester, 2006. Topological Measure Spaces. Part I, II, Corrected second printing of the 2003 originalGoogle Scholar
  24. 24.
    Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in \(n\) variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)Google Scholar
  25. 25.
    Kellerer, H.G.: Duality theorems for marginal problems. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 67(4), 399–432 (1984)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Keyfitz, B.L., Kranzer, H.C.: A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Ration. Mech. Anal. 72(3), 219–241 (1979/80)Google Scholar
  27. 27.
    Morse, A.P.: Perfect blankets. Trans. Am. Math. Soc. 61(3), 418–442 (1947)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Paolini, E., Stepanov, E.: Decomposition of acyclic normal currents in a metric space. J. Funct. Anal. 263(11), 3358–3390 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Paolini, E., Stepanov, E.: Structure of metric cycles and normal one-dimensionalcurrents. J. Funct. Anal. 264(6), 1269–1295 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Smirnov, S.K.: Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents. St. Petersb. Math. J. 5(4), 841–867 (1994)Google Scholar
  31. 31.
    Ziemer, W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics. Springer, New York (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S.I.S.S.A.TriesteItaly

Personalised recommendations