Abstract
This paper provides an overview of the theory of flows associated to nonsmooth vector fields, describing the main developments from the seminal paper (DiPerna and Lions in Invent Math 98:511–547, 1989) till now. The problem of well posedness of ODE’s associated to vector fields arises in many fields, for instance conservation laws (via the theory of characteristics) and fluid mechanics (when looking for consistence between Eulerian and Lagrangian points of view). The theory developed so far covers many classes of vector fields and, besides uniqueness, also more quantitative aspects, as stability estimates and differentiability of the flow. Detailed lecture notes on this topic are given in Ambrosio (in: Dacorogna, Marcellini (eds) Lecture Notes in Mathematics “Calculus of variations and non-linear partial differential equations” (CIME Series, Cetraro, 2005), vol 1927, pp 2–41, 2008), Ambrosio and Crippa (Lect Notes UMI 5:3–54, 2008), Ambrosio and Crippa (Proc R Soc Edinb Sect A 144:1191–1244 2014), Ambrosio and Trevisan (Ann Fac Sci Toulouse, 2016).
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References
Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math. 107, 287–296 (1978)
Alberti, G.: Rank-one properties for derivatives of functions with bounded variation. Proc. R. Soc. Edinb. Sect. A 123, 239–274 (1993)
Alberti, G., Ambrosio, L.: A geometric approach to monotone functions in \({\mathbb{R}}^n\). Math. Z. 230, 259–316 (1999)
Alberti, G., Bianchini, S., Crippa, G.: A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. 16, 201–234 (2014)
Alberti, G., Bianchini, S., Crippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 12, 863–902 (2013)
Alberti, G., Bianchini, S., Crippa, G.: On the \(L^p\) differentiability of certain classes of functions. Revista Matemática Iberoamericana 30, 349–367 (2014)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000)
Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158, 227–260 (2004)
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)
Ambrosio, L.: Transport equation and Cauchy problem for non-smooth vector fields. In: Dacorogna, B., Marcellini, P. (eds.) Lecture Notes in Mathematics “Calculus of Variations and Non-Linear Partial Differential Equations” (CIME Series, Cetraro, 2005), vol. 1927, pp. 2–41 (2008)
Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. Lect. Notes UMI 5, 3–54 (2008)
Ambrosio, L., Bouchut, F., De Lellis, C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Commun. PDE 29, 1635–1651 (2004)
Ambrosio, L., Crippa, G., Maniglia, S.: Traces and fine properties of a \(BD\) class of vector fields and applications. Ann. Sci. Toulouse XIV(4), 527–561 (2005)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the Wasserstein space of probability measures. Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005, second edition in 2008
Ambrosio, L., Friesecke, G., Giannoulis, J.: Passage from quantum to classical molecular dynamics in the presence of Coulomb interactions. Commun. PDE 35, 1490–1515 (2009)
Ambrosio, L., Figalli, A., Friesecke, G., Giannoulis, J., Paul, T.: Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data. Commun. Pure Appl. Math. 64, 1199–1242 (2011)
Ambrosio, L., Lecumberry, M., Maniglia, S.: Lipschitz regularity and approximate differentiability of the DiPerna–Lions flow. Rendiconti del Seminario Fisico Matematico di Padova 114, 29–50 (2005)
Ambrosio, L., Malý, J.: Very weak notions of differentiability. Proc. R. Soc. Edinb. 137 A, 447–455 (2007)
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: in honour of Haim Brezis, Contemporary Mathematics, vol. 446, pp. 31–67. AMS (2007)
Ambrosio, L., Figalli, A.: On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna–Lions. J. Funct. Anal. 256, 179–214 (2009)
Ambrosio, L., Figalli, A.: Almost everywhere well-posedness of continuity equations with measure initial data. C. R. Acad. Sci. Paris 348, 249–252 (2010)
Ambrosio, L., Colombo, M., De Philippis, G., Figalli, A.: Existence of Eulerian solutions to the semigeostrophic equations in physical space: the two-dimensional periodic case. Commun. Partial Differ. Equ. 37, 2209–2227 (2012)
Ambrosio, L., Colombo, M., De Philippis, G., Figalli, A.: A global existence result for the semigeostrophic equations in three dimensional convex domains. Preprint, 2012, to appear on Discrete and Continuous Dynamical Systems
Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A 144, 1191–1244 (2014)
Ambrosio, L., Trevisan, D.: Well posedness of Lagrangian flows and continuity equations in metric measure spaces. Anal. PDE 7, 1179–1234 (2014)
Ambrosio, L., Colombo, M., Figalli, A.: Existence and uniqueness of maximal regular flows for non-smooth vector fields. Arch. Ration. Mech. Anal. 218, 1043–1081 (2015)
Ambrosio, L., Colombo, M., Figalli, A.: On the Lagrangian structure of transport equations: the Vlasov–Poisson system. ArXiv Preprint (2015)
Ambrosio, L., Trevisan, D.: Lecture notes on the DiPerna–Lions theory in abstract measure spaces. ArXiv preprint, to appear in the Ann. Fac. Sci. Toulouse (2016)
Ambrosio, L., Stra, F., Trevisan, D.: Weak and strong convergence of derivations and stability of flows with respect to MGH convergence. J. Funct. Anal. 272, 1182–1229 (2017)
Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der mathematischen Wissenschaften, vol. 348. Springer, Berlin (2014)
Balder, E.J.: New fundamentals of Young measure convergence. In: CRC Res. Notes in Math., vol. 411 (2001)
Benamou, J.-D., Brenier, Y.: Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem. SIAM J. Appl. Math. 58, 1450–1461 (1998)
Bianchini, S., Gusev, N.A.: Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization. Arch. Ration. Mech. Anal. 222, 451–505 (2016)
Bianchini, S., Bonicatto, P., Gusev, N.A.: Renormalization for autonomous nearly incompressible \(BV\) vector fields in two dimensions. SIAM J. Math. Anal. 48, 1–33 (2016)
Bianchini, S., Bonicatto, P.: A uniqueness result for the decomposition of vector fields in \({\mathbb{R}}^d\). Preprint (2017)
Bourgain, J.: Invariant measures for NLS in infinite volume. Commun. Math. Phys. 210, 605–620 (2000)
Bogachev, V., Wolf, E.M.: Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions. J. Funct. Anal. 167, 1–68 (1999)
Bohun, A., Bouchut, F., Crippa, G.: Lagrangian flows for vector fields with anisotropic regularity. Ann. Inst. H. Poincaré 33, 1409–1429 (2016)
Bohun, A., Bouchut, F., Crippa, G.: Lagrangian solutions to the Vlasov–Poisson system with \(L^1\) density. J. Differ. Equ. 260, 3576–3597 (2016)
Bouchut, F., Crippa, G.: Uniqueness, renormalization, and smooth approximations for linear transport equations. SIAM J. Math. Anal. 38, 1316–1328 (2006)
Bouchut, F., Crippa, G.: Equations de transport à coefficient dont le gradient est donné par une intégrale singulière. (French) [Transport equations with a coefficient whose gradient is given by a singular integral]. Séminaire: Équations aux Dérivées Partielles. 2007–2008, Exp. No. I, 15 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2009)
Bouchut, F., Crippa, G.: Lagrangian flows for vector fields with gradient given by a singular integral. J. Hyperbolic Differ. Equ. 10, 235–282 (2013)
Bouchut, F., James, F.: One dimensional transport equation with discontinuous coefficients. Nonlinear Anal. 32, 891–933 (1998)
Bouchut, F.: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Ration. Mech. Anal. 157, 75–90 (2001)
Bouchut, F., James, F., Mancini, S.: Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficients. Annali Scuola Normale Superiore 4, 1–25 (2005)
Brenier, Y.: The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2, 225–255 (1989)
Brenier, Y.: The dual least action problem for an ideal, incompressible fluid. Arch. Ration. Mech. Anal. 122, 323–351 (1993)
Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52, 411–452 (1999)
Bressan, A.: A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110, 97–102 (2003)
Bressan, A.: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova 110, 103–117 (2003)
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)
Caffarelli, L.A.: Some regularity properties of solutions of Monge Ampère equation. Commun. Pure Appl. Math. 44, 965–969 (1991)
Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99–104 (1992)
Caffarelli, L.A.: Boundary regularity of maps with convex potentials. Ann. Math. 144, 453–496 (1996)
Capuzzo Dolcetta, I., Perthame, B.: On some analogy between different approaches to first order PDE’s with nonsmooth coefficients. Adv. Math. Sci Appl. 6, 689–703 (1996)
Cellina, A.: On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Anal. TMA 25, 899–903 (1995)
Cellina, A., Vornicescu, M.: On gradient flows. J. Differ. Equ. 145, 489–501 (1998)
Cullen, M.: A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow. Imperial College Press, London (2006)
Champagnat, N., Jabin, P.-E.: Well posedness in any dimension for Hamiltonian flows with non BV force terms. Commun. Partial Differ. Equ. 35, 786–816 (2010)
Colombini, F., Lerner, N.: Uniqueness of continuous solutions for \(BV\) vector fields. Duke Math. J. 111, 357–384 (2002)
Colombini, F., Lerner, N.: Uniqueness of \(L^\infty \) solutions for a class of conormal \(BV\) vector fields. Contemp. Math. 368, 133–156 (2005)
Colombini, F., Luo, T., Rauch, J.: Uniqueness and no nuniqueness for non smooth divergence-free transport. Seminaire: Équations aux Dérivées Partielles, 20022003, Exp. No. XXII, 21 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2003)
Crippa, G.: Ordinary Differential Equations and Singular Integrals. HYP2012 Proceedings, in press (2012)
Crippa, G., De Lellis, C.: Oscillatory solutions to transport equations. Indiana Univ. Math. J. 55, 1–13 (2006)
Crippa, G., De Lellis, C.: Estimates for transport equations and regularity of the DiPerna–Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)
Cullen, M.: On the accuracy of the semi-geostrophic approximation. Q. J. R. Metereol. Soc. 126, 1099–1115 (2000)
Cullen, M., Gangbo, W.: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Anal. 156, 241–273 (2001)
Cullen, M., Feldman, M.: Lagrangian solutions of semigeostrophic equations in physical space. J. Math. Anal. 37, 1371–1395 (2006)
Dafermos, C.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)
De Lellis, C.: Blow-up of the \(BV\) norm in the multidimensional Keyfitz and Kranzer system. Duke Math. J. 127, 313–339 (2004)
DePauw, N.: Non unicité des solutions bornées pour un champ de vecteurs \(BV\) en dehors d’un hyperplan. C.R. Math. Sci. Acad. Paris 337, 249–252 (2003)
De Philippis, G., Figalli, A.: \(W^{2,1}\) regularity for solutions to the Monge-Ampére equation. Invent. Math. 192, 55–69 (2013)
De Philippis, G., Figalli, A., Savin, O.: A note on \(W^{2,1+\epsilon }\) interior regularity for solutions to the Monge-Ampére equation. Math. Ann. 357, 11–22 (2013)
DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88, 223–270 (1985)
DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
Evans, L.C., Gariepy, R.F.: Lecture Notes on Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)
Figalli, A.: Existence and uniqueness of martingale solutions for SDE with rough or degenerate coefficients. J. Funct. Anal. 254, 109–153 (2008)
Flandoli, F.: Random perturbation of PDEs and fluid dynamic models. Lectures from the 40th Probability Summer School held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2015. cole d’t de Probabilités de Saint-Flour
Gigli, N.: Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below. ArXiv preprint arXiv:1407.0809. To appear on Mem. Am. Math. Soc
Hauray, M.: On Liouville transport equation with potential in \(BV_{{\rm loc}}\). Commun. PDE 29, 207–217 (2004)
Hauray, M.: On two-dimensional Hamiltonian transport equations with \(L^p_{{\rm loc}}\) coefficients. Ann. IHP Nonlinear Anal. Non Linéaire 20, 625–644 (2003)
Jabin, P.-E.: Differential equations with singular fields. J. Math. Pures Appl. 94, 597–621 (2010)
Keyfitz, B.L., Kranzer, H.C.: A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Ration. Mech. Anal. 72, 219–241 (1980)
Le Bris, C., Lions, P.-L.: Renormalized solutions of some transport equations with partially \(W^{1,1}\) velocities and applications. Annali di Matematica 183, 97–130 (2003)
Lerner, N.: Transport equations with partially BV velocities. Annali Scuola Normale Superiore 3, 681–703 (2004)
Lions, P.-L.: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci. Paris Sér. I(326), 833–838 (1998)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. I: Incompressible Models. Oxford Lecture Series in Mathematics and Its Applications, vol. 3. Oxford University Press, Oxford (1996)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, vol. 10. Oxford University Press, Oxford (1998)
Maniglia, S.: Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl. 87, 601–626 (2007)
Panov, E.Y.: On strong precompactness of bounded sets of measure-valued solutions of a first order quasilinear equation. Math. Sb. 186, 729–740 (1995)
Petrova, G., Popov, B.: Linear transport equation with discontinuous coefficients. Commun. PDE 24, 1849–1873 (1999)
Poupaud, F., Rascle, M.: Measure solutions to the linear multidimensional transport equation with non-smooth coefficients. Commun. PDE 22, 337–358 (1997)
Schmidt, T.: \(W^{1,2+\epsilon }\) estimates for solutions to the Monge-Ampére equation. Adv. Math. 240, 672–689 (2013)
Smirnov, S.K.: Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents. St. Petersburg Math. J. 5, 841–867 (1994)
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Trevisan, D.: Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21, 41 (2016)
Trevisan, D.: Lagrangian flows driven by BV fields in Wiener spaces. Probab. Theory Relat. Fields 163, 123–147 (2015)
Urbas, J.I.E.: Global Hölder estimates for equations of Monge-Ampère type. Invent. Math. 91, 1–29 (1988)
Urbas, J.I.E.: Regularity of generalized solutions of Monge-Ampère equations. Math. Z. 197, 365–393 (1988)
Villani, C.: Topics in Mass Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2004)
Villani, C.: Optimal Transport: Old and New. Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2009)
Weaver, N.: Lipschitz algebras and derivations. II. Exterior differentiation. J. Funct. Anal. 178, 64–112 (2000)
Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, Philadelphia (1969)
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Ambrosio, L. Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications. Rev Mat Complut 30, 427–450 (2017). https://doi.org/10.1007/s13163-017-0244-3
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DOI: https://doi.org/10.1007/s13163-017-0244-3