Skip to main content
SpringerLink
Log in

Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields

  • Chapter
Transport Equations and Multi-D Hyperbolic Conservation Laws

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 5))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MathSciNet  Google Scholar 

  2. G.Alberti: Rank-one properties for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239–274.

    MATH  MathSciNet  Google Scholar 

  3. G.Alberti & L.Ambrosio: A geometric approach to monotone functions in ℝ n . Math. Z., 230 (1999), 259–316.

    Article  MATH  MathSciNet  Google Scholar 

  4. G.Alberti & S.Müller: A new approach to variational problems with multiple scales. Comm. Pure Appl. Math., 54 (2001), 761–825.

    Article  MATH  MathSciNet  Google Scholar 

  5. F.J.Almgren: The theory of varifolds – A variational calculus in the large, Princeton University Press, 1972.

    Google Scholar 

  6. L.Ambrosio: Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae, 158 (2004), 227–260.

    Article  MATH  MathSciNet  Google Scholar 

  7. L.Ambrosio: Lecture notes on transport equation and Cauchy problem for BV vector fields and applications. Preprint, 2004 (available at http://cvgmt.sns.it).

  8. L.Ambrosio: Lecture notes on transport equation and Cauchy problem for non-smooth vector fields and applications. Preprint, 2005 (available at http://cvgmt.sns.it).

  9. L.Ambrosio, F.Bouchut & C.De Lellis: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm. PDE, 29 (2004), 1635–1651.

    Article  MATH  MathSciNet  Google Scholar 

  10. L.Ambrosio, G.Crippa & S.Maniglia: Traces and fine properties of a BD class of vector fields and applications. Ann. Sci. Toulouse, XIV (4) (2005), 527–561.

    MathSciNet  Google Scholar 

  11. L.Ambrosio & C.De Lellis: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. International Mathematical Research Notices, 41 (2003), 2205–2220.

    Article  MathSciNet  Google Scholar 

  12. L.Ambrosio, C.De Lellis & J.Malý: On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems. To appear in the forthcoming book by the AMS series in contemporary mathematics “Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis” (available at http://cvgmt.sns.it).

  13. L.Ambrosio, N.Fusco & D.Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000.

    Google Scholar 

  14. L.Ambrosio, N.Gigli & G.Savaré: Gradient flows in metric spaces and in the Wasserstein space of probability measures. Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005.

    Google Scholar 

  15. L.Ambrosio, M.Lecumberry & S.Maniglia: Lipschitz regularity and approximate differentiability of the DiPerna–Lions flow. Rendiconti del Seminario Fisico Matematico di Padova, 114 (2005), 29–50.

    MATH  MathSciNet  Google Scholar 

  16. L.Ambrosio, S.Lisini & G.Savaré: Stability of flows associated to gradient vector fields and convergence of iterated transport maps. Manuscripta Math., 121 (2006), 1–50.

    Article  MATH  MathSciNet  Google Scholar 

  17. L.Ambrosio & J.Malý: Very weak notions of differentiability. Proceedings of the Royal Society of Edinburgh, 137A (2007), 447–455.

    Google Scholar 

  18. E.J.Balder: New fundamentals of Young measure convergence. CRC Res. Notes in Math. 411, 2001.

    Google Scholar 

  19. J.Ball & R.James: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal., 100 (1987), 13–52.

    Article  MATH  MathSciNet  Google Scholar 

  20. V.Bangert: Minimal measures and minimizing closed normal one-currents. Geom. funct. anal., 9 (1999), 413–427.

    Article  MATH  MathSciNet  Google Scholar 

  21. J.-D.Benamou & Y.Brenier: Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem. SIAM J. Appl. Math., 58 (1998), 1450–1461.

    Article  MATH  MathSciNet  Google Scholar 

  22. P.Bernard & B.Buffoni: Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS), 9 (2007), 85–121.

    Article  MATH  MathSciNet  Google Scholar 

  23. M.Bernot, V.Caselles & J.M.Morel: Traffic plans. Publ. Mat., 49 (2005), 417–451.

    MATH  MathSciNet  Google Scholar 

  24. V.Bogachev & E.M.Wolf: Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions. J. Funct. Anal., 167 (1999), 1–68.

    Article  MATH  MathSciNet  Google Scholar 

  25. F.Bouchut: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Rational Mech. Anal., 157 (2001), 75–90.

    Article  MATH  MathSciNet  Google Scholar 

  26. F.Bouchut & G.Crippa: Uniqueness, Renormalization, and Smooth Approximations for Linear Transport Equations. SIAM J. Math. Anal., 38 (2006), 1316–1328.

    Article  MATH  MathSciNet  Google Scholar 

  27. F.Bouchut, F.Golse & M.Pulvirenti: Kinetic equations and asymptotic theory. Series in Appl. Math., Gauthiers-Villars, 2000.

    Google Scholar 

  28. F.Bouchut & F.James: One dimensional transport equation with discontinuous coefficients. Nonlinear Analysis, 32 (1998), 891–933.

    Article  MATH  MathSciNet  Google Scholar 

  29. F.Bouchut, F.James & S.Mancini: Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficients. Annali Scuola Normale Superiore, Ser. 5, 4 (2005), 1–25.

    Google Scholar 

  30. Y.Brenier: The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Amer. Mat. Soc., 2 (1989), 225–255.

    Article  MATH  MathSciNet  Google Scholar 

  31. Y.Brenier: The dual least action problem for an ideal, incompressible fluid. Arch. Rational Mech. Anal., 122 (1993), 323–351.

    Article  MATH  MathSciNet  Google Scholar 

  32. Y.Brenier: A homogenized model for vortex sheets. Arch. Rational Mech. Anal., 138 (1997), 319–353.

    Article  MATH  MathSciNet  Google Scholar 

  33. Y.Brenier: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math., 52 (1999), 411–452.

    Article  MathSciNet  Google Scholar 

  34. A.Bressan: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova, 110 (2003), 103–117.

    MATH  MathSciNet  Google Scholar 

  35. H.Brezis: Analyse fonctionnelle. Théorie et applications. Masson, Paris, 1983.

    Google Scholar 

  36. L.A.Caffarelli: Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math., 44 (1991), 965–969.

    Article  MATH  MathSciNet  Google Scholar 

  37. L.A.Caffarelli: Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math., 45 (1992), 1141–1151.

    Article  MATH  MathSciNet  Google Scholar 

  38. L.A.Caffarelli: The regularity of mappings with a convex potential. J. Amer. Math. Soc., 5 (1992), 99–104.

    Article  MATH  MathSciNet  Google Scholar 

  39. L.A.Caffarelli: Boundary regularity of maps with convex potentials. Ann. of Math., 144 (1996), 453–496.

    Article  MATH  MathSciNet  Google Scholar 

  40. I.Capuzzo Dolcetta & B.Perthame: On some analogy between different approaches to first order PDE’s with nonsmooth coefficients. Adv. Math. Sci Appl., 6 (1996), 689–703.

    MATH  MathSciNet  Google Scholar 

  41. A.Cellina: On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Analysis, TMA, 25 (1995), 899–903.

    Article  MATH  MathSciNet  Google Scholar 

  42. A.Cellina & M.Vornicescu: On gradient flows. Journal of Differential Equations, 145 (1998), 489–501.

    Article  MATH  MathSciNet  Google Scholar 

  43. F.Colombini, G.Crippa & J.Rauch: A note on two dimensional transport with bounded divergence. Comm. Partial Differential Equations, 31 (2006), 1109–1115.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  45. F.Colombini & N.Lerner: Uniqueness of L ∞ solutions for a class of conormal BV vector fields. Contemp. Math. 368 (2005), 133–156.

    MathSciNet  Google Scholar 

  46. F.Colombini, T. Luo & J.Rauch: Nearly Lipschitzean diverge free transport propagates neither continuity nor BV regularity. Commun. Math. Sci., 2 (2004), 207–212.

    MATH  MathSciNet  Google Scholar 

  47. G.Crippa & C.De Lellis: Oscillatory solutions to transport equations. Indiana Univ. Math. J., 55 (2006), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  48. G.Crippa & C.De Lellis: Estimates and regularity results for the DiPerna–Lions flow. Preprint, 2006 (available at http://cvgmt.sns.it). Accepted by J. Reine Angew. Math.

  49. A.B.Cruzeiro: Équations différentielles ordinaires: non explosion et mesures quasi-invariantes. J. Funct. Anal., 54 (1983), 193–205.

    Article  MATH  MathSciNet  Google Scholar 

  50. A.B.Cruzeiro: Équations différentielles sur l’espace de Wiener et formules de Cameron-Martin non linéaires. J. Funct. Anal., 54 (1983), 206–227.

    Article  MATH  MathSciNet  Google Scholar 

  51. A.B.Cruzeiro: Unicité de solutions d’équations différentielles sur l’espace de Wiener. J. Funct. Anal., 58 (1984), 335–347.

    Article  MATH  MathSciNet  Google Scholar 

  52. M.Cullen: On the accuracy of the semi-geostrophic approximation. Quart. J. Roy. Metereol. Soc., 126 (2000), 1099–1115.

    Article  Google Scholar 

  53. M.Cullen & M.Feldman: Lagrangian solutions of semigeostrophic equations in physical space. SIAM J. Math. Anal., 37 (2006), 1371–1395.

    Article  MATH  MathSciNet  Google Scholar 

  54. M.Cullen & W.Gangbo: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Rational Mech. Anal., 156 (2001), 241–273.

    Article  MATH  MathSciNet  Google Scholar 

  55. C.Dafermos: Hyperbolic conservation laws in continuum physics. Springer Verlag, 2000.

    Google Scholar 

  56. N.G.de Bruijn: On almost additive functions. Colloq. Math. 15 (1966), 59–63.

    MATH  MathSciNet  Google Scholar 

  57. C.De Lellis: Blow-up of the BV norm in the multidimensional Keyfitz and Kranzer system. Duke Math. J., 127 (2004), 313–339.

    Article  MathSciNet  Google Scholar 

  58. L.De Pascale, M.S.Gelli & L.Granieri: Minimal measures, one-dimensional currents and the Monge-Kantorovich problem. Calc. Var. Partial Differential Equations, 27 (2006), 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  59. N.Depauw: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C.R. Math. Sci. Acad. Paris, 337 (2003), 249–252.

    MATH  MathSciNet  Google Scholar 

  60. R.J.DiPerna: Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal., 88 (1985), 223–270.

    Article  MATH  MathSciNet  Google Scholar 

  61. R.J.DiPerna & P.L.Lions: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98 (1989), 511–547.

    Article  MATH  MathSciNet  Google Scholar 

  62. R.J.DiPerna & P.L.Lions: On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. of Math., 130 (1989), 312–366.

    Article  MathSciNet  Google Scholar 

  63. L.C.Evans: Partial Differential Equations and Monge–Kantorovich Mass Transfer. Current Developments in Mathematics (1997), 65–126.

    Google Scholar 

  64. L.C.Evans: Partial Differential Equations. Graduate studies in Mathematics, 19 (1998), American Mathematical Society.

    Google Scholar 

  65. L.C.Evans & W.Gangbo: Differential equations methods for the Monge-Kantorovich mass transfer problem. Memoirs AMS, 653, 1999.

    Google Scholar 

  66. L.C.Evans, W.Gangbo & O.Savin: Diffeomorphisms and nonlinear heat flows. SIAM J. Math. Anal., 37 (2005), 737–751.

    Article  MATH  MathSciNet  Google Scholar 

  67. L.C.Evans & R.F.Gariepy: Lecture notes on measure theory and fine properties of functions, CRC Press, 1992.

    Google Scholar 

  68. H.Federer: Geometric measure theory, Springer, 1969.

    Google Scholar 

  69. M.Hauray: On Liouville transport equation with potential in BV loc . Comm. Partial Differential Equations, 29 (2004), 207–217.

    Article  MATH  MathSciNet  Google Scholar 

  70. M.Hauray: On two-dimensional Hamiltonian transport equations with L loc p coefficients. Ann. IHP Nonlinear Anal. Non Linéaire, 20 (2003), 625–644.

    Article  MATH  MathSciNet  Google Scholar 

  71. W.B.Jurkat: On Cauchy’s functional equation. Proc. Amer. Math. Soc., 16 (1965), 683–686.

    Article  MATH  MathSciNet  Google Scholar 

  72. B.L.Keyfitz & H.C.Kranzer: A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal. 1980, 72, 219–241.

    Article  MATH  MathSciNet  Google Scholar 

  73. C.Le Bris & P.L.Lions: Renormalized solutions of some transport equations with partially W 1,1 velocities and applications. Annali di Matematica, 183 (2003), 97–130.

    Article  Google Scholar 

  74. N.Lerner: Transport equations with partially BV velocities. Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 681–703.

    MATH  MathSciNet  Google Scholar 

  75. P.L.Lions: Mathematical topics in fluid mechanics, Vol. I: incompressible models. Oxford Lecture Series in Mathematics and its applications, 3 (1996), Oxford University Press.

    Google Scholar 

  76. P.L.Lions: Mathematical topics in fluid mechanics, Vol. II: compressible models. Oxford Lecture Series in Mathematics and its applications, 10 (1998), Oxford University Press.

    Google Scholar 

  77. P.L.Lions: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci. Paris Sér. I, 326 (1998), 833–838.

    MATH  Google Scholar 

  78. J.Lott & C.Villani: Weak curvature conditions and functional inequalities. J. Funct. Anal., in press.

    Google Scholar 

  79. S.Maniglia: Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl., 87 (2007), 601–626.

    MATH  MathSciNet  Google Scholar 

  80. J.N.Mather: Minimal measures. Comment. Math. Helv., 64 (1989), 375–394.

    Article  MATH  MathSciNet  Google Scholar 

  81. J.N.Mather: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207 (1991), 169–207.

    Article  MATH  MathSciNet  Google Scholar 

  82. E.Y.Panov: On strong precompactness of bounded sets of measure-valued solutions of a first order quasilinear equation. Math. Sb., 186 (1995), 729–740.

    Article  MATH  MathSciNet  Google Scholar 

  83. G.Petrova & B.Popov: Linear transport equation with discontinuous coefficients. Comm. PDE, 24 (1999), 1849–1873.

    Article  MATH  MathSciNet  Google Scholar 

  84. F.Poupaud & M.Rascle: Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Comm. PDE, 22 (1997), 337–358.

    Article  MATH  MathSciNet  Google Scholar 

  85. A.Pratelli: Equivalence between some definitions for the optimal transport problem and for the transport density on manifolds. Ann. Mat. Pura Appl., 184 (2005), 215–238.

    Article  MATH  MathSciNet  Google Scholar 

  86. S.K.Smirnov: Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents. St. Petersburg Math. J., 5 (1994), 841–867.

    MathSciNet  Google Scholar 

  87. E.M.Stein: Singular integrals and differentiability properties of functions. Princeton University Press, 1970.

    Google Scholar 

  88. L. Tartar: Compensated compactness and applications to partial differential equations. Research Notes in Mathematics, Nonlinear Analysis and Mechanics, ed. R. J. Knops, vol. 4, Pitman Press, New York, 1979, 136–211.

    Google Scholar 

  89. R.Temam: Problémes mathématiques en plasticité. Gauthier-Villars, Paris, 1983.

    Google Scholar 

  90. J.I.E.Urbas: Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  91. J.I.E.Urbas: Regularity of generalized solutions of Monge-Ampère equations, Math. Z., 197 (1988), 365–393.

    Article  MATH  MathSciNet  Google Scholar 

  92. A.Vasseur: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 160 (2001), 181–193.

    Article  MATH  MathSciNet  Google Scholar 

  93. C.Villani: Topics in mass transportation. Graduate Studies in Mathematics, 58 (2004), American Mathematical Society.

    Google Scholar 

  94. C.Villani: Optimal transport: old and new. Lecture Notes of the 2005 Saint-Flour Summer school.

    Google Scholar 

  95. L.C.Young: Lectures on the calculus of variations and optimal control theory, Saunders, 1969.

    Google Scholar 

Download references

Authors
  1. Luigi Ambrosio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluca Crippa
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ambrosio, L., Crippa, G. (2008). Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76781-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation