Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

Francois Bouchut[1]; Francois James[2]; Simona Mancini[3]

  • [1] DMA, Ecole Normale Supérieure et CNRS 45 rue d’Ulm 75230 Paris cedex 05, France
  • [2] Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France
  • [3] Laboratoire J.-L. Lions, UMR 7598 Université Pierre et Marie Curie, BP 187 4 place Jussieu 75252 Paris cedex 05, France current address: Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 1, page 1-25
  • ISSN: 0391-173X

Abstract

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The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.

How to cite

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Bouchut, Francois, James, Francois, and Mancini, Simona. "Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.1 (2005): 1-25. <http://eudml.org/doc/84554>.

@article{Bouchut2005,
abstract = {The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.},
affiliation = {DMA, Ecole Normale Supérieure et CNRS 45 rue d’Ulm 75230 Paris cedex 05, France; Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France; Laboratoire J.-L. Lions, UMR 7598 Université Pierre et Marie Curie, BP 187 4 place Jussieu 75252 Paris cedex 05, France current address: Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France},
author = {Bouchut, Francois, James, Francois, Mancini, Simona},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {1-25},
publisher = {Scuola Normale Superiore, Pisa},
title = {Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient},
url = {http://eudml.org/doc/84554},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Bouchut, Francois
AU - James, Francois
AU - Mancini, Simona
TI - Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 1
SP - 1
EP - 25
AB - The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.
LA - eng
UR - http://eudml.org/doc/84554
ER -

References

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  1. [1] G. Alberti, Rank-one properties for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh, Sect. A 123 (1993), 239-274. Zbl0791.26008MR1215412
  2. [2] G. Alberti and L. Ambrosio, A geometric approach to monotone functions in ℝ n , Math. Z. 230 (1999), 259-316. Zbl0934.49025MR1676726
  3. [3] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227-260. Zbl1075.35087MR2096794
  4. [4] J.-P. Aubin and A. Cellina, “Differential Inclusions”, Springer-Verlag, 1984. Zbl0538.34007MR755330
  5. [5] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 42 (1977), 337-403. Zbl0368.73040MR475169
  6. [6] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal. 157 (2001), 75-90. Zbl0979.35032MR1822415
  7. [7] F. Bouchut and L. Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients, Differential Integral Equations 14 (2001), 1015-1024. Zbl1028.35042MR1827101
  8. [8] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), 891-933. Zbl0989.35130MR1618393
  9. [9] F. Bouchut and F. James, Differentiability with respect to initial data for a scalar conservation law, In: “Hyperbolic problems: theory, numerics, applications”, Vol. I (Zürich, 1998), 113–118, Internat. Ser. Numer. Math., Vol. 129, Birkhäuser, Basel, 1999. Zbl0928.35097MR1715739
  10. [10] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations 24 (1999), 2173-2189. Zbl0937.35098MR1720754
  11. [11] E. D. Conway, Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. Math. Anal. Appl. 18 (1967), 238-251. Zbl0163.12103MR206474
  12. [12] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097-1119. Zbl0377.35051MR457947
  13. [13] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. Zbl0696.34049MR1022305
  14. [14] A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Transl. (2) 42 (1964), 199-231. Zbl0148.33002
  15. [15] A. F. Filippov, Differential equations with discontinuous righthand sides, In: “Coll. Mathematics and Its Applications”, Kluwer Academic Publishers Dordrecht-Boston-London, 1988. Zbl0664.34001MR1028776
  16. [16] E. Godlewski, M. Olazabal and P.-A. Raviart, On the linearization of hyperbolic systems of conservation laws. Application to stability, In:“Équations aux dérivées partielles et applications”, articles dédiés à J.-L. Lions, Gauthier-Villars, Paris, 1998, 549-570. Zbl0912.35103MR1648240
  17. [17] E. Godlewski, M. Olazabal and P.-A. Raviart, On the linearization of systems of conservation laws for fluids at a material contact discontinuity, J. Math. Pures Appl. (9) 78 (1999), 1013-1042. Zbl0968.76073MR1732051
  18. [18] M. Hauray, On two-dimensional Hamiltonian transport equations with L loc p coefficients, Ann. Inst. H. Poincaré Anal. Non Lin. 20 (2003), 625-644. Zbl1028.35148MR1981402
  19. [19] D. Hoff, The sharp form of Oleinik’s entropy condition in several space variables, Trans. Amer. Math. Soc. 276 (1983), 707-714. Zbl0528.35062MR688972
  20. [20] F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law, SIAM J. Control Optim. 37 (1999), 869-891. Zbl0970.35161MR1680830
  21. [21] B. L. Keyfitz and H. C. Krantzer, A strictly hyperbolic system of conservation laws admitting singular shocks, In :“Nonlinear Evolution Equations that Change Type”, IMA Volumes in Mathematics and its Applications, Vol. 27, 1990, Springer-Verlag, New-York, 107-125. Zbl0718.76071MR1074189
  22. [22] P. LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, In: “Nonlinear Evolution Equations that Change Type”, IMA Volumes in Mathematics and its Applications, Vol. 27, 1990, Springer-Verlag, New-York, 126-138. Zbl0727.35083MR1074190
  23. [23] C. T. McMullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8 (1998), 304-314. Zbl0941.37030
  24. [24] O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95-172. Zbl0131.31803
  25. [25] B. Popov and G. Petrova, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), 1849-1873. Zbl0992.35104
  26. [26] B. Popov and G. Petrova, Linear transport equations with μ -monotone coefficients, J. Math. Anal. Appl. 260 (2001), 307-324. Zbl0980.35003
  27. [27] F. Poupaud and M. Rascle, Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Partial Differential Equations 22 (1997), 337-358. Zbl0882.35026
  28. [28] T. Qin, Symmetrizing nonlinear elastodynamics system, J. Elasticity 50 (1998), 245-252. Zbl0919.73015MR1651340
  29. [29] Y. Reshetnyak, On the stability of conformal mappings in multidimensional spaces, Sibirsk. Mat. Zh. 8 (1967), 91-114. Zbl0172.37801MR209469
  30. [30] Y. Reshetnyak, Stability theorems for mappings with bounded excursions, Sibirsk. Mat. Zh. 9 (1968), 667-684. Zbl0176.03503MR230901
  31. [31] D. H. Wagner, Conservation laws, coordinate transformations, and differential forms, In: “Hyperbolic Problems: Theory, Numerics, Applications”, J. Glimm, J. W. Grove, M. J. Graham, B. J. Plohr (Eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1996, 471-477. Zbl0923.35096MR1446061
  32. [32] Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data, Comm. Pure Appl. Math. 47 (1994), 1365-1401. Zbl0809.35088MR1295933

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