Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 1, p. 1-25

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.

Classification:  35F10,  34A36,  35D05,  35B35
@article{ASNSP_2005_5_4_1_1_0,
author = {Bouchut, Francois and James, Francois and Mancini, Simona},
title = {Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 4},
number = {1},
year = {2005},
pages = {1-25},
zbl = {1170.35363},
mrnumber = {2165401},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2005_5_4_1_1_0}
}

Bouchut, Francois; James, Francois; 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, Serie 5, Volume 4 (2005) no. 1, pp. 1-25. http://www.numdam.org/item/ASNSP_2005_5_4_1_1_0/

[1] G. Alberti, Rank-one properties for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh, Sect. A 123 (1993), 239-274. | MR 1215412 | Zbl 0791.26008

[2] G. Alberti and L. Ambrosio, A geometric approach to monotone functions in ${ℝ}^{n}$, Math. Z. 230 (1999), 259-316. | MR 1676726 | Zbl 0934.49025

[3] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), 227-260. | MR 2096794 | Zbl 1075.35087

[4] J.-P. Aubin and A. Cellina, “Differential Inclusions”, Springer-Verlag, 1984. | MR 755330 | Zbl 0538.34007

[5] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 42 (1977), 337-403. | MR 475169 | Zbl 0368.73040

[6] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal. 157 (2001), 75-90. | MR 1822415 | Zbl 0979.35032

[7] F. Bouchut and L. Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients, Differential Integral Equations 14 (2001), 1015-1024. | MR 1827101 | Zbl 1028.35042

[8] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), 891-933. | MR 1618393 | Zbl 0989.35130

[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. | MR 1715739 | Zbl 0928.35097

[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. | MR 1720754 | Zbl 0937.35098

[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. | MR 206474 | Zbl 0163.12103

[12] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097-1119. | MR 457947 | Zbl 0377.35051

[13] R. J. Diperna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. | MR 1022305 | Zbl 0696.34049

[14] A. F. Filippov, Differential equations with discontinuous right-hand side, Amer. Math. Soc. Transl. (2) 42 (1964), 199-231. | Zbl 0148.33002

[15] A. F. Filippov, Differential equations with discontinuous righthand sides, In: “Coll. Mathematics and Its Applications”, Kluwer Academic Publishers Dordrecht-Boston-London, 1988. | MR 1028776 | Zbl 0664.34001

[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. | MR 1648240 | Zbl 0912.35103

[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. | MR 1732051 | Zbl 0968.76073

[18] M. Hauray, On two-dimensional Hamiltonian transport equations with ${L}_{\mathrm{loc}}^{p}$ coefficients, Ann. Inst. H. Poincaré Anal. Non Lin. 20 (2003), 625-644. | Numdam | MR 1981402 | Zbl 1028.35148

[19] D. Hoff, The sharp form of Oleinik's entropy condition in several space variables, Trans. Amer. Math. Soc. 276 (1983), 707-714. | MR 688972 | Zbl 0528.35062

[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. | MR 1680830 | Zbl 0970.35161

[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. | MR 1074189 | Zbl 0718.76071

[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. | MR 1074190 | Zbl 0727.35083

[23] C. T. Mcmullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8 (1998), 304-314. | Zbl 0941.37030

[24] O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95-172. | Zbl 0131.31803

[25] B. Popov and G. Petrova, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), 1849-1873. | Zbl 0992.35104

[26] B. Popov and G. Petrova, Linear transport equations with $\mu$-monotone coefficients, J. Math. Anal. Appl. 260 (2001), 307-324. | Zbl 0980.35003

[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. | Zbl 0882.35026

[28] T. Qin, Symmetrizing nonlinear elastodynamics system, J. Elasticity 50 (1998), 245-252. | MR 1651340 | Zbl 0919.73015

[29] Y. Reshetnyak, On the stability of conformal mappings in multidimensional spaces, Sibirsk. Mat. Zh. 8 (1967), 91-114. | MR 209469 | Zbl 0172.37801

[30] Y. Reshetnyak, Stability theorems for mappings with bounded excursions, Sibirsk. Mat. Zh. 9 (1968), 667-684. | MR 230901 | Zbl 0176.03503

[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. | MR 1446061 | Zbl 0923.35096

[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. | MR 1295933 | Zbl 0809.35088