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

Bouchut, Francois; James, Francois; Mancini, Simona

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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

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

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

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

[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 | Zbl

[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 | Zbl