One-dimensional transport equations with discontinuous coefficients. (English) Zbl 0989.35130

From the introduction: This paper is devoted to one-dimensional homogeneous linear transport equations \[ \partial_t u+a(t,x) \partial_x u=0\quad \text{in } ]0,T[ \times\mathbb{R}, \tag{1} \] with \(T>0\) and \(a\) a given bounded coefficient. This equation will be referred to as the nonconservative problem. By differentiating (1) with respect to \(x\), we obtain the conservative problem \[ \partial_t \mu+\partial_x \bigl(a(t,x)\mu\bigr)=0 \quad\text{in }]0, T [\times \mathbb{R},\tag{2} \] with \(\mu= \partial_xu\).
In Section 2, we prove our main lemma which states that conservative and nonconservative equations are equivalent. We also prove two sharp results of uniqueness. Section 3 is concerned with the case of a piecewise continuous \(a\). Finally, Section 4 is devoted to the case where \(a\) satisfies the one-sided Lipschitz condition. In 4.1 we study the backward problem and Lipschitz solutions, in 4.2 we define duality solutions for the forward problem, 4.3 contains more sophisticated results and the relation with the generalized Filippov flow, and 4.4 is devoted to some comments about viscous problems.


35R05 PDEs with low regular coefficients and/or low regular data
35L65 Hyperbolic conservation laws
Full Text: DOI


[1] Kranzer, H.C.; Keyfitz, B.L., A strictly hyperbolic system of conservation laws admitting singular shocks, () · Zbl 0718.76071
[2] Le Floch, P., An existence and uniqueness result for two nonstrictly hyperbolic systems, () · Zbl 0727.35083
[3] Tan, D.; Zhang, T.; Zheng, Y., Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. diff. eq., 112, 1-32, (1994) · Zbl 0804.35077
[4] Zheng, Y.; Majda, A., 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. on pure and appl. math., 47, 1365-1401, (1994) · Zbl 0809.35088
[5] Bouchut, F., On zero pressure gas dynamics, (), 171-190 · Zbl 0863.76068
[6] Grenier, E., Existence globale pour le système des gaz sans pression, C.R. acad. sci. Paris, 321, 171-174, (1995), Serie I · Zbl 0837.35088
[7] Brenier, Y.; Grenier, E., On the model of pressureless gases with sticky particles, ()
[8] E, W.; Rykov, Y.G.; Sinai, Y.G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. math. phys., 177, 349-380, (1996) · Zbl 0852.35097
[9] James, F.; Sepúlveda, M., Convergence results for the flux identification in a scalar conservation law, (1994), Université d’Orléans, Preprint
[10] DiPerna, R.J.; Lions, P.L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. math., 98, 511-547, (1989) · Zbl 0696.34049
[11] Capuzzo Dolcetta, I.; Perthame, B., On some analogy between different approaches to first order PDE’s with nonsmooth coefficients, Advances in math. sci. and appl., 6, 689-703, (1996) · Zbl 0865.35032
[12] Dal Maso, G.; Le Floch, P.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068
[13] Oleinik, O.A., Discontinuous solutions of nonlinear differential equations, Amer. math. soc. transl., 26, 2, 95-172, (1963) · Zbl 0131.31803
[14] Conway, E.D., Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws, J. of math. anal. and appl., 18, 238-251, (1967) · Zbl 0163.12103
[15] Hoff, D., The sharp form of Oleinik’s entropy condition in several space variables, Trans. of the A.M.S., 276, 707-714, (1983) · Zbl 0528.35062
[16] Tadmor, E., Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. numer. anal., 28, 891-906, (1991) · Zbl 0732.65084
[17] Le Floch, P.; Xin, Z., Uniqueness via the adjoint problems for systems of conservation laws, Comm. on pure and applied maths., XLVI, 11, 1499-1533, (1993) · Zbl 0797.35116
[18] Filippov, A.F., Differential equations with discontinuous righthand side, A.M.S. transl., 42, 2, 199-231, (1964) · Zbl 0148.33002
[19] Aubin, J.P.; Cellina, A., Differential inclusions, (1984), Springer-Verlag Berlin
[20] Filippov, A.F., Differential equations with discontinuous righthand sides, () · Zbl 1098.34006
[21] Dafermos, C.M., Generalized characteristics and the structure of solutions of hyperabolic conservation laws, Indiana univ. math. J., 26, 1097-1119, (1977) · Zbl 0377.35051
[22] Tadmor, E.; Tassa, T., On the piecewise smoothness of entropy solutions to scalar conservation laws, Comm. in P.D.E., 18, 1631-1652, (1993) · Zbl 0807.35091
[23] Bouchut, F.; James, F., Equations de transport unidimensionnelles à coefficients discontinus, C.R. acad. sci. Paris, 320, 1097-1102, (1995), Serie I · Zbl 0829.35139
[24] Liu, T.-P.; Pierre, M., Source-solutions and asymptotic behavior in conservation laws, J. diff. eq., 51, 419-441, (1984) · Zbl 0545.35057
[25] Caroff, N., Generalized solutions of linear partial differential equations with discontinuous coefficients, (1991), CEREMADE Paris, Preprint
[26] Serfati, P., Thesis université Paris 6, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.