Linear transport equations with discontinuous coefficients. (English) Zbl 0992.35104

From the introduction: We consider the one-dimensional homogeneous linear transport equation \[ u_t(t,x)+ a(t,x)u_x=0, \quad (t,x)\in \Omega:= [0,T]\times \mathbb{R},\tag{1} \] with initial condition \[ u(0,x)= u^0(x),\tag{2} \] where the velocity \(a\) is bounded and satisfies the one-sided Lipschitz condition \[ \bigl(a(t,x) -a(t,y)\bigr) (x-y)\geq-m (t)(x-y)^2,\quad m\in L^1 [0,T]. \] Here the coefficient \(a\) is possibly discontinuous and the theory of R. DiPerna and P. Lions cannot be applied.
Our main result is to introduce an entropy condition which selects a unique solution from the set of continuous weak solutions to (1), (2). This entropy solution has certain minimality properties and can also be described by using the Filippov flow for the characteristic equation. When \(u_0\in \text{Lip}1\), our entropy condition is equivalent to the ones introduced by Bouchut and James in the case of Lipschitz initial conditions.


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


[1] Bouchut F, Nonlinear Analysis, Theory, Methods and Applications 32 (7) pp 891– (1998) · Zbl 0989.35130
[2] Conway E.D, J . of Math. Anal. and Appl 18 (7) pp 238– (1967) · Zbl 0163.12103
[3] DeVore R, Acta Numerica 18 (7) pp 1– (1998)
[4] Devore R, Grundlehren 303 (7) (1993)
[5] Diperna R, Transport Theoryand Sobolev Spaces, Invent. Math 98 (7) pp 511– (1989) · Zbl 0696.34049
[6] Capuzzo Dolcetta I, Advances in Math. Sci. and Appl 6 (7) pp 689– (1996)
[7] Filippov A.F, Side, A.M.S. Transl 42 (2) pp 199– (1964)
[8] James F. Sepdveda M. Convergence results for the flux identification in a scalar conservation law Université d’Orléans 1994
[9] Peller V, applications, Mat. Sbornik 122 (2) pp 481– (1980)
[10] Petrushev, P and popov, V. 1987. ”Rational Approximation of Real Functions”. Cambridge: Cambridge University Press. · Zbl 0644.41010
[11] Poupand F, Coeficients, Comm. 22 pp 337– (1997)
[12] Tadmor E, Equations, SIAM J . Numer. Anal 28 pp 891– (1991) · Zbl 0732.65084
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.