A FEW REMARKS ON ORDINARY DIFFERENTIAL-EQUATIONS

We present in this note existence and uniqueness results for solutions oi` ordinary differential equations and linear transport equations wi th discontinuous coefficients in a bounded open subset Omega of R(N) o r in the whole space R(N) (N greater than or equal to 1). R.J. Di Pern a and P.L. Lions studied the case of vector fields b with coefficients in Sobolev spaces and bounded divergence. We want to show that simila r results hold for more general b: we assume in the bounded autonomous case that b belongs to W-1,W-1(Omega), b.n = 0 on partial derivative Omega, and that there exists T-0 > 0 such that exp(T-0\div b\) is an e lement of L(1)(Omega). Furthermore, we establish results on transport equations with initial values in L(p) spaces (p > 1).