Ordinary differential equations, transport theory and Sobolev spaces

Abstract

Summary.We obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.

Convergent difference schemes for the Hunter–Saxton equation

“…The next lemma contains an improved integrability estimate showing that v ∆x is uniformly bounded in L p for any p ∈ [1,3). This estimate is important, as it prevents v 2 ∆x from exhibiting concentrations as ∆x → 0.…”

“…In all the computations we have used ∆x = 1 · 10 −9 . In Figure 2 we show a contour plot of the computed v(x, t) for (x, t) ∈ [0, 1.1] × [0,3]. We see that the Engquist-Osher scheme produces an approximation which does not seem close either to the conservative or to the dissipative solution.…”

“…Strong convergence of v ∆x is necessary if we want to recover the Hunter-Saxton equation when we take the limit in the finite difference schemes. Strong convergence of v ∆x is obtained by analyzing various renormalizations (in the sense of DiPerna and Lions [3,12,13]) of the numerical schemes and corresponding defect measures. In addition, to prevent v 2 ∆x from exhibiting concentrations as ∆x → 0, we need to derive higher (than L 2 ) integrability estimates for v ∆x .…”

“…Set v ε = v ω ε , w ε = w ω ε , where ω ε is a standard mollifier acting on the spatial variable. Then according to the DiPerna-Lions folklore lemma [3], as well as (3.24) and (3.26), v ε solves…”

“…To make this argument rigorous one appeals to the DiPerna-Lions folklore lemma [3] and the local L 3 estimate on v.…”