On ordinary differential equations and transport equations. (Sur les équations différentielles ordinaires et les équations de transport.) (French. Abridged English version) Zbl 0919.34028

Lions, Pierre-Louis

C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 7, 833-838 (1998).

The author considers ordinary differential equations associated with divergence free vector fields on the torus \(\Pi^n\) \((n\geq 2):\dot X=b(X)\) for \(t\in\mathbb{R}\), \(X|_{t=0}= x\in\Pi^n\). Observing that it is equivalent to solve the associated transport equation (i.e. Liouville equation) the author shows existence, uniqueness and stability results for generic vector fields in \(L^1\) or for piecewise \(W^{1,1}\) vector fields.

Reviewer: Abdelkader Boucherif (Tlemcen)

MSC:

34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
37G99	Local and nonlocal bifurcation theory for dynamical systems
34C40	Ordinary differential equations and systems on manifolds
37C10	Dynamics induced by flows and semiflows
34A34	Nonlinear ordinary differential equations and systems

Keywords:

ordinary differential equations; divergence free vector fields; almost everywhere flow solution; transport equation; generic vector fields; associated transport equation; Liouville equation