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

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.

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
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