**Title:** ODE invariants and 2d Hamiltonian vector fields

**Speaker:** Professor Stefano Bianchini

**Speaker Info:** SISSA-ISAS, Trieste, Italy

**Brief Description:**

**Special Note**:

**Abstract:**

We study the ODE\[ \dot x = b(t,x) \]

where $b$ is an $L^\infty(\R^d,\R^d)$ vector field with bounded divergence. We assume that there is a Lipschitz function $H : \R^d \to \R^{d-k}$ such that $ nabla H \cdot b = 0$, i.e. it is invariant for the flow.

The main result is a contruction to reduce the ODE on each connected component of the level sets of $H$, assuming a condition on the function $H$ which resembles the Sard condition.

In the case $d = 2$, $H : \R^2 \to \R^1$ our weak Sard condition is equivalent to the well posedness of the flow.

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