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| In this work, we prove a new functional inequality of Hardy-Littlewood type for generalized rearrangements of functions.
We then show how this inequality yields a quantitative stability result for dynamical systems that essentially have the important property to preserve the rearrangement and the Hamiltonian. In particular we derive a quantitative stability result for a large class of steady state solutions to Vlasov-Poisson systems by providing a quantitative control of the $L^1$ norm of the perturbation (uniformly in time) by the perturbed Hamiltonian and the $L^1$ norm of the perturbation at initial time . In fact, such non linear stability
has already been recently obtained, but our proof was based in a crucial way on compactness arguments which by construction provide no quantitative
control of the perturbation. We finally investigate the application of our inequality to other contexts such as the relativistic Vlasov-Poisson and 2D-Euler systems. |
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