Stability estimates for magnetized Vlasov equations

Abstract: We present two results related to magnetized Vlasov equations. Our first contribution concerns the stability of solutions to the magnetized Vlasov-Poisson system with a non-uniform magnetic field using the optimal transport approach introduced by Loeper [24]. We show that the extra magnetized terms can be suitably controlled by imposing stronger decay in velocity on one of the distribution functions, illustrating how the external magnetic field creates anisotropy in the evolution. This allows us to generalize the classical 2-Wasserstein stability estimate by Loeper [24, Theorem 1.2] and the recent stability estimate using a kinetic Wasserstein distance by Iacobelli [20, Theorem 3.1] to the magnetized Vlasov-Poisson system. In our second result, we extend the improved Dobrushin estimate by Iacobelli [20, Theorem 2.1] to the magnetized Vlasov equation with a uniform magnetic field.