Large data global solution of the 3D RVM system with cylindrical symmetry I: Iterative smoothing scheme

This is the first part of a two-paper sequence establishing the global existence of $3D$ relativistic Vlasov-Maxwell system (RVM) for arbitrarily large smooth localized initial data with cylindrical symmetry. This paper employs Part II's pointwise estimates, developed independently of Part I, as fundamental tools to demonstrate the iterative smoothing scheme (ISS), which is originated and inspired by the work of Klainerman-Staffilani(2002). Using ISS and a novel singular weighted space-time estimate for the distribution function, we prove upper bounds for the projection and full velocity characteristics via a standard bootstrap argument. Using the classic momentum method, we find that the high order momentum at most grows polynomially in time. This further implies that the $L^\infty_{x}$-norm of the electromagnetic field, $\|(E(t),B(t)\|_{L^\infty_x}$, also grows at most polynomially over time. Consequently, this verifies the continuation criteria obtained by Luk-Strain (2014). As a result, the energy function of $3D$ RVM system doesn't blow up in finite time, thereby establishing global existence.