Strichartz Estimates and Moment Bounds for the Relativistic Vlasov–Maxwell System

Abstract

We consider the relativistic Vlasov–Maxwell system with data of unrestricted size and without compact support in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, Glassey–Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the ${L^\infty_x}$ norms of the electromagnetic fields compared to Glassey and Schaeffer (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three-dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as ${{\|p_0^{\theta} f \|_{L^{q}_{x}L^1_{p}}}}$ remains bounded for ${\theta > \frac{2}{q}}$, ${2 < q \leqq \infty}$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).