Three new results on continuation criteria for the 3D relativistic Vlasov-Maxwell system

We consider continuation criteria for the three-dimensional relativistic Vlasov-Maxwell system. When the particle density, f(t,x,p), is compactly supported at t=0, we prove ‖p0185r−1+βf‖Lt∞LxrLp1≲1, where 1≤r≤2 and β>0 is arbitrarily small, is a continuation criteria. Our continuation criteria is an improvement in the 1≤r≤2 range to the previously best known criteria ‖p04r−1+βf‖Lt∞LxrL1p≲1 due to Kunze [7]. We also consider continuation criteria when f(0,x,p) has noncompact support. In this regime, Luk-Strain [9] proved that ‖p0θf‖Lx1Lp1≲1 is a continuation criteria for θ>5. We improve this result to θ>3. Finally, we build on another result by Luk-Strain [8]. The authors proved boundedness of momentum support on a fixed two-dimensional plane is a sufficient continuation criteria. We prove the same result even if the plane varies continuously in time.