Compensated integrability. Applications to the Vlasov--Poisson equation and other models in mathematical physics

TL;DR

This paper advances the theory of divergence-free positive symmetric tensors, revealing their unique integrability properties, and applies these insights to models in mathematical physics including Maxwell's equations and the Vlasov--Poisson system.

Contribution

It refines the analysis of divergence-free positive symmetric tensors and demonstrates their relevance to various physical models, notably linking singularities to Minkowski's problem.

Findings

Determinant is the only quantity with improved integrability among DPTs.

Singular DPTs relate to Minkowski's problem for convex bodies.

Vlasov--Poisson equation fits within the developed framework.

Abstract

We extend our analysis of divergence-free positive symmetric tensors (DPT) begun in a previous paper. On the one hand, we refine the statements and give more direct proofs. Next, we study the most singular DPTs, and use them to prove that the determinant is the only quantity that enjoys an improved integrability. Curiously, these singularities are intimately related to the Minkowski's Problem for convex bodys with prescribed Gaussian curvature. We then cover a list of models of mathematical physics that display a divergence-free symmetric tensor ; the most interesting one is probably that of nonlinear Maxwell's equations in a relativistic frame. The case of the wave equation is the occasion to highlight the role of the positivity assumption. Last, but not least, we show that the Vlasov--Poisson equation for a plasma is eligible for our theory.