Projective Properties of Divergence-Free Symmetric Tensors, and New Dispersive Estimates in Gas Dynamics

Abstract

The class of Divergence-free symmetric tensors is ubiquitous in Continuum Mechanics. We show its invariance under projective transformations of the independent variables. This action, which preserves the positiveness, extends Sophus Lie’s group analysis of Newtonian dynamics. When applied to models of gas dynamics—such as Euler system or Boltzmann equation,—in combination with Compensated Integrability, this yields new dispersive estimates. The most accurate one is obtained for mono-atomic gases. Then the space-time integral of \(t\rho ^\frac{1}{d} p\) is bounded in terms of the total mass and moment of inertia alone.