Entropy conditions for quasilinear first order equations on nonlinear fiber bundles with special emphasis on the equation of 2D flat projective structure. I

TL;DR

This paper develops a coordinate-free formulation of entropy inequalities for first-order quasilinear equations, emphasizing the geometric structure of 2D flat projective equations and deriving entropy densities from shock conditions.

Contribution

It introduces an independent, geometric approach to entropy inequalities for quasilinear PDEs, especially for 2D flat projective structures, using projective geometry and Rankine-Hugoniot rules.

Findings

Coordinate-free entropy inequalities formulated using characteristics.

Explicit expression of entropy density for 2D flat projective structure.

Connection between entropy densities and shock conditions via projective geometry.

Abstract

Taking only the characteristics as absolute, in the spirit of Arnold's "Geometrical Methods in the Theory of Ordinary Differential Equations" (Springer, 1988), we give an independent of coordinates formulation of general variational entropy inequalities for quasilinear equations of first order, that locally read as Kruzhkov inequalities, in terms of certain "entropy densities", and in the case of the equation of 2D flat projective structure we get the expression of the general entropy density from its abstract Rankine-Hugoniot rule for shocks using the projective geometry of the plane.