An infinite family of conservation laws for perfect compressible fluids

Abstract

We prove the existence of an infinite family of conservation laws and, more generally, balance laws, satisfied by plane-symmetric flows of perfect compressible fluids, allowing all components of the velocity vector to be possibly non-vanishing. We cover general pressure laws satisfying solely hyperbolicity and asymptotic conditions. We exhibit an almost quadratic formulation of the Euler equations, which we refer to as the J-formulation, and introduce a reduced Euler system based on an effective mass-energy density and an effective pressure law which account for transverse velocities. We establish the existence of a fundamental kernel which we use to generate all spacetime (quasi-)currents, also called mathematical entropy pairs. We derive explicit formulas for their density and flux components, together with the source-terms of the corresponding balance laws. Furthermore, we establish regularity and asymptotic estimates covering the whole range of physical variables, including vacuum. We prove that the kernel is positive on the interior of its support provided the pressure law satisfies a genuine nonlinearity condition, and we also investigate the future-convexity of the spacetime currents. Throughout, the fluid momentum parallel to the orbits of plane-symmetry is kept arbitrary. In turn, this family of spacetime quasi-currents provides a key ingredient in our companion study of torus-symmetric Einstein-Euler flows with weak regularity.