A New Formulation of the 3D Compressible Euler Equations with Dynamic Entropy: Remarkable Null Structures and Regularity Properties

Abstract

We derive a new formulation of the 3D compressible Euler equations exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work here is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity (defined to be vorticity divided by density), transport equations for the entropy and its gradient, and some additional transport–divergence–curl-type equations involving special combinations of the derivatives of the solution variables. The good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the “wave part” of the system. In a forthcoming application, we will use the new formulation to give a sharp, constructive proof of finite-time shock formation, tied to the intersection of acoustic “wave characteristics,” for solutions with nontrivial vorticity and entropy at the singularity. In the present article, we derive the new formulation and provide an overview of the central role that it plays in the proof of shock formation. Although the equations are significantly more complicated than they are in the barotropic case, they enjoy many of same remarkable features, including: (i) all derivative-quadratic inhomogeneous terms are null forms relative to the acoustical metric, which is the Lorentzian metric driving the propagation of sound waves and (ii) the transport–divergence–curl-type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity, assuming that the initial data enjoy the same gain in regularity. This represents a gain of one derivative compared to standard estimates. This gain of a derivative, which seems to be new for the entropy, is essential for closing the energy estimates in our forthcoming proof of shock formation, since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations.