HYPERBOLIC BALANCE LAWS WITH ENTROPY ON A CURVED SPACETIME: THE WEAK–STRONG UNIQUENESS THEORY

Abstract

We are interested in nonlinear partial differential equations of hyperbolic type, written as first-order balance laws posed on a curved spacetime, whose unknown is typically a section of the bundle of future-oriented, timelike vectors. We are especially interested in dealing with systems and solutions with low regularity and, to this end, we introduce here the notion of "strictly convex entropy field". We then prove that a system of balance laws endowed with such an entropy admits a symmetrization which, however, may be "degenerate" if the entropy is not uniformly convex or if the constitutive laws defining the balance laws have limited regularity. Next, we establish a weak–strong uniqueness and stability theorem for the initial value problem associated with balance laws endowed with a strictly convex entropy field. We compare two solutions with limited regularity: on the one hand, a continuous solution that need not be Lipschitz continuous and, on the other hand, a weak solution that satisfies an "entropy inequality". Finally, we apply our theory to the Euler system for compressible fluid flows on a curved spacetime, and we exhibit an entropy field, which is strictly convex but fails to be uniformly convex as vacuum is approached. This leads us to a uniqueness theorem for (both the relativistic and non-relativistic versions of) the Euler system, which applies to a continuous solution with vacuum and an entropy weak solution with vacuum.