A note on weak solutions of conservation laws and energy/entropy conservation

Abstract

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak solutions. We want to answer the question what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality. An archetypal example of such result was derived for the incompressible Euler system by Constantin et al. ([8]) in the context of the seminal Onsager's conjecture. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

References in corpus (1)

• An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

Cited by in corpus (7)

• Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum
• On the Extension of Onsager's Conjecture for General Conservation Laws
• On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations
• The role of density in the energy conservation for the isentropic compressible Euler equations
• Onsager's Conjecture for Subgrid Scale $\alpha$-Models of Turbulence
• An endpoint case of the renormalization property for therelativistic Vlasov-Maxwell system
• Energy conservation in the limit of filtered solutions for the 2D Euler equations