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 cases 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 the role of physical admissibility conditions 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 a result was derived for the incompressible Euler system in the context of Onsager’s conjecture in the early nineties. 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.
Highlights
The current decade has been preoccupied with solving the famous conjecture of Onsager, which says that solutions to the incompressible Euler system conserve total kinetic energy as long as they are Hölder continuous with a Hölder exponent α > 1/3, and that otherwise they may dissipate the energy
A sufficient regularity for the energy to be conserved has been established for a variety of models, including the incompressible inhomogeneous Euler system and the compressible Euler in [18], the incompressible inhomogeneous Navier–Stokes system in [24], compressible Navier–Stokes in [27] and equations of magnetohydrodynamics in [6]
Let us comment in more detail on results related to the question of energy conservation for weak solutions of some conservation laws
Summary
The current decade has been preoccupied with solving the famous conjecture of Onsager, which says that solutions to the incompressible Euler system conserve total kinetic energy as long as they are Hölder continuous with a Hölder exponent α > 1/3, and that otherwise they may dissipate the energy. Let us comment in more detail on results related to the question of energy conservation for weak solutions of some conservation laws. Both parts of Onsager’s conjecture for the incompressible inviscid Euler system have been resolved. To the best of our knowledge, this was the first result treating nonlinearity which is not in a multilinear form We extend this approach to a general class of conservation laws of the form (1). We observe that in case we consider hyperbolic systems, the opposite direction of the Onsager’s hypothesis is almost trivial This is a completely different situation to the case of incompressible Euler system, which is not a hyperbolic conservation law and where the construction of solutions dissipating the energy was a challenge.
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