## A note on weak solutions of conservation laws and energy/entropy conservation.(English)Zbl 1398.35168

What is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? The authors answered the question by extending the approach in [E. Feireisl et al., Arch. Ration. Mech. Anal. 223, No. 3, 1375–1395 (2017; Zbl 1365.35113)] for a general class of conservation laws. 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. This general scenario might be of independent interest for some experts.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35L65 Hyperbolic conservation laws 35Q31 Euler equations 76W05 Magnetohydrodynamics and electrohydrodynamics 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q79 PDEs in connection with classical thermodynamics and heat transfer 35D30 Weak solutions to PDEs

Zbl 1365.35113
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### References:

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