Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1223–1238 | Cite as

A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation

  • Piotr Gwiazda
  • Martin Michálek
  • Agnieszka Świerczewska-Gwiazda
Open Access


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.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Mathematics, Polish Academy of SciencesWarsawPoland
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  3. 3.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland

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