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 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.

On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids

“…Fan et al [17] established the existence of global-in-time weak solution to the resistive planar MHD system with zero shear viscosity. Recently, Gwiazda et al [23] obtained a sufficient condition for the energy conservation for weak solutions to inviscid non-resistive compressible MHD equations.…”

Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws

“…Since in the formulas (2.10) and then (2.13) only the second derivative of U → G(U ) appears, such formulas are trivial when this function is affine. Therefore the Corollaries 4.1 -4.3 of [21] transfer directly to the present situation giving the following results which will be used in Sections 5.…”

“…Moreover, the functions U ∈ B 1/3 3,VMO are characterized by a simple property in the physical space which makes this space well adapted to localized formulation of an extra conservation law. This makes this space a good tool to deal with the case of domains with boundary, extending the results of [4,5,14,21] and in particular relaxing the Hölder α > 1 3 regularity hypothesis. At the end of the day the use of the Besov-V MO space leads to a very concise proof of our main theorem (see formulas (3.6) and (3.5) in the proof of Theorem 3.1).…”

“…To apply the previous lemma to systems of (not necessarily hyperbolic) conservation laws in the full generality of [21], we consider the problem: div X (G(U (X ))) = 0 for X ∈ X .…”

“…The study of such more general nonlinearities was initiated in [18], where the isentropic compressible Euler equations (for which the density appears in a non-quadratic way) were studied, but only in the absence of physical boundaries. The general framework for conservation laws of the type (1.1) was introduced in [21], and considered for bounded domains (but not in the optimal functional setting) in [5].…”