Strict positivity of density in weak solutions

Prove strict positivity of the fluid density in the context of weak solutions of the compressible magnetohydrodynamics system.

Background

The paper imposes non-negativity of the density throughout its weak-solution analysis. It explicitly distinguishes this established requirement from the substantially more difficult question of strict positivity, which is identified as a major open problem in the theory of weak solutions.

References

In fact, the strict positivity of $$ in the context of weak solutions is a major open problem.

— On the existence of a weak martingale solution for a stochastic magnetohydrodynamics system with noise acting in the magnetic field  (2609.29747 - Púček, 24 Sep 2026) in Section 1, Introduction to the classical formulation of the problem