Characterization of initial data admitting absolute entropy maximizers

Characterize the class of initial data for which the DMV solution set \mathcal{U}[_0, v_0, S_0, \mathcal{E}_0] contains an absolute entropy maximizer, i.e., a DMV solution that minimizes the exponentially weighted functional \int_0^\infty e^{-\lambda t} ( \int_{\Omega} [ E(\rho, v, S) - \Theta S ] ) \,dt for all sufficiently large \lambda > 0, where \Theta is the equilibrium temperature determined by the conserved total energy and mass.

Background

The authors propose a single-step selection principle minimizing a strictly convex functional combining total energy and entropy relative to the background equilibrium. If a DMV solution minimizes this functional for all large discount factors \lambda, Theorem TS2 implies it is a weak solution.

The open problem asks for a structural description of initial data sets for which such ‘absolute entropy maximizers’ exist within the DMV solution set, linking the selection principle to weak-solution selection.