Global existence of large strong solutions to the 1D Brenner–Navier–Stokes–Fourier system

Establish global-in-time existence of large strong solutions to the one-dimensional Brenner–Navier–Stokes–Fourier system in Lagrangian mass coordinates, as defined by equation (1.10) with temperature-dependent coefficients (μ(θ), κ(θ), τ(θ)), for general large initial data, producing solutions that belong to the function space X_T specified in the paper (with the stated H^1/H^2 regularity and positivity conditions on v and θ) so that the shift construction for the stability analysis is applicable.

Background

The paper’s main stability and uniqueness theorem for Riemann shocks in the full Euler system is proved in a class of vanishing dissipation limits arising from solutions to the Brenner–Navier–Stokes–Fourier (BNSF) system. To apply the a-contraction with shifts method, the analysis assumes solutions to the BNSF system lie in a strong solution class X_T that ensures sufficient regularity and positivity for the variables and enables the construction of a time-dependent shift via an ODE.

The authors explicitly note that, up to the time of writing, global existence of large strong solutions to the BNSF system with the required properties is not known. They point to an ongoing work for establishing such solutions. Resolving this problem would close a key gap needed to fully ground the stability framework in a rigorous existence theory for the viscous approximations.