Global Stability and Non-Vanishing Vacuum States of the 3D Full Compressible Navier–Stokes equations

We are concerned with the global stability and non-vanishing vacuum states of large strong solutions to the full compressible Navier–Stokes equations on the torus $${\mathbb {T}}^3$$ , and the main goal of this work is twofold. First, it is shown that the global strong solutions converge to an equilibrium state exponentially in $$L^2$$ in the presence of vacuum provided that the density $$\rho $$ and the temperature $$\theta $$ are bounded uniformly in $$L^\infty $$ . This improves the previous related works in (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 37 (2020), no. 2, 457–488) and (J. Math. Fluid Mech., 24 (2022), no. 2, Paper No. 31), where both $$\rho (x, t)$$ and $$\theta (x, t)$$ possess uniform-in-time positive lower and upper bounds, and $$\rho (x,t)$$ is bounded uniformly in the Hölder space $$C^\alpha $$ for some $$0<\alpha <1$$ . Moreover, we remove the extra restriction $$2\mu >\lambda $$ in their results. Second, by employing some new ideas, we show that the density and temperature converge to their equilibrium states exponentially in the $$L^\infty $$ -norm if additionally the initial density has positive lower bound, which extends the isentropic case in (SIAM J. Math. Anal., 55 (2023), no. 2, 882–899) to the non-isentropic case. As a by-product, we get that the vacuum state will persist for any time as long as the initial density contains vacuum.