Global stability of large solutions to the three-dimensional full compressible Navier–Stokes equations with vacuum

This paper is concerned with large-time behavior of global solutions to the Cauchy problem of three-dimensional (3D) full compressible Navier–Stokes equations with large data and initial vacuum. It is shown that if the Serrin’s type criterion is satisfied, i.e., the quantity ‖ρ‖L∞(0,∞;L∞)+‖u‖L(2r)/(r−3)(0,∞;Lr) is bounded for any 3 < r ≤ ∞, then the problem has a global unique strong solution (ρ, u, θ) on R3×(0,∞). The exponential decay estimates for the lower-higher order norms of both velocity and temperature are also derived. It is worth pointing out that the L2-Lq-norms of the gradient of density with 3 < q < 6 are uniformly bounded for all t ≥ 0, which is in sharp contrast to that in Li et al. [arXiv:2208.11925v1 (2022)] and Li et al. [arXiv:2207.00441v1 (2022)] for the initial-boundary value problem with the Navier’s slip boundary condition for velocity and the thermal-insulated boundary condition for temperature.