On Outer Pressure Problem of Compressible Navier-Stokes System with Degenerate Heat-Conductivity in Unbounded Domains
Manyu Liu, Yanfang Peng, Zhilun Peng
TL;DR
This work addresses global well-posedness and long-time behavior for the 1D compressible Navier–Stokes system with degenerate heat conductivity in unbounded domains under an outer pressure boundary condition. The authors deploy energy methods and refined a priori estimates to obtain time-uniform bounds for the specific volume $v$ and temperature $\theta$ in the presence of degeneracy $\kappa=\tilde{\kappa}\theta^{\beta}$ with $\beta>0$, and to manage the boundary influence. For $\gamma=0$, $\beta>0$, they establish the existence and uniqueness of a global strong solution with $(v-1,u,\theta-1)\in L^{\infty}(0,\infty;H^{1}(\Omega))$ and $u_x,\theta_x\in L^{2}(0,\infty;L^{2}(\Omega))$, and prove that $\|(v-1,u,\theta-1)(t)\|_{L^{p}(\Omega)}\to 0$ as $t\to\infty$ for $p>2$. This extends prior results in the nondegenerate setting to degenerate heat conduction on unbounded domains and demonstrates global stability of the outer-pressure problem.
Abstract
The compressible Navier-Stokes system with the constant viscosity and the nonlinear heat conductivity which is proportional to a positive power of the temperature and may be degenerate is considered. Under the outer pressure boundary conditions in one-dimensional unbounded spatial domains, the global existence of the strong solutions is obtained after proving that both the specific volume and temperature are bounded from below and above independently of time and space. Moreover, the asymptotically stability of global solutions is established as time tends to infinity.