Unconditional stability of stationary flows of compressible heat-conducting fluids driven by large external forces (Q1303801)

The authors discuss the long-time behaviour of a one-dimensional version of the Navier Stokes equation which involves five scalars \(\rho,u,p,e,q\) with \(\rho\) the fluid density, \(p\) the pressure, \(e\) the specific internal energy and \(q\) the heat flux. Set \(D=\rho (e+{1\over 2} u^2)\). The system in question is: \[ \rho_t+ (\rho u)_x=0, \quad (\rho u)_t +(\rho u^2)_x-\nu u_{xx}+ p_x=\rho f, \quad D_t+ (uD+up)_x-\nu (uu_x)_x+ q_x=\rho fu. \tag{1} \] Here \(\nu\) is the viscosity and \(f=f(x)\) an outer force. There are two constitutive laws: \[ p(\rho,e)= (\gamma-1) \rho e,\quad q(e,e_x)= -k(e)e_x,\tag{2} \] with \(k(z)\) a given function and \(\gamma>1\) the adiabatic constant. One assumes: \[ u(t,0)=u(t,1)= q(t,0)= q(t,1)=0,\;t>0.\tag{3} \] There is an equilibrium solution \(\rho_s(x)\), \(u_s\equiv 0\) whose explicit form is omitted here for simplicity. The aim of the paper is the proof of Theorem 1.1, whose simplified form is as follows: Theorem. Let \(f\) be bounded and Hölder continuous on \((0,1)\). Let \(k(z)\geq k_0>0\) for \(z>0\) and \(\lim k(z)=\infty\) as \(z\to\infty\). Let \(\rho>0\), \(e>0\), \(u\) be classical solutions of (1)--(3). Then there exist constants \(\rho_0,\rho_1\) such that \(0<\rho_0\leq \rho(t,x)\leq\rho_1\) for \(t>0\), \(x\in (0,1)\). In addition we have: (a) \(\rho\to\rho_s\) strongly in \(L^1 (0,1)\) as \(t\to\infty\), (b) \(u\to 0\) strongly in \(L^2(0,1)\) as \(t\to \infty\). The proof is based on a series of delicate estimates which are obtained by exploiting the specific structure of the system.