Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor
The author investigates the compressible Navier-Stokes equations for isothermal fluids \[ \begin{gathered} (\rho u)_t+ \text{div}(\rho u\otimes u)+ \nabla\rho- \text{div\,}P(Du)=\rho f,\quad x\in\Omega,\\ \rho_t+ \text{div}(\rho u)= 0,\quad u(x,t)= 0\quad\text{for }x\in \Omega.\end{gathered}\tag{1} \] Here \(Du\) is the symmetric part of the velocity gradient \(\nabla u\), while \(P\) represents the nonlinear dependence of the stress tensor on the velocity field. Moreover, one is given initial data \(\rho(x,0)= \rho_0(x)\geq 0\), \((\rho u)(x, 0)= q_0(x)\), \(x\in \Omega\). The function spaces, in which weak solutions of (1) are found, are Orlicz spaces \(L_\psi(\Omega)\) defined in terms of a Young function \(\psi\) and its complementary function \(\overline\psi\), and provided with a norm \(\|\;\|_\psi\). The notion of Orlicz space is then extended to vector fields, giving rise to a space \(E_\psi\), with respect to equation (1). Based on this definitions and a series of basic assumptions (A) involving the stress tensor \(P\) among others, a Definition 4.4 of weak solutions to (1) is given. The basic aim of the paper is to prove Theorem 1.1 which states the following. Let \(\rho\), \(u\) be a weak solution of Definition 4.4, let \(f=\nabla g\), \(g\in W^{2\infty}(\Omega)\). Under assumption (A) there is a unique \(\rho_\infty\in L_{\phi_1}(\Omega)\) such that \[ \lim\|\rho(t)- \rho_\infty(t)\|_{\phi_1}= 0,\quad t\to\infty, \] where \(\nabla\rho_\infty= \rho_\infty f\) a.e. in \(\Omega\), and \(\int_\Omega\rho_\infty dx= \int_\Omega \rho_0 dx\), \(\rho_\infty\geq 0\); moreover for any \(a>0\) we have \[ \lim\int^{t+a}_{t-a}\| Du(s)\|^2_{\psi_2} \,ds= 0,\quad \lim\int^{t+a}_{t-a} \int_\Omega\rho|u|^2 \,dx\,ds= 0,\quad t\to\infty. \] Here, \(\phi_1\), \(\psi_2\) are certain Young functions. The proof is based on a large number of lemmas, some of them are really interesting.
compressible Navier-Stokes equations for isothermal fluids
scientific article; zbMATH DE number 2103745