Large time behavior of solutions to the compressible Navier-Stokes equation in an infinite layer.

*(English)*Zbl 1151.35072The object of the paper is well expressed in its title. The infinite layer is \(\Omega\subset\mathbb{R}^n\) and it is defined as \(\Omega= \{x= (x', x_n)\); \(x'= (x_1,\dots, x_{n-1})\in \mathbb{R}^{n-1}\), \(0< x_n< a\}\), \(n\geq 2\). The boundary problem is formulated in the following terms: solve the system of equations \(\partial_t\rho+ \text{div}(\rho v)= 0\), \(\partial_t(\rho v)+ \text{div}(\rho v\otimes v)+ \nabla P(\rho)= \mu\Delta v+ (\mu+ \mu')\nabla\text{\,div\,}v\), \(v|_{x_n= 0,a}= 0\), \(\rho|_{t= 0}= \rho_0(x)\), \(v|_{t= 0}= v_0(x)\), where: \(\rho= \rho(x,t)\) is an unknown density, \(v= (v^1(x,t),\dots, v^n(x, t))\) is an unknown velocity at time \(t\geq 0\) and \(x\in\Omega\), \(P= P(\rho)\) is the pressure, \(\mu\) and \(\mu'\) are the viscosity coefficients satisfying the conditions \(\mu> 0\), \((2/n)\mu+ \mu'\geq 0\); the notation \(\text{div}(\rho v\otimes v)\) means that its \(j\)th component is given by \(\text{div}(\rho v^j v)\).

This problem was previously studied by other authors, minutely presented in the introduction of the paper. The author studies the above formulated boundary problem in the case of large time behaviour of solutions, when the initial value \((\rho_0, v_0)\) is sufficiently close to a given constant state \((\rho_*,0)\), where \(\rho_*\) is a given positive number.

The main result of the paper is contained in two theorems. The first of them gives the conditions in which the above boundary problem has a unique global solution for all \(t\geq 0\) and its limit when \(t\to\infty\) is zero. The second theorem gives the magnitude orders of the \(\| u(t)\|_p\), \(\|\partial_x u(t)\|_p\) and of the \(\| u(t)- u^+(t)\|_p\) for any \(1\leq p<\infty\); \(u(t)= (\rho(t)- \rho_*, v(t))\). For the proof of the first theorem, the author gives only some summary indications, but for the proof of the second theorem he assigns near the content of the entire paper. The paper ends with an appendix where the author gives the proof of estimates for the solutions of the Stokes problem, used previously in the proof the energy estimates.

This problem was previously studied by other authors, minutely presented in the introduction of the paper. The author studies the above formulated boundary problem in the case of large time behaviour of solutions, when the initial value \((\rho_0, v_0)\) is sufficiently close to a given constant state \((\rho_*,0)\), where \(\rho_*\) is a given positive number.

The main result of the paper is contained in two theorems. The first of them gives the conditions in which the above boundary problem has a unique global solution for all \(t\geq 0\) and its limit when \(t\to\infty\) is zero. The second theorem gives the magnitude orders of the \(\| u(t)\|_p\), \(\|\partial_x u(t)\|_p\) and of the \(\| u(t)- u^+(t)\|_p\) for any \(1\leq p<\infty\); \(u(t)= (\rho(t)- \rho_*, v(t))\). For the proof of the first theorem, the author gives only some summary indications, but for the proof of the second theorem he assigns near the content of the entire paper. The paper ends with an appendix where the author gives the proof of estimates for the solutions of the Stokes problem, used previously in the proof the energy estimates.

Reviewer: Vasile Ionescu (Bucureşti)

##### MSC:

35Q30 | Navier-Stokes equations |

76N15 | Gas dynamics, general |

35B40 | Asymptotic behavior of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |