## Introduction to the mathematical theory of compressible flow.(English)Zbl 1088.35051

Oxford Lecture Series in Mathematics and its Applications 27. Oxford: Oxford University Press (ISBN 0-19-853084-6/hbk). xx, 506 p. (2004).
The monograph is a detailed description of mathematical models of Newtonian compressible fluids in the steady and unsteady regime.
The contents of the book may be divided into three parts. The first part (chapters 1, 3 and 6) surveys the basic mathematical tools used in the other parts of the book. The second part contains the mathematical investigation of the Euler system for inviscid compressible fluids. \begin{aligned} &\frac{\partial \rho}{\partial t} +\text{div}\,(\rho\bar{v})=0\\ &\frac{\partial}{\partial t}\,(\rho\bar{v})+\text{div}\,(\rho\bar{v}\otimes\bar{v})+\nabla p=0,\tag{1}\\ &\frac{\partial E}{\partial t}+\text{div}\,(\bar{v}(E+p))=0. \end{aligned}
Here $$E=\rho(\frac{| \bar{v}| }{2}+e)$$ is the total energy, $$x\in \mathbb R^n$$, $$n\geq1$$, $$>0$$, $$\rho(x,t)$$ is the density, $$\bar{v}$$ the velocity, $$p$$ is the pressure and $$e$$ the internal energy. $$e,\;\rho$$ and $$p$$ are connected by an appropriate constitutive equation $e=e(\rho,p).$ The Cauchy problem to the system (1) with initial condition $\rho(x,0)=\rho_0(x),\quad \bar{v}(x,0)=\bar{v}_0(x)\tag{2}$ is the subject of chapter 2. This problem is considered as Cauchy problem to a nonlinear hyperbolic system. Local and global existence results are proved. Few particular cases (including the Riemann problem) are analyzed in detail.
The third part of the monograph (chapters 4, 5, 7, 8, 9) is the largest one. This part is devoted to the steady and unsteady Navier-Stokes systems for compressible fluids. Steady compressible Navier-Stokes equations in the barotropic regime is the subject of the chapters 4 and 5.
\begin{aligned} &\sum\limits_{j=1}^3\frac{\partial}{\partial x_j}(\rho\bar{v}v_j)-\mu\Delta\bar{v} +(\mu+\lambda)\nabla\text{div}\,\bar{v}+\nabla p(\rho)=\rho \bar{f}+g\quad \text{in}\;\Omega,\\ &\text{div}\,(\rho\bar{v})=0\quad \text{in}\;\Omega,\tag{3}\\ &\bar{v}(x)=0,\quad x\in\partial\Omega, \end{aligned}
$$p(\rho)$$ is the prescribed function $p(\rho)=\rho^{\gamma}.\tag{4}$ Here $$\Omega$$ is a bounded or unbounded domain of $$\mathbb R^3$$. The behaviour of $$(\rho,\,\bar{v})$$ is described at infinity for unbounded $$\Omega$$
$\bar{v}(x)\to 0,\quad \rho(x)\to\rho_{\infty}= \text{const}>0\quad \text{as}\;| x| \to\infty .$ The constants $$\mu$$ and $$\lambda$$ satisfy the inequalities $\mu>0,\quad \lambda+\frac23\,\mu\geq 0.\tag{5}$ The existence of a weak solution to the problem (3) is established in chapter 4 without the assumption of small data. The existence of a regular solution is proved in chapter 5 for small data.
The unsteady Navier-Stokes equations of compressible barotropic flow are studied in chapters 7, 8, 9. \begin{aligned} &\frac{\partial}{\partial t}\,(\rho\bar{v})+\sum\limits_{j=1}^3\frac{\partial}{\partial x_j}(\rho\bar{v}v_j)-\mu\Delta\bar{v} +(\mu+\lambda)\nabla\text{div}\,\bar{v}+\nabla p(\rho)=\\ &=\rho \bar{f}+g\quad \text{in}\;\Omega_T=\Omega\times(0,T),\\ &\frac{\partial \rho}{\partial t} +\text{div}\,(\rho\bar{v})=0\quad \text{in}\;\Omega_T\tag{6}\\ &\bar{v}(x,t)=0\quad \text{on}\;\partial\Omega\times(0,T),\quad \rho(x,0)=\rho_0(x),\;\bar{v}(x,0)=\bar{v}_0(x), x\in\Omega. \end{aligned} Equation (4) and inequalities (5) are assumed. The existence of a weak solution to problem (6) is studied in chapter 7 both for bounded and unbounded domains $$\Omega$$. The cases of nonmonotone pressure laws, nonsmooth boundaries, nonhomogeneous boundary conditions are discussed.
The global in time behaviour of solutions to the problem (6) is investigated in chapter 8. The existence of a global in time strong solution to the problem (6) in a bounded domain $$\Omega$$ is established in chapter 9 for sufficiently small initial data.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q30 Navier-Stokes equations 35L60 First-order nonlinear hyperbolic equations 35Q05 Euler-Poisson-Darboux equations