Supersonic Euler-Poisson flows in divergent nozzles (Q6111022)

The paper is concerned with the steady state of an isotropic the Euler-Poisson system \[ \begin{cases} \operatorname{div}(\rho u)=0, \\ \operatorname{div}(\rho u\otimes u)+ \nabla p=-\rho \nabla \Psi, \\ \Delta \Psi =\rho- b, \end{cases} \] here \(\rho\) denotes the density, \( p \) the pressure, \(u\) the velocity field and \(\Delta \) is the Laplace operator. The system is closed by an equation of state \[ p=e^S\rho^\gamma, \qquad \gamma>1, \] and \(S\) is called entropy. The authors study the problem in the two-dimensional plane and assume that the fluid is irrotational, which means that velocity has a potential. They use polar coordinates and construct a radial supersonic solution that serves as a background solution. Then, they consider suitable small perturbations of the radial background solution in angular domains. In this framework, the Euler-Poisson the system is transformed into a second-order quasilinear hyperbolic-elliptic coupled system with nonlinear boundary conditions. The result achieved is a well-posedness of the quasilinear boundary value problem that preserves the fluid's supersonic character. First, the well-posedness of the boundary value problem for the associated linearized hyperbolic-elliptic coupled system is proved via a priori estimates. Then, the proof of the nonlinear structural stability of supersonic flows for the Euler-Poisson system is obtained by an iteration method, and the estimates for linearized hyperbolic-elliptic system. Despite that the equation of state includes the entropy \(S\), the authors refrain from consider it as an independent variable that satisfies an additional conservation law, and which should be included in the system.