On periodic transonic flows of a viscous gas
The equation considered in the paper under review was proposed by \textit{O. S. Ryzhov} [Prikl. Mat. Mekh. 29, 1004-1014 (1965; Zbl 0173.28104)]\ for describing transonic flow of a viscous gas: \[ Au\equiv-\mu u_{xxx}+u_xu_{xx}-\Delta_yu=f(x,y), \] where the parameter \(\mu>0\) characterizes viscosity and thermal conductivity of the gas. The author studies the problem of finding a solution to the equation in the cylinder \(Q=(0,L)\times D\), subject to the conditions \[ D_x^j|_{x=0}=D_x^j|_{x=L},\quad j=0,1,2,\quad{\partial u\over\partial n}|_{(0,L)\times\gamma}=0, \] where \(D\subset \mathbb{R}^2\) is a bounded domain with \(C^2\)-smooth boundary \(\gamma\). It is proved that there exists a solution, provided that \(|f|_{L_2(Q)}+|f_x|_{L_2(Q)}<\alpha\), where \(\alpha\) is some positive number depending on \(\mu\) and \(Q\). It is shown that the solution is unique in some function class. The proofs are accomplished by considering a linearized problem and fixed point arguments.
nonlinear third-order partial differential equation
scientific article; zbMATH DE number 1037962