Studies on transonic flow problems by nonlinear variational inequalities
The author considers an irrotational, steady and isentropic flow of a non-viscous, compressible fluid in a bounded, simply connected domain \(\Omega \subset R^ N\) (N\(\geq 2)\), governed by the standard partial differential equation for the velocity potential u, to which a suitable entropy condition, a bound for the gas velocity, and the boundary conditions for u, must be added. Weak formulations of the boundary value problems having physical interest are given. There is to find an \(u\in V\) such that: \[ \int_{\Omega}\rho (| \nabla u|^ 2)\nabla u\nabla v dx=\int_{R}gv do\quad for\quad all\quad v\in V, \] where V has different expressions according to the particular considered problem. The above equation is the Euler- Lagrange-equation for the variational problem: \[ F(v):=\int_{\Omega}(\int^{| \nabla v|^ 2}_{0}\rho (q)dq)dx-\int_{R}gv do\to_{v\in V}. \] Now F(v) must be minimized over all \(v\in V\), with additional constraints, that means: F(v)\(\to_{v\in K}\), where K is a well defined subset of V. If \(u\in K\) is a minimum point of this variational problem, then u satisfies for all \(v\in K\) the variational inequality: \[ DF(u,u-v)=\int_{\Omega}\rho (| \nabla u|^ 2)\nabla u\nabla (u-v)dx-\int_{R}g(u-v)do\leq 0. \] Using Katchanov's method and the compactness of the convex set this variational inequality is solved. Furthermore, a result on flows in the subsonic region is given.
scientific article; zbMATH DE number 4070525