On the Solvability of Transonic Potential Flow Problems

Abstract

The paper is devoted to the study of the solvability of transonic potential flow problems. The velocity potential equation governing irrotational non-viscous transonic flows is nonlinear, second order and of mixed type. There exist a series of numerical methods for the solution of the transonic flows. However, the existence of the solution has not yet been proved.

Here, with the use of the secant-modulus method and a convenient optimal control principle, we construct a functional $\psi$, whose minimization is equivalent to the solution of the problem. Since the so-called shocks, represented by jumps in the velocity, density and pressure, occur in the flow field, we consider weak solutions from the space $W^{1,2}(\Omega )$. From the physical point of view the entropy condition across the shock is very important. There exist various approaches how to embody this condition into the numerical method. Here we consider its simplified version by Glowinski, Pironneau etc. and besides, we propose its new natural, more complex formulation.

We show that these conditions introduce the missing compactness into the problem and allow to prove the existence of the solution in the following sense: If the minimizing sequence of the functional $\psi$ satisfies (a posteriori) the entropy and bounded velocity conditions and converges weakly to a function $u$, then it converges strongly to $u$ and $u$ is a solution of the transonic flow problem.

The paper contains also some results concerning subsonic flows and the regularity of the minimizing sequence.