Navier-Stokes equations are one of the problems of the millennium. How, then, are CFD programs doing simulation, calculating forces, etc if these equations are unsolvable?
The Navier Stokes cannot be solved analytically, they can however be computed numerically. Which is the technology behind flight simulation, using a determined time interval to compute all dynamic (differential) equations, then use the result as input into the next time interval equation. Real-time digital computation.
The technology is quite mature, as demonstrated by the Newton-Raphson method which dates back a couple of centuries. The Navier-Stokes equations form a matrix of multi-dimensional differential equations, and need to compute the effects of non-deterministic turbulence effects, so they require a lot of computation power with fast iteration loops. Less and less of a problem with increasing computer capacity.
The equations solved for the most of the CFD software are the Navier-Stokes equations (let's stay in the general case of compressible ones). The Problem of Navier Stokes equations solution existence means that (only in case of the 3D problem) the solution of Navier-Stokes equations is not proven to "exist and be unique" alias the Lax-Millgram lemma is not verified.
Said so the equation is still solvable. In particular in 2D and 1D this problem does not poses because the lemma is verified.
The most of the CFD software still solve the equations numerically. Even if the well posedness is not verified, it is still possible to solve the equations. Not proving the well posedness it means only that:
"in 3D, solving the Navier-Stokes system, a regular solution is not guaranteed that exist and it is not guaranteed to be unique starting from an initial state and a set of boundary conditions"
I have been fast here, for details have a look at this book for example, in the first chapter there are also basics about the mathematical spaces basis and variational formulation behind Lax-Millgram theorem.