Short story:
The inability to prove well-posedness, or even existence of solutions, for the Navier-Stokes equations may well be a consequence of how a compressible fluid is modelled. Hence, I have studied the basic assumptions upon which the Navier-Stokes equations are derived. This has led me to propose a model of compressible viscous flows that is entirely derived in an Eulerian (fixed in space), rather than Lagrangian frame (moving fluid parcel), with diffusion as the main physical principle rather than viscous stresses. Preliminary comparisons indicate that the new model is as accurate as the standard Navier-Stokes model, but many more test cases must be investigated, as well as its mathematical properties.
Journal publications:
Validation,NACA0012 (with Vít Dolejší)
Validation, sound attenuation (with Karl Munthe)
By other research groups:
Validation, aerodynamics (by group at KAUST)
Validation, incompressible limit (by researchers at the Polish Academy of Sciences)