On the Integrability of Compressible Potential Flow in Multiple Spatial Dimensions

This work analyzes the local integrability of the potential equation for unsteady, compressible flow in multiple spatial dimensions. Techniques from coordinate-free differential geometry are applied to cast the compressible potential flow equation as an exterior differential system with an independence condition. The resulting exterior differential system is a linear Pfaffian system which converts the difficult analysis question of local solvability about non-singular points to a simpler problem in linear algebra. The Cartan test is applied to the tableau and torsion tensors from the linear Pfaffian system associated with the potential flow equation to analyze local integrability. The Cartan test gives the conditions needed for which the Cartan-Kähler generalization of the Cauchy-Kowalevski existence theorem may be applied to guarantee that unsteady, compressible, irrotational flow fields have local Taylor series solutions.