On the evolution equation of compressible vortex sheets

We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the Mach number $\mathsf{M}$ satisfies , the symbol is elliptic and the problem is ill-posed. On the contrary, if then the problem is weakly stable, and we are able to derive a wave-type a priori energy estimate for the solution, with no loss of regularity with respect to the data. Then we prove the well-posedness of the problem, by showing the existence of the solution in weighted Sobolev spaces.     

The differentiation operator in the space of uniformly convergent Dirichlet series

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes for each . The properties of the formal inverse of the differentiation are also investigated.