  We analyze from a far field the evolution of an interface that separates
ideal incompressible fluids of different densities and has an interfacial mass
flux. We develop and apply the general matrix method to rigorously solve the
boundary value problem involving the governing equations in the fluid bulk and
the boundary conditions at the interface and at the outside boundaries of the
domain. We find the fundamental solutions for the linearized system of
equations, and analyze the interplay of interface stability with flow fields
structure, by directly linking rigorous mathematical attributes to physical
observables. New mechanisms are identified of the interface stabilization and
destabilization. We find that interfacial dynamics is stable when it conserves
the fluxes of mass, momentum and energy. The stabilization is due to inertial
effects causing small oscillations of the interface velocity. In the classic
Landau dynamics, the postulate of perfect constancy of the interface velocity
leads to the development of the Landau-Darrieus instability. This
destabilization is also associated with the imbalance of the perturbed energy
at the interface, in full consistency with the classic results. We identify
extreme sensitivity of the interface dynamics to the interfacial boundary
conditions, including formal properties of fundamental solutions and
qualitative and quantitative properties of the flow fields. This provides new
opportunities for studies, diagnostics, and control of multiphase flows in a
broad range of processes in nature and technology.
