On velocity selection and the Saffman-Taylor problem.

A new formulation of the velocity selection criterion is presented. It establishes a relation between the existence of propagating steady states as a function of a parameter and the solvability of a certain inhomogeneous linear problem. Velocity selection, determined by the solvability condition, is associated with the presence of an extra marginal mode. The method is illustrated on the example of a geometrical model of solidification. It is used to explain analytically the fact that at large velocity the Saffman-Taylor "fingers" have width close to 1/2 and to predict the scaling exponent for the dependence of the finger width on velocity.