Unique continuation type theorem for the self-similar Euler equations

We prove a unique continuation type theorem for the self-similar Euler equations in R-3, assuming the time periodicity. Namely, if a time periodic solution V(y, s) of the time dependent self-similar Euler equations has the property that V(0, s) = 0 for all s is an element of [0, S-0] where S-0 is the time period, and = 0 is a local extremal point of V(y, s) near the origin, then, V(y, s) = 0 for all (y, s) is an element of R-3 x [0, S-0]. A similar result holds for more general system with arbitrary coefficients, and also for the inviscid incompressible magnetohydrodynamic (MHD) system. As a consequence we obtain new criteria for the absence of the discretely self-similar singularities for the 3D Euler equations and the inviscid 3D MHD. (C) 2015 Elsevier Inc. All rights reserved.