Unique continuation type theorem for the self-similar Euler equations

We prove a unique continuation type theorem for the self-similar Euler equations in R3R3, assuming the time periodicity. Namely, if a time periodic solution V(y,s)V(y,s) of the time dependent self-similar Euler equations has the property that V(0,s)=0V(0,s)=0 for all s∈[0,S0]s∈[0,S0], where S0S0 is the time period, and y=0y=0 is a local extremal point of V(y,s)V(y,s) near the origin, then, V(y,s)=0V(y,s)=0 for all (y,s)∈R3×[0,S0](y,s)∈R3×[0,S0]. 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.