Digital Object Identifier DOI s00205 x Arch Rational Mech Anal

Digital Object Identifier (DOI) 10.1007/s00205-008-0149-x Arch. Rational Mech. Anal. 192 (2009) 375–401 Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations Lorenzo Brandolese Communicated by V. Sverak Abstract We study the solutions of the nonstationary incompressible Navier–Stokes equations in Rd , d 2, of self-similar form u(x, t) = 1√ t U ( x √ t ) , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic for- mula relating the self-similar profile U (x) of the velocity field to its corresponding initial datum a(x). 1. Introduction In this paper we are concernedwith the study of solutions of the elliptic problem ? ? ? ? 1 2U ? 1 2 (x · ?)U ? ∆U + (U · ?U ) + ? P = 0 ? · U = 0, x ? Rd , (1) where U = (U1, . . . , Ud) is a vector field in Rd , d 2, ? = (∂1, . . . , ∂d), and P is a scalar function definedonRd . Such a systemarises from the nonstationaryNavier– Stokes equations (NS), for an incompressible viscous fluid filling the whole Rd , when looking for a velocity field u(x, t) and pressure p(x, t) of forward self-similar form: u(x

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