Digital Object Identifier (DOI) 10.1007/s002080100185 Math. Ann. 319, 809–815 (2001) Mathematische Annalen Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system Judith R. Miller ⋆ · Mike O’Leary · Maria Schonbek Received March 8, 2000 / Published online February 5, 2001 – © Springer-Verlag 2001 Abstract. We show that there are no singular pseudo-self-similar solutions of the Navier-Stokes system with finite energy. 1 Introduction In his 1934 pioneering paper, Jean Leray [1] asked whether it is possible to construct a self-similar solution to the Navier-Stokes system in R 3 ∂ u ∂t -△u + (u ·∇ )u +∇ p = 0, (1) div u = 0 (2) of the form u(x,t) = 1 √ T - t U  x √ T - t  , (3) p(x,t) = 1 T - t P  x √ T - t  . (4) The motivation for studying such of solutions is that they would possess a sin- gularity when t = T ; indeed ||∇ u(·,t)|| L 2 (R 3 ) = 1 √ T -t ||∇ U|| L 2 (R 3 ) . This ques- tion was first answered in 1996 by Ne ˇ cas, R˚ u ˇ zi ˇ cka, and ˇ Sver´ ak in the nega- tive. Specifically, in [3], they showed that the only self-similar solution with U ∈ L 3 (R 3 ) ∩ W 1 2,loc (R 3 ) is the trivial solution. Later, M´ alek, Ne ˇ cas, Pokorn´ y, and Schonbek [2] showed that any self-similar solution with U ∈ W 1 2 (R 3 ) was J.R. Miller Department of Mathematics, Georgetown University,Washington D.C. 20057, USA M. O’Leary Department of Mathematics, Towson University, Towson, MD 21252, USA M. Schonbek Mathematics Department, University of California Santa Cruz, Santa Cruz, CA 95064, USA ⋆ Research partially supported by NSF grant DMS-9804814