Leray's problem for the Navier–Stokes equations revisited

Abstract

In a seminal work in 1934, J. Leray constructed solutions u(⋅,t)∈L∞([0,∞),Lσ2(R3))∩Cw0([0,∞),L2(R3))∩L2([0,∞),H˙1(R3)) of the Navier–Stokes equations for arbitrary initial data u(⋅,0)∈Lσ2(R3) and left it open whether ‖u(⋅,t)‖L2(R3) would necessarily tend to zero as t→∞. This question was answered positively fifty years later by T. Kato, using a different approach. Here, we reexamine Leray's problem and solve this and other important related questions using Leray's original ideas and some standard tools (Fourier transform, Duhamel's principle, heat kernel estimates) already in use in his time.