A local smoothness criterion for solutions of the 3D Navier-Stokes equations

  • James C. Robinson

    University of Warwick, Coventry, UK
  • Witold Sadowski

    University of Warsaw, Poland


We consider the three-dimensional Navier–Stokes equations on the whole space R3\mathbb{R}^3 and on the three-dimensional torus T3\mathbb{T}^3. We give a simple proof of the local existence of (finite energy) solutions in L3L^3 for initial data u0L2L3u_0\in L^2\cap L^3, based on energy estimates and regularisation of the initial data with the heat semigroup.

We also provide a lower bound on the existence time of a strong solution in terms of the solution v(t)v(t) of the heat equation with such initial data: there is an absolute constant ε>0\varepsilon>0 such that solutions remain regular on [0,T][0,T] if u0L330TR3v(s)2v(s)dxdtε\|u_0\|_{L^3}^3\int_0^T\int_{\mathbb R^3}|\nabla v(s)|^2|v(s)|\,\rm d x\,\rm d t\leq \varepsilon. This implies the uC0([0,T];L3)u\in C^0([0,T];L^3) regularity criterion due to von Wahl. We also derive simple a priori estimates in LpL^p for p>3p>3 that recover the well known lower bound u(Tt)Lpct(p3)/2p\|u(T-t)\|_{L^p}\ge ct^{-(p-3)/2p} on any solution that blows up in LpL^p at time TT.

The key ingredients are a calculus inequality uL3ppcup2u2\|u\|_{L^{3p}}^p\le c\int|u|^{p-2}|\nabla u|^2 (valid on R3\mathbb R^3 and for functions on bounded domains with zero average) and the bound on the pressure pLrcruL2r2\|p\|_{L^r}\le c_r\|u\|_{L^{2r}}^2, valid both on the whole space and for periodic boundary conditions.

Cite this article

James C. Robinson, Witold Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 131 (2014), pp. 159–178

DOI 10.4171/RSMUP/131-9