Regularity of a Suitable Weak Solution to the Navier-Stokes Equations as a Consequence of Regularity of One Velocity Component

On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion

“…Proof for the borderline case in various settings was obtained in [17,29,43,44]. Similar results concerning the 3D Navier-Stokes, Boussinesq and MHD equations were obtain in [2,3,6,5,24,25,32,36,34,35,55,56,41]. In particular, in [50,51], regularity criteria for MHD equations involving only two velocity components was proved but in a smaller Lebesgue space.…”

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

“…We recall a fact proved by Neustupa and Penel in [20] on the epoch of possible irregularity for suitable weak solutions.…”

“…After that there are numerous conditional regularity D. Chae JMFM results on the weak solutions, imposing integrability conditions on the velocity or the vorticity, which guarantee regularity of the weak solutions (see e.g. [25,21,23,17,10,26,1,2,3,28,9,13,15,16,20,5,6]). For the local analysis of the regularity properties of weak solutions Caffarelli-Kohn-Nirenberg introduced the notion of suitable weak solutions and proved its partial regularity as well as global in time existence ([4]).…”

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

“…In the three dimensional case, a large gap remains between the regularity available in the existence results and the additional regularity required in the sufficient conditions to guarantee the smoothness of weak solutions of the standard Navier-Stokes equations. This gap has been narrowed by the works of Iskauriaza-Seregin-Sverak [18], LadayzhenskayaSeregin [19], Scheffer [25], Serrin [27], Struwe [29], see also [2], [3], [4], [5], [6], [8], [9], [10], [13], [14], [15], [16], [20], [22], [23], [24], [26], [31], [32] and the references therein, which bring about a deeper understanding of the regularity. In particular, some local partial regularity results and Hausdorff dimension estimates on the possible singular set have been obtained for a class of suitable weak solutions defined and constructed in [7], where the principal tools are the so-called generalized energy inequality and a scaling argument.…”