An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range

Abstract We show that a suitable weak solution to the incompressible Navier–Stokes equations on R 3 × ( − 1 , 1 ) is regular on R 3 × ( 0 , 1 ] if ∂ 3 u belongs to M 2 p ∕ ( 2 p − 3 ) , α ( ( − 1 , 0 ) ; L p ( R 3 ) ) for any α > 1 and p ∈ ( 3 ∕ 2 , ∞ ) , which is a logarithmic-type variation of a Morrey space in time. For each α > 1 this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces L q ( ( − 1 , 0 ) ; L p ( R 3 ) ) that are subcritical, that is for which 2 ∕ q + 3 ∕ p 2 .

[1]  Zdenek Skalak.  The end-point regularity criterion for the Navier–Stokes equations in terms of ∂3u , 2020 .

[2]  J. Neustupa,et al.  REGULARITY OF A SUITABLE WEAK SOLUTION TO THE NAVIER-STOKES EQUATIONS AS A CONSEQUENCE OF REGULARITY OF ONE VELOCITY COMPONENT , 2002 .

[3]  Zhifei Zhang,et al.  Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations , 2020, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[4]  W. Ozanski.  The Partial Regularity Theory of Caffarelli, Kohn, and Nirenberg and its Sharpness , 2019, Advances in Mathematical Fluid Mechanics.

[5]  Robinson Rodrigo Sadowski.  Three-Dimensional Navier–Stokes Equations , 2016 .

[6]  Zdenek Skalak,et al.  A note on coupling of velocity components in the Navier‐Stokes equations , 2004 .

[7]  Igor Kukavica,et al.  Navier-Stokes equations with regularity in one direction , 2007 .

[8]  Luigi C. Berselli,et al.  Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations , 2002 .

[9]  Zdenek Skalak.  On the regularity of the solutions to the Navier–Stokes equations via the gradient of one velocity component , 2014 .

[10]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[11]  Milan Pokorný,et al.  On Anisotropic Regularity Criteria for the Solutions to 3D Navier–Stokes Equations , 2011 .

[12]  J. Wolf.  A direct proof of the Caffarelli-Kohn-Nirenberg theorem , 2008 .

[13]  G. Prodi.  Un teorema di unicità per le equazioni di Navier-Stokes , 1959 .

[14]  Zhifei Zhang,et al.  Blow-up of critical norms for the 3-D Navier-Stokes equations , 2015, 1510.02589.

[15]  Benjamin C. Pooley,et al.  Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace” , 2017, Partial Differential Equations in Fluid Mechanics.

[16]  J. Serrin.  On the interior regularity of weak solutions of the Navier-Stokes equations , 1962 .

[17]  Hi Jun Choe,et al.  Regularity of Solutions to the Navier-Stokes Equation , 1999 .

[18]  Milan Pokorny.  On the result of He concerning the smoothness of solutions to the Navier-Stokes equations , 2003 .

[19]  Patrick Penel,et al.  Anisotropic and Geometric Criteria for Interior Regularity of Weak Solutions to the 3D Navier—Stokes Equations , 2001 .

[20]  Zhifei Zhang,et al.  On the interior regularity criteria and the number of singular points to the Navier-Stokes equations , 2014 .

[21]  Cheng He.  Regularity for solutions to the Navier-Stokes equations with one velocity component regular , 2002 .

[22]  Ping Zhang,et al.  On the Critical One Component Regularity for 3-D Navier-Stokes System: General Case , 2013, Archive for Rational Mechanics and Analysis.

[23]  Igor Kukavica,et al.  Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations , 2017, J. Nonlinear Sci..

[24]  Alexis Vasseur,et al.  A new proof of partial regularity of solutions to Navier-Stokes equations , 2007 .

[25]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[26]  Da Veiga,et al.  A new regularity class for the Navier-Stokes equations in R^n , 1995 .

[27]  Milan Pokorný,et al.  Some New Regularity Criteria for the Navier-Stokes Equations Containing Gradient of the Velocity , 2004 .