Regularity Criteria Based On Pressure In Lorentz Spaces For The 3D Navier-Stokes Equations

Three-dimensional incompressible Navier-Stokes system describing the motion of incompressible Newtonian fluids is one of the basic equations in fluid mechanics.In this thesis,we mainly derive the new regularity criteria based on pressure and related quantities,and gradient of velocity and eigenvalues of its symmetric part in Lorentz spaces in this system.The thesis consists of five chapters.In Chapter 1,we briefly present the background of Navier-Stokes equations,recent progress of some known regularity criteria for Navier-Stokes equations,and the main results.Chapter 2 is devoted to preliminary work and auxiliary lemmas.In Chapter 3 and 4,we conclude the following results for the Leray-Hopf weak solutions(1)regularity criteria based on pressure or gradient of pressure in Lorentz space??Lp,?(0,T;Lq,?(R3)),and???Lp,?(0,T;Lq,?(R3))??1?with2/p+3/q=2,3/2<q<?.???Lp,?(0,T;Lq,?(R3)),and????Lp,?(0,T;Lq,?(R3))??2?with2/p+3/q=3,1?q<?.(2)regularity criteria via the horizontal component of gradient of the pressure or one direction of gradient of pressure in Lorentz space?h??Lp(0,T;Lq,?(R3)),2/p+3/q=2+3/?,??3,3?/2?+1<q??.?3??Lp(0,T;Lq,?(R3)),2/p+3/q=2+3/?,?>3,2?/2?+1<q??.(3)regularity criteria in terms of gradient of velocity in Lorentz space?u?Lp,?(0,T;Lq,?(R3)),and??u?Lp,?(0,T;Lq,?(R3))??3,with 2/p+3/q=2,3/2<q<?.(4)regularity criteria in terms of the eigenvalue of deformation tensor in Lorentz space??Lp,?(0,T;Lq,?(R3)),and???Lp,?(0,T;Lq,?(R3))??4,with 2/p+3/q=2,3/2<q<?.In Chapter 5,we prove regularity criteria based on geometric in Lorentz spaces for the 3D Navier-Stokes equations.In particular,our first result gives an affirmative answer to a question proposed by Suzuki(Nonlinear Anal.75:3849-3853,2012,Remark 2.4,p.3850).These regularity criteria extend the known corresponding results.Finally,we summarize the work of this thesis and mention some challenging problems.