Some Studies On The Regularity Issue Of The Incompressible Navier-Stokes And MHD Equations

The aim of this thesis is to study the regularity issue of the three dimensional incompressible Navier-Stokes and MHD equations.The thesis contains three chapters. In Chapter One, we briefly survey the back-ground of the problems studied and introduce the related definitions, known results, notations, as well as some lemmas which will play crucial roles in the remaining parts. In Chapter Two and Chapter Three, we establish some new regularity criteria for the weak solutions of these two models, respectively. Also, some essential improvements to the known results are made.For the Navier-Stokes equations, we obtain the following results:1. An anisotropic regularity criterion is proved which improves a result of Zhou and Pokorny. In particular, our condition added on the diagonal component of the gradient of velocity belongs to the Ladyzhenskaya-Prodi-Serrin’s class.2. Two "combinational" type regularity criteria are obtained which involve only one component of velocity and one component of the gradient of velocity with (at least) one of them belonging to the Ladyzhenskaya-Prodi-Serrin’s class.3. An important method is introduced which can be used to remove the smallness assumptions for a large class of "sub-Ladyzhenskaya-Prodi-Serrin" type regular-ity conditions. As an application, we apply this method to improve three results of Zhou and Pokorny.4. We prove a (?)iuj∈LT∞,2-type regularity criterion which is very important in the sense that:(1) it contains the case of strong solution;(2) it indicates that a strong solution blows up at time t=T*if and only if all components of the strong solution blow up in all directions at the same time;(3) it obviously improves Cao-Qin-Titi’s result (▽ui∈LT∞,2), significantly. For the MHD equations, we obtain the following results:5. We establish an regularity criterion for the MHD equations involving one velocity component, which, as far as we known, is the first result in this direction.6. A new regularity criterion in terms of partial components of the gradient of ve-locity is proved which improves the related results of Cao-Wu, Jia-Zhou, Du-Lin and Yamazaki.7. We get a regularity criterion via only one component of the gradient of pressure which combining with a recent result of Zhang-Li-Yu sharpens a theorem of Cao and Wu.8. We show that the Ladyzhenskaya-Prodi-Serrin-type criterion via any two compo-nents of the gradient of pressure is "almost true"(in the case when9/7≤γ≤∞) and "sometimes true"(in the case when9/7≤γ≤3). In fact, the later improves a theorem of Berselli and Galdi concerning the regularity criterion for the Navier-Stokes equations.