Research Of Regularity To The Weak Solutions Of The MHD Equations

In this paper, the regularity of the weak solutions to the magneto-hydrodynamic (MHD)equations is studied.Here u,b, p are nondimensional qualities corresponding to the velocity of the fluid, magnetic field and its pressure.f is force;ν,ηare viscous coefficients;u0(x) and b0(x)are initial velocity and magnetic field.When b=0,the equations(0.1)is Navier-Stokes equations.In 1933,Leray used the energy estimates to prove the existence of weak solutions to the Navier-Stokes equations for the first time. Subsequently, Leray raised his famous questions about the existence of backward self-similar solutions for the Navier-Stokes equations in 1934.And this question was answered by Necas,Ruzicka and Sverak in 1996.They showed that the only backward self-similar solution satisfying the global energy estimates is zero.Similar results were also obtained by Tsai in 1998 under the conditions that self-similar solutions satisfy the local energy estimates.He and Xin showed that,for the 3D MHD equations,there exists only trivial backward self-similar solutions in LP for p≥3,under some smallness assumption on the kinetic energy of the self-similar solutions related to the velocity field or the magnetic field.In this paper,we talk about the regularity of the weak solutions to the MHD equations from three aspects.Firstly, considering regularity of the weak solutions of the MHD equations.Serrin's uniqueness criterion tells us the solution after regularizing weak solution coincides with smooth solution,so we study the continuation criteria to the smooth solutions.For this purpose,axisymmetrizating the nonlinear terms.On the one hand,we obtain the blow-up criteria on(wθ,Jθ) or (wθ,▽(uθeθ))by using the properties of the axisymmetric solutions and the boundedness of singular integral operators in Lp(1 0) on the basis of actual situations,and have generalized MHD(GMHD) equations. In this paper,we study the continuation criteria to the smooth solutions of GMHD equa-tions.On the one hand, obtaining some continuation criteria to axisymmetric smooth solutions by the similar way to the first part. On the other hand,talking about other continuation criteria through Fourier localization and Bony paraproduct decomposition. First of all,localizing the equations and making use of Bony paraproduct decomposi-tion to get the estimate of commutator. When p<∞,using the equivalence between B2,2s and Hs,interpolation inequality, Young inequality and Gronwall inequality to ob-tain Hm estimate of(u,b).Combining with energy inequality, we deduce continua-tion criteria with respect to(u,b) or(w,J).When p=∞,linking with logarithmic Sobolev inequality to receive the continuation criterion which is characterized by the norm‖(w,J)‖L1(0,T;B∞,∞0).