The Blow-up Criteria Of Strong And Smooth Solutions To The Compressible Magneto-hydrodynamic System

In this dissertation, we consider the following full compressible magnetohydrodynamic?MHD? system on ? ?R^d which describes the motion of electrically conducting media in the presence of the magnetic field. Here ? is either a whole space or a bounded domain with smooth boundary. ?, u=?u¹,u²,…,u^d?^T,b=?b¹,b²,…,b^d?^T,e=c_v? and p =R?? are the unknown density, velocity, magnetic field, specific internal energy and pressure respectively. Positive constants c_v, ?, ? are the specific heat, the thermal conductivity coefficient and the coefficient of diffusion of the magnetic field respectively. ? is the absolute temperature, R is the gas constant, ? and ? are, respectively, the shear viscosity coefficient and the bulk viscosity coefficient satisfying the conditions ? >0, 2? +d ? ?0.In this paper, the blow-up problems to the compressible MHD system are studied from the following three fields.Firstly, the smooth solution to the full compressible MHD system with zero resistivity blows up in finite time if the initial density is compactly supported on d?d ?1?. On the one hand, we prove that if the initial density is compactly supported then the density compactly supported all the time. On the other hand, by the energy method, we can get the relationships of the basic physical quantities. Finally, combining the formulas of the basic physical quantities, one can derive the inequality of the life span of the smooth solution. Through the three steps above, we can prove that the existence time of smooth solution is finite if the initial density is compactly supported. Secondly, two blow-up criteria of a strong solution to the full compressible MHD system?0.0.2? on a two-dimensional bounded smooth domain ? are established. If ?T 0||divu||_L^?dt+||b||_L^??0,T,L^?? or ?T 0||?u||_L^?dt is finite, then the strong solution to thefull compressible MHD system?0.0.2? can be continued beyond the time T. Our main method is reductio ad absurdum. By means of a new logarithmic Sobolev inequality?2.2.2?, Gagliardo-Nirenberg, Young and H???lder's inequalities, standard ellipse estimate, Gronwall's lemma and so on, we get a priori estimate of the strong solution at the maximal existence time T. Applying the local existence theorem of the strong solution, the local strong solution can be extended beyond the time T. That is to say, the strong solution can not blow up at the time T.Thirdly, if ? is a two-dimensional bounded smooth domain and T^? is the maximal existence time of the strong solution to the full compressible MHD system with zero resistivity?? =0?, then lim T?T^? ?T 0||?u||_L^?dt=?.