Global Well-posedness To The Compressible Magneto-micropolar Fluid Equations With Vacuum

The mathematical model of magnetohydrodynamics is used to simulate the mo-tion of electrically conducting fluids such as liquid metals,plasmas and electrolytes.It mainly describes the interaction between magnetic field and velocity field.When the fluid contains particles,the interaction between particles can change the state of motion of the fluid.We need to add the micro-rotation effect to the magnetohy-drodynamics equations to obtain a new set of magneto-micropolar fluid equations.This new model is widely used in complex fluids such as electro-rheological fluids,blood,and liquid crystals.The compressible magneto-micropolar fluid equations have strong coupling and nonlinearity,and the global well-posedness of the classical solutions has made some progress in recent years.However,the global well-posedness of classical solutions with vacuum still needs to be studied urgently,which is also the main research content of this paper.In this paper,we study the three-dimensional Cauchy problem of compressible isentropic magneto-micropolar fluid equations with initial density containing vacuum states by virtue of the precise priori estimates and time-independent upper bound of the density.Based on energy method and the structural characteristics of the model,we can obtain the lower order priori estimates which is independent of time,thereby obtaining the time-independent upper bound of the density.The higher order estimates of the velocity field,the micro-rotational velocity field,and the magnetic field are obtained by using the estimates of the effective viscous flux and rotation,we show the global existence of classical solutions provided that ［（γ-1）^1/9+ν^-1/4］E₀ is suitably small,where γ,ν,and E₀represent the adiabatic exponent,resistivity coefficient,and initial energy,respectively.It is worth noting that the solution in this paper is a Nishida-Smoller type solution.Our result is an extension of the work of Wei–Guo–Li（J.Differential Equations,263:2457–2480,2017）,where the global existence of smooth solutions was established under the condition that the initial data are small perturbations of some given constant state.