Existence Of Stationary Solutions To Two Types Of Compressible Fluids Under Generalized Boundary Conditions

The paper concerns the steady solutions to compressible NavierStokes/Allen-Cahn systems and the steady equations to compressible nonNewtonian fluids on three dimensional bounded regions.Using the fixed point theory and the method of weak convergence,we obtain the existence of steady solutions to compressible Navier-Stokes/Allen-Cahn systems under generalized boundary conditions and the existence of weak solutions to steady equations of compressible non-Newtonian fluids under generalized boundary conditions.The difficulties come from the nonzero velocity in generalized boundary conditions and nonlinear terms.The main results of this paper include two aspects.On the one hand,the approximation system for the compressible NavierStokes/Allen-Cahn steady system is constructed.On the reasonable assumptions on the pressure and appropriate extension of boundary velocity,we get the uniform estimates of the approximation solutions with help of the energy method.Combined with the method of weak convergence,the existence of weak solutions to the compressible Navier-Stokes/Allen-Cahn steady system under generalized boundary conditions is obtained.On the other hand,we concern on the steady equations of compressible non-Newtonian fluid.First,the approximation system is constructed.Then the existence of the solutions for the approximation system is obtained by using the fixed point method.According to the uniform estimates of the approximation solutions,the existence of weak solutions to the steady systems of compressible non-Newtonian fluids under generalized boundary conditions is obtained,by using the method of weak convergence.