Asymptotic Stability Of One-dimensional Periodic Solutions For Non-ideal Fluids Of Bird-carreau Type Viscosity

In recent decades,with the rapid expansion of industrial technology,non-Newtonian fluids have become more and more widely used in practical production in various fields such as petrochemicals,food processing and aerospace and water conservancy.With the widespread application of non-Newtonian fluids in these industrial processes and the incorporation of a large number of mechanical equipment,it is particularly important to study the flow characteristics of fluids to improve mechanical properties and production efficiency.Based on the above description,in this paper,the periodic boundary value problems of isotropic and non-isentropic models for one-dimensional compressible non-Newtonian fluids are discussed,in which the viscosity coefficient is a nonlinear function that satisfies the Bird-Carreau rheological model,and the fluid pressure is a non-convex function that satisfies the van der Waals equation.The main difficulty in this issue is the non-linearity of the viscosity coefficient and the non-convexity of the pressure.In this paper,the existence and uniqueness of local solutions are obtained by using the fixed point theorem and monotone operator theory.Then the difficulty of non-convex pressure is overcome by constructing the energy functional,and then the related energy estimates are obtained,and the difficulty of nonlinearity of viscosity coefficient is overcome.The main conclusions are as follows:For isentropic model of compressible non-Newtonian fluids,it is proved:when the average of the initial value is located in the stable region,if the viscosity coefficient is large enough,then there is a unique global solution that asymptotically converges to the average of the initial value;when the average of the initial value is located in the metastable region,if the viscosity coefficient is large enough and the initial value is in the vicinity of its average,then there is a unique global solution that asymptotically converges to the average of the initial value.For the non-isentropic model of compressible non-Newtonian fluids,it is proved that when the average of the initial value is located in the stable region,if the viscosity coefficient is large enough,then there is a unique global solution that asymptotically converges to the average of the initial value.