Existence Of Global Weak Solutions And Asymptotic Behavior To 1D Navier-Stokes Equations With Density-dependent Viscosity

The paper is to study the equation model, which has very far-reaching influ-ence in the field of mathematical and physics research. That is one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. We dis-cussed the property of global weak solutions with free boundary value problems. The specific form of the model is as follows where viscosity is μ(ρ)=θρθ+1, p denotes the density of the fluids and 0<θ< γ is a constant,γ> 1.In this paper, some new ideas and techniques are used to overcome many difficulties of the system. We study the following problems:1. When the density function is continuously connected to the vacuum, we constructed the special energy function, and discuss the upper and lower bounds of the density function. And then prove the existence of global weak solution by analysis higher order estimates of solutions wherc 0< θ< γ,γ> 1.2. We construct the auxiliary function, rewrite the original problem and finally study the solution of the asymptotic behavior and decay rate with t → ∞.