Flux Approximation To The Zero Dissipation Limit To Rarefaction Wave For 1D Compressible Isentropic Navier-Stokes Equations

Navier-Stokes equations are very important nonlinear partial differential equations in fluid mechanics theory,and the existence,uniqueness,stability,singularity,blow-up criterion as well as zero dissipation limit of the solutions have always been highly concerned by the international mathematics community.In this paper,we study the flux approximation to the zero dissipation limit to rarefaction wave for the one-dimensional compressible isentropic Navier-Stokes equations.The equations can be stated as follows where (?)?[x,t),u(x,t)represent the density and the velocity,respectively,and p(x,t)?0,them p(p)=??/? denotes the pressure,?>1 is isentropic exponent,and ?>0 is the constant viscosity coefficient.Given that the solution of the Euler equations corresponding to the 1D compressible isentropic Navier-Stokes equations is rarefaction wave with one-side vacuum state,we employ the flux approximation method to control the degeneracies caused by the vacuum in the rarefaction wave,then we adopt the elementary energy analysis to prove that a sequence of solutions constructed to the compressible isentropic Navier-Stokes equations which converge to the rarefaction wave with vacuum as the viscosity vanishes.In addition,the uniform convergence rate with respect to viscosity ? is obtained.