The Global Existence And Stability Of Compressible Fluids In An Infinitely Expanding Circle

We are concerned with the global existence and stability of compressible fluids with fixed mass in an infinitely expanding circle.In a physical point of view,this problem should be globally well-posed,however,this requires a strictly mathematical proof.There are three typical models to describe the movement of compressible fluids:Euler equations(viscosity of fluid is neglected),Navier-Stokes equations(viscosity of fluid is considered)and Boltzmann equations(microcosmic factors of fluid particles are considered),where Euler equations are the hyperbolic systems,Navier-Stokes equations are the hyperbolic-parabolic coupled systems,and Boltzmann equations are the scalar transport equations with nonlocal collision operator.In a mathematical point of view,the properties and research methods on these three equations are rather different.But for the same physical phenomenon,three different mathematical models should have the same global well-posed results.This paper considered the behavior of compressible inviscid fluid in a 2-D infinitely expanding circle.We assumed that the motion of the fluid in the circle was controlled by the 2-D isentropic compressible Euler system and met the corresponding initial boundary condition.The purpose of this paper was to prove the global existence of the solution.The Euler system can be transformed into a second-order quasilinear equation by Bernoulli's law and implicit function theorem.The second-order quasilinear equation is expressed in polar coordinates according to the geometric properties of the circle.We have already known that(1/(1+Lt)2)t,Lx/1+Lt)was a special solution to this problem,and the local solvability of the problem could be obtained by standard Picard iterations.Further,we linearized the original equation near the background solution.A suitable operator was found to find a consistent first-order energy estimate for the corresponding linearization problem,and a consistent high-order energy estimate was obtained by analyzing the radial derivative and the angular derivative.When performing high-order energy estimation,we divided the domain into three parts,which were the area containing only the tangent derivative,the area containing only the radial derivation,and both contained the area of the tangent derivative and the radial derivation.The linearized operator had an approximate form(?)t2-?/(1+Lt)2(?-1)((?)12+(?)22)+2L(?-1)/1+Lt(?)t,which was degenerate at infinite time.Due to the conservation of mass,the fluid in the expanding circle became rarefied and eventually tended to a vacuum state.Meanwhile there are no appearances of vacuum domains in any part of the expansive circle,which was easily observed in finite time.We would confirm this physical phenomenon by obtaining the exact lower and upper bound on the density function that there are no appearances of vacuum domains for the compressible inviscid and irrotational gases.