Analytical Solution to 1D Compressible Navier-Stokes Equations

Changsheng Dou,Zishu Zhao,Hans G Feichtinger

doi:10.1155/2021/6339203

Changsheng Dou, Zishu Zhao + Show 1 more

Open Access

https://doi.org/10.1155/2021/6339203

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Abstract

There exist complex behavior of the solution to the 1D compressible Navier-Stokes equations in half space. We find an interesting phenomenon on the solution to 1D compressible isentropic Navier-Stokes equations with constant viscosity coefficient on x , t ∈ 0 , + ∞ × R + , that is, the solutions to the initial boundary value problem to 1D compressible Navier-Stokes equations in half space can be transformed to the solution to the Riccati differential equation under some suitable conditions.

Highlights

Where ρðx, tÞ, uðx, tÞ stand for the density and velocity of compressible flow. μ is the constant viscosity coefficient. p = pðρÞ means the pressure of the flow
As viscosity μ depends on density and has a positive constant lower bound, [5,6,7,8] gave the global well-posedness and large time behavior of solutions to the system without initial vacuum
Zhang and Zhu [20] derived the global existence of classical solution to the initial boundary value problem for the onedimensional Navier-Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature

Summary

Introduction

Zhang and Zhu [20] derived the global existence of classical solution to the initial boundary value problem for the onedimensional Navier-Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. For two-dimensional case, global well-posedness of classical solutions to the Cauchy problem or periodic domain problem of compressible NavierStokes equations with vacuum was obtained in [22,23,24] when the first and second viscosity coefficients are μ and λðρÞ, respectively. Journal of Function Spaces and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density, provided the smooth initial data are of small total energy and the viscosity coefficients are two constants. We can see the boundary condition of velocity gðtÞ ≠ 0

The Proof of Main Result

Example