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Structure stability of steady supersonic shear flow with inflow boundary conditions

Song Jiang, Chunhui Zhou

Abstract

We study the existence and zero viscous limit of smooth solutions to steady compressible Navier-Stokes equations near plane shear flow between two moving parallel walls. Under the assumption $0<L\ll1$, we prove that for any plane supersonic shear flow $\mathbf{U}^0=(μ(x_2),0)$, there exist smooth solutions near $\mathbf{U}^0$ to steady compressible Navier-Stokes equations in a 2-dimension domain $Ω=(0,L)\times (0,2)$. Moreover, based on the uniform-in-$\varepsilon$ estimates, we establish the zero viscosity limit of the solutions obtained above to the solutions of the steady Euler equations.

Structure stability of steady supersonic shear flow with inflow boundary conditions

Abstract

We study the existence and zero viscous limit of smooth solutions to steady compressible Navier-Stokes equations near plane shear flow between two moving parallel walls. Under the assumption , we prove that for any plane supersonic shear flow , there exist smooth solutions near to steady compressible Navier-Stokes equations in a 2-dimension domain . Moreover, based on the uniform-in- estimates, we establish the zero viscosity limit of the solutions obtained above to the solutions of the steady Euler equations.

Paper Structure

This paper contains 13 sections, 16 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Formal asymptotic expansion around shear flows
  3. Euler correctors
  4. Weak boundary layer correctors
  5. The approximate solution $(u_s,v_s,\rho_s)$
  6. The Linearized System
  7. Existence of Approximate Solutions in Hilbert Space
  8. Apriori Estimates
  9. Existence of Solutions to the Approximate System
  10. Estimates of the Approximate Solutions in $W^{2,p}(\Omega)$
  11. Solutions to the Linear System
  12. Solutions to the Nonlinear System
  13. Weight estimates for a linear parabolic equation

Key Result

Theorem 1.1

For $2<p<p^0$, $\mathbf{U}^0=(\mu(x_2),0)$, $\rho^*>0$, we assume that $\mu(x_2)\in C^6([0,2])$ satisfying: and there is no mismatch between the basic flow $\mathbf{U}^0$ and the moving boundaries, then there exists a triple $(u_s,v_s,\rho_s)$ defined in (app.0) such that if there exists a unique solution $(\mathbf{u}^\varepsilon,\rho^\varepsilon)\in W^{2,p}(\Omega)\times W^{1,p}(\Omega)$ to the

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • ...and 20 more