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Existence of Large Boundary Layer Solutions to Inflow Problem of 1D Full Compressible Navier-Stokes Equations

Yi Wang, Yong-Fu Yang, Qiuyang Yu

TL;DR

This paper resolves the existence/nonexistence of large-amplitude boundary layer solutions for the 1D full compressible Navier–Stokes inflow problem on a half-line by employing a global phase-plane analysis of the stationary boundary-layer ODEs in Lagrangian coordinates. It provides a complete Mach-number regime characterization: no solution for $M_+>1$, a unique curve $\Sigma$ for $M_+=1$ with a strict boundary-data condition, and a pair of curves $\Gamma_1\cup\Gamma_2$ for $0<M_+<1$ with precise tangency and positivity requirements, all without smallness assumptions on amplitude. The proof hinges on reducing to a planar autonomous system, analyzing equilibria, and applying a phase-plane argument together with a specialized lemma to obtain the $\Sigma$, $\Gamma_1$, and $\Gamma_2$ criteria, thereby fully addressing the inflow problem in this setting. The work also discusses stability aspects and notes substantial open problems for time-asymptotic stability of large BLs and for the outflow problem, indicating directions for future research.

Abstract

We present the existence/non-existence criteria for large-amplitude boundary layer solutions to the inflow problem of the one-dimensional (1D) full compressible Navier-Stokes equations on a half line $\mathbb{R}_+$. Instead of the classical center manifold approach for the existence of small-amplitude boundary layer solutions in the previous results, the delicate global phase plane analysis, based on the qualitative theory of ODEs, is utilized to obtain the sufficient and necessary conditions for the existence/non-existence of large boundary layer solutions to the half-space inflow problem when the right end state belongs to the supersonic, transonic, and subsonic regions, respectively, which completely answers the existence/non-existence of boundary layer solutions to the half-space inflow problem of 1D full compressible Navier-Stokes equations.

Existence of Large Boundary Layer Solutions to Inflow Problem of 1D Full Compressible Navier-Stokes Equations

TL;DR

This paper resolves the existence/nonexistence of large-amplitude boundary layer solutions for the 1D full compressible Navier–Stokes inflow problem on a half-line by employing a global phase-plane analysis of the stationary boundary-layer ODEs in Lagrangian coordinates. It provides a complete Mach-number regime characterization: no solution for , a unique curve for with a strict boundary-data condition, and a pair of curves for with precise tangency and positivity requirements, all without smallness assumptions on amplitude. The proof hinges on reducing to a planar autonomous system, analyzing equilibria, and applying a phase-plane argument together with a specialized lemma to obtain the , , and criteria, thereby fully addressing the inflow problem in this setting. The work also discusses stability aspects and notes substantial open problems for time-asymptotic stability of large BLs and for the outflow problem, indicating directions for future research.

Abstract

We present the existence/non-existence criteria for large-amplitude boundary layer solutions to the inflow problem of the one-dimensional (1D) full compressible Navier-Stokes equations on a half line . Instead of the classical center manifold approach for the existence of small-amplitude boundary layer solutions in the previous results, the delicate global phase plane analysis, based on the qualitative theory of ODEs, is utilized to obtain the sufficient and necessary conditions for the existence/non-existence of large boundary layer solutions to the half-space inflow problem when the right end state belongs to the supersonic, transonic, and subsonic regions, respectively, which completely answers the existence/non-existence of boundary layer solutions to the half-space inflow problem of 1D full compressible Navier-Stokes equations.
Paper Structure (6 sections, 3 theorems, 50 equations, 5 figures)

This paper contains 6 sections, 3 theorems, 50 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of main result
  4. Further discussions
  5. Stability of large boundary layer solution
  6. Large boundary layer solutions to outflow problem

Key Result

Theorem 2.1

For the inflow problem Lag-equation with $v_\pm>0$, $u_->0$, $\theta_\pm>0$.

Figures (5)

  • Figure 1: Transonic case.
  • Figure 2: Subsonic case.
  • Figure 3: Saddle-node point.
  • Figure 4: Region I in transonic case.
  • Figure 5: Region I and Region II in subsonic case.

Theorems & Definitions (4)

  • Theorem 2.1
  • Remark 1
  • Proposition 3.1
  • Lemma 3.1