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Stationary flows for viscous heat-conductive fluid in a perturbed half-space

Mingjie Li, Masahiro Suzuki, Katherine Zhiyuan Zhang

Abstract

We consider the non-isentropic compressible Navier-Stokes equation in a perturbed half space with an outflow boundary condition as well as the supersonic condition. This equation models a compressible viscous, heat-conductive, and Newtonian polytropic fluid. We show the unique existence of stationary solutions for the perturbed half-space. The stationary solution constructed in this paper depends on all directions and has multidirectional flow. We also prove the asymptotic stability of this stationary solution.

Stationary flows for viscous heat-conductive fluid in a perturbed half-space

Abstract

We consider the non-isentropic compressible Navier-Stokes equation in a perturbed half space with an outflow boundary condition as well as the supersonic condition. This equation models a compressible viscous, heat-conductive, and Newtonian polytropic fluid. We show the unique existence of stationary solutions for the perturbed half-space. The stationary solution constructed in this paper depends on all directions and has multidirectional flow. We also prove the asymptotic stability of this stationary solution.

Paper Structure

This paper contains 27 sections, 42 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main results
  4. Reformulation
  5. Time-global solvability
  6. $L^2$ estimate
  7. Time-derivative estimates
  8. Spatial-derivative estimates
  9. Cattabriga estimates
  10. Elliptic estimates
  11. Completion of the a priori estimates
  12. Construction of stationary solutions
  13. Time-periodic solutions
  14. Uniqueness
  15. Existence
  16. ...and 12 more sections

Key Result

Proposition 1.1

For the supersonic case, i.e. when super1 holds, if $\tilde{\delta}<\varepsilon_0$ for a certain positive constant $\varepsilon_0$, there exists a unique smooth solution $(\tilde{\rho},\tilde{u}_1,\tt)$ to the problem ste. Moreover, there exist a positive constant $\alpha$ such that the stationary s

Theorems & Definitions (65)

  • Proposition 1.1: knnz-10
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 3.1
  • ...and 55 more