In this paper, we investigate the nonlinear stability of the Couette flow for the two-dimensional compressible Navier–Stokes equations at high Reynolds numbers (Re) regime. It was proved that if the initial data <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\rho _{in},\varvec{u}_{in})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow> <mi mathvariant="italic">in</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> <mrow> <mi mathvariant="italic">in</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Vert (\rho _{in},\varvec{u}_{in})-(1, y, 0)\Vert _{H^4({\mathbb {T}}\times {\mathbb {R}})}\le \epsilon Re^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow> <mi mathvariant="italic">in</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> <mrow> <mi mathvariant="italic">in</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>H</mi> <mn>4</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≤</mo> <mi>ϵ</mi> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for some small <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> independent of Re, then the corresponding solution exists globally and remains close to the Couette flow for all time. Formal asymptotics indicate that this stability threshold is sharp within the class of Sobolev perturbations. The proof relies on the Fourier-multiplier method and exploits three essential ingredients: (i) the introduction of “good unknowns” that decouple the perturbation system; (ii) the construction of a carefully designed Fourier multiplier that simultaneously captures the enhanced dissipation and inviscid-damping effects while taming the lift-up mechanism; and (iii) the design of distinct energy functionals for the incompressible and compressible modes.

Nonlinear Stability of 2-D Couette Flow for the Compressible Navier–Stokes Equations at High Reynolds Number

期刊论文
发表日期：

2026年8月1日

查看全文

Minling Li,
Chao Wang,
Zhifei Zhang

摘要

In this paper, we investigate the nonlinear stability of the Couette flow for the two-dimensional compressible Navier–Stokes equations at high Reynolds numbers (Re) regime. It was proved that if the initial data $(\rho _{in},\varvec{u}_{in})$ $(ρ_{in}, u_{in})$ satisfies $\Vert (\rho _{in},\varvec{u}_{in})-(1, y, 0)\Vert _{H^4({\mathbb {T}}\times {\mathbb {R}})}\le \epsilon Re^{-1}$ $‖ (ρ_{in}, u_{in}) - (1, y, 0) ‖_{H^{4} (T \times R)} \leq ϵ R e^{- 1}$ for some small $\epsilon $ $ϵ$ independent of Re, then the corresponding solution exists globally and remains close to the Couette flow for all time. Formal asymptotics indicate that this stability threshold is sharp within the class of Sobolev perturbations. The proof relies on the Fourier-multiplier method and exploits three essential ingredients: (i) the introduction of “good unknowns” that decouple the perturbation system; (ii) the construction of a carefully designed Fourier multiplier that simultaneously captures the enhanced dissipation and inviscid-damping effects while taming the lift-up mechanism; and (iii) the design of distinct energy functionals for the incompressible and compressible modes.