<p>In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equations. We observe that a lift-up effect similar to that in the three-dimensional Navier–Stokes equation near Couette flow destabilizes the system, while the inviscid-damping-type decay due to the Couette flow <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(Y,0,0)^\top $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mi>⊤</mi> </msup> </math></EquationSource> </InlineEquation>, together with the damping structure induced by the decreasing background density <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varrho _\textrm{s}(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϱ</mi> <mtext>s</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey-<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{s}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </math></EquationSource> </InlineEquation> class with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{1}{2}< s\leqq 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>s</mi> <mo>≦</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, namely, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Vert \varrho _\textrm{in}(X,Y,Z)-\varrho _\textrm{s}(Y)\Vert _{\mathcal {G}^{s}}\leqq \varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>ϱ</mi> <mtext>in</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>ϱ</mi> <mtext>s</mtext> </msub> <msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mrow> <mi mathvariant="script">G</mi> </mrow> <mi>s</mi> </msup> </msub> <mo>≦</mo> <mi>ε</mi> </mrow> </math></EquationSource> </InlineEquation>, then the perturbed density remains close to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varrho _\textrm{s}(Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϱ</mi> <mtext>s</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the associated velocity field converges to Couette flow <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(Y, 0, 0)^{\top }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mi>⊤</mi> </msup> </math></EquationSource> </InlineEquation> with a convergence rate of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\frac{1}{\langle t\rangle ^3}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <msup> <mrow> <mo stretchy="false">⟨</mo> <mi>t</mi> <mo stretchy="false">⟩</mo> </mrow> <mn>3</mn> </msup> </mfrac> </math></EquationSource> </InlineEquation>.</p>

Asymptotic Stability of the Three-Dimensional Couette Flow for the Stokes-Transport Equations

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发表日期：

2026年7月2日

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Daniel Sinambela,
Weiren Zhao,
Ruizhao Zi

摘要

In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equations. We observe that a lift-up effect similar to that in the three-dimensional Navier–Stokes equation near Couette flow destabilizes the system, while the inviscid-damping-type decay due to the Couette flow $(Y,0,0)^\top $ ${(Y, 0, 0)}^{⊤}$ , together with the damping structure induced by the decreasing background density $\varrho _\textrm{s}(Y)$ $ϱ_{s} (Y)$ , stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey- $\frac{1}{s}$ $\frac{1}{s}$ class with $\frac{1}{2}< s\leqq 1$ $\frac{1}{2} < s ≦ 1$ , namely, $\Vert \varrho _\textrm{in}(X,Y,Z)-\varrho _\textrm{s}(Y)\Vert _{\mathcal {G}^{s}}\leqq \varepsilon $ $‖ ϱ_{in} (X, Y, Z) - ϱ_{s} {(Y) ‖}_{G^{s}} ≦ ε$ , then the perturbed density remains close to $\varrho _\textrm{s}(Y)$ $ϱ_{s} (Y)$ . Moreover, the associated velocity field converges to Couette flow $(Y, 0, 0)^{\top }$ ${(Y, 0, 0)}^{⊤}$ with a convergence rate of $\frac{1}{\langle t\rangle ^3}$ $\frac{1}{{⟨ t ⟩}^{3}}$ .