<p>We investigate the stability and large-time behavior of solutions to the three-dimensional incompressible Navier–Stokes equations with horizontal fractional dissipation of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> near a constant background state <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$v^{(0)} = A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>. For sufficiently small initial perturbations in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^3(\mathbb {R}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we establish global existence and uniform bounds for the solutions. Moreover, for a suitable range of parameters <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, we prove that these solutions exhibit optimal time-decay rates, with the third component decaying faster than the horizontal components, reflecting an enhanced dissipation phenomenon. The proofs rely on a combination of anisotropic inequalities, energy estimates, and integral representations associated with the fractional horizontal heat semigroup. These results extend classical well-posedness and decay theory for the Navier–Stokes equations to the setting of partial dissipation and provide a framework for studying related geophysical and anisotropic fluid models.</p>

Stability and optimal Decay for the 3D Navier–Stokes equations around a uniform flow

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发表日期：

2026年5月6日

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Wen Luo,
Jiahong Wu,
Xiaojing Xu

摘要

We investigate the stability and large-time behavior of solutions to the three-dimensional incompressible Navier–Stokes equations with horizontal fractional dissipation of order $\alpha \in (0,1]$ $α \in (0, 1]$ near a constant background state $v^{(0)} = A$ $v^{(0)} = A$ . For sufficiently small initial perturbations in $H^3(\mathbb {R}^3)$ $H^{3} (R^{3})$ , we establish global existence and uniform bounds for the solutions. Moreover, for a suitable range of parameters $\alpha $ $α$ and $\sigma $ $σ$ , we prove that these solutions exhibit optimal time-decay rates, with the third component decaying faster than the horizontal components, reflecting an enhanced dissipation phenomenon. The proofs rely on a combination of anisotropic inequalities, energy estimates, and integral representations associated with the fractional horizontal heat semigroup. These results extend classical well-posedness and decay theory for the Navier–Stokes equations to the setting of partial dissipation and provide a framework for studying related geophysical and anisotropic fluid models.