<p>In this manuscript, we consider the Navier-Stokes equations with damping term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|u|^{\beta -1}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>β</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). First, for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove the weak solutions to Navier-Stokes equations with damping decay to 0 as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Then, by using Fourier splitting method, we obtain decay rates with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta \ge \frac{7}{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> respectively. Finally, for the <i>n</i> dimensional case, by using Zhou’s method, we derive the decay of solutions comparing with the heat equation with the same initial data.</p>

Large Time Behavior of Solutions to the Navier-Stokes Equations with Damping

期刊论文
发表日期：

2025年12月8日

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Wenjie Zhou,
Yong Zhou

摘要

In this manuscript, we consider the Navier-Stokes equations with damping term $|u|^{\beta -1}u$ ${| u |}^{β - 1} u$ ( $\beta > 0$ $β > 0$ ). First, for any $\beta >0$ $β > 0$ , we prove the weak solutions to Navier-Stokes equations with damping decay to 0 as $t \rightarrow \infty $ $t \to \infty$ . Then, by using Fourier splitting method, we obtain decay rates with $\beta =1$ $β = 1$ and $\beta \ge \frac{7}{3}$ $β \geq \frac{7}{3}$ respectively. Finally, for the n dimensional case, by using Zhou’s method, we derive the decay of solutions comparing with the heat equation with the same initial data.