The existence of a global attractor is established for generalized Navier-Stokes equations incorporating damping term within the periodic domain Ω = [−π, π]n. Initially, we show the existence and uniqueness of strong solutions. Subsequently, we verify the continuity of the associated semigroup when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\max \{{{2n + 1} \over {n - 1}},{{5n + 2} \over {3n - 2}}\} < \beta < {{3n + 2} \over {n - 2}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo form="prefix" movablelimits="true">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">}</mo> <mo><</mo> <mi>β</mi> <mo><</mo> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Finally, we establish the existence of both Hα-global attractor and H2α-global attractor.

Well-posedness and attractor for the multi-dimensional Navier-Stokes equations with fractional dissipation and damping

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2025年12月1日

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Subha Pal

摘要

The existence of a global attractor is established for generalized Navier-Stokes equations incorporating damping term within the periodic domain Ω = [−π, π]ⁿ. Initially, we show the existence and uniqueness of strong solutions. Subsequently, we verify the continuity of the associated semigroup when $\max \{{{2n + 1} \over {n - 1}},{{5n + 2} \over {3n - 2}}\} < \beta < {{3n + 2} \over {n - 2}}$ $max {\frac{2 n + 1}{n - 1}, \frac{5 n + 2}{3 n - 2}} < β < \frac{3 n + 2}{n - 2}$ . Finally, we establish the existence of both H^α-global attractor and H^2α-global attractor.