<p>We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier–Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1057_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$(H^k\cap V)\times H^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mi>k</mi> </msup> <mo>∩</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mi>H</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1057_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$k\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1057_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$H^s\times H^s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mo>×</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1057_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0<s<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.</p>

Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains

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2025年2月9日

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Mustafa Sencer Aydın,
Pranava Chaitanya Jayanti

摘要

We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier–Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to $(H^k\cap V)\times H^k$ $(H^{k} \cap V) \times H^{k}$ for $k\in \mathbb {N}$ $k \in N$ and $H^s\times H^s$ $H^{s} \times H^{s}$ for $0<s<2$ $0 < s < 2$ . The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.