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- W4376137394 abstract "Abstract The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space <?CDATA $mathbb R^3$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> , as <?CDATA $tto infty$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>t</mml:mi> <mml:mo stretchy=false>→</mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> </mml:math> , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is <?CDATA $mathbb{T}^{2}timesmathbb{R}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:math> . Our results reveal that the large-time behaviour for <?CDATA $mathbb{T}^{2}timesmathbb{R}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:math> is quite different from that for <?CDATA $mathbb R^3$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . We show that any small initial velocity field <?CDATA $u_0in H^{,2}(Omega)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:math> leads to a unique global solution u that remains small in <?CDATA $H^{,2}(Omega)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:math> . More importantly, as <?CDATA $tto infty$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>t</mml:mi> <mml:mo stretchy=false>→</mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> </mml:math> , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast." @default.
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- W4376137394 date "2023-05-11" @default.
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- W4376137394 title "3D anisotropic Navier–Stokes equations in T2×R : stability and large-time behaviour" @default.
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