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• W4287025254 abstract "Abstract We investigate the problem of classification of solutions for the steady Navier–Stokes equations in any cone-like domain. In the form of separated variables, <?CDATA begin{align*}u(x,y)=left( begin{array}{c} varphi_1(r)v_1(theta) varphi_2(r)v_2(theta) end{array} right) ,end{align*}?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=block overflow=scroll> <mml:mtable columnalign=right left right left right left right left right left right left columnspacing=0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em 2em 0.2777777777777778em rowspacing=3pt> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mfenced close=) open=(> <mml:mtable columnspacing=1em rowspacing=4pt> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>φ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>φ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <?CDATA $x = rcostheta$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mi>cos</mml:mi> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> and <?CDATA $y = rsintheta$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mo form=prefix>sin</mml:mo> <mml:mi>θ</mml:mi> </mml:math> in the polar coordinates, we obtain the expressions of all smooth solutions with C 0 Dirichlet boundary condition. In particular, we find some solutions which are Hölder continuous on the boundary but their gradients blow up at the corner, show that all solutions in the entire plane <?CDATA $mathbb{R}^2$?> <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>2</mml:mn> </mml:msup> </mml:math> must be polynomials, and prove a sharp Liouville type theorem." @default.
• W4287025254 created "2022-07-25" @default.
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• W4287025254 date "2023-04-11" @default.
• W4287025254 modified "2023-10-17" @default.
• W4287025254 title "Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains" @default.
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