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- W4386864770 abstract "Abstract We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $$v=(y,0)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a stratified temperature $$theta =alpha y$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> with $$alpha >frac{1}{4}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . We show that for any $$epsilon >0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> there exist non-trivial explicit solutions, which are initially perturbations of size $$epsilon $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϵ</mml:mi> </mml:math> , and grow to size 1 on a time scale $$epsilon ^{-2}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than $$epsilon ^{-4}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> ." @default.
- W4386864770 created "2023-09-20" @default.
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- W4386864770 date "2023-09-19" @default.
- W4386864770 modified "2023-10-13" @default.
- W4386864770 title "On Stability Estimates for the Inviscid Boussinesq Equations" @default.
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- W4386864770 doi "https://doi.org/10.1007/s00332-023-09958-2" @default.
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