Transition Threshold for the 3D Couette Flow in a Finite Channel
Qi Chen, Dongyi Wei, Zhifei Zhang
Abstract
In this paper, we study nonlinear stability of the 3D plane Couette flow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis y comma 0 comma 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(y,0,0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at high Reynolds number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{Re}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a finite channel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T times left-bracket negative 1 comma 1 right-bracket times double-struck upper T"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo> × </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T}\times [-1,1]\times \mathbb {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v 0"> <mml:semantics> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">v_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar v 0 minus left-parenthesis y comma 0 comma 0 right-parenthesis double-vertical-bar Subscript upper H squared Baseline less-than-or-equal-to c 0 upper R e Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0 greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c_0>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Re</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sense, and rapidly converges to a streak solution for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math