Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case
Jacob Bedrossian, Pierre Germain, Nader Masmoudi
Abstract
We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number <bold>Re</bold> . We prove that for sufficiently regular initial data of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon less-than-or-equal-to c 0 bold upper R bold e Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mi> ϵ </mml:mi> <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 mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">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">\epsilon \leq c_0\mathbf {Re}^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some universal <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> , the solution is global, remains within <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis c 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(c_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Couette flow in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and returns to the Couette flow as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t \rightarrow \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For times <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equivalent-to bold upper R bold e Superscript 1 slash 3"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ≳ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t \gtrsim \mathbf {Re}^{1/3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of “2.5 dimensional” streamwise-independent solutions referred to as <italic>streaks</italic> . Our analysis contains perturbations that experience a transient growth of kinetic energy from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis bold upper R bold e Superscript negative 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> <mml:mi mathvariant="bold">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:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\mathbf {Re}^{-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis c 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(c_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> due to the algebraic linear instability known as the <italic>lift-up effect</italic> . Furthermore, solutions can exhibit a direct cascade of energy to small scales. The behavior is very different from the 2D Couette flow, in which stability is independent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R bold e"> <mml:semantics> <mml:m