Stability Threshold of the 2D Couette Flow in a Homogeneous Magnetic Field Using Symmetric Variables
Michele Dolce
Abstract
Abstract We consider a 2D incompressible and electrically conducting fluid in the domain $${\mathbb {T}}\times {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . The aim is to quantify stability properties of the Couette flow ( y , 0) with a constant homogenous magnetic field $$(\beta ,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> when $$|\beta |>1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>β</mml:mi> <mml:mo>|</mml:mo> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The focus lies on the regime with small fluid viscosity $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> , magnetic resistivity $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> and we assume that the magnetic Prandtl number satisfies $$\mu ^2\lesssim \textrm{Pr}_{\textrm{m}}=\nu /\mu \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≲</mml:mo> <mml:msub> <mml:mtext>Pr</mml:mtext> <mml:mtext>m</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ν</mml:mi> <mml:mo>/</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We establish that small perturbations around this steady state remain close to it, provided their size is of order $$\varepsilon \ll \nu ^\frac{2}{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>≪</mml:mo> <mml:msup> <mml:mi>ν</mml:mi> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msup> </mml:mrow> </mml:math> in $$H^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:math> with N large enough. Additionally, the vorticity and current density experience a transient growth of order $$\nu ^{-\frac{1}{3}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> while converging exponentially fast to an x -independent state after a time-scale of order $$\nu ^{-\frac{1}{3}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> . The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system’s dynamic behavior.