Magnetohydrodynamic Equations Around Couette Flow

This thesis studies the stability of the magnetohydrodynamic equations around an affine shear flow and constant magnetic field. The dynamics of the equations change drastically depending on the fluid viscosity ν ≥ 0 and magnetic resistivity κ ≥ 0. The main goal of this thesis is to establish results on (nonlinear) stability and instability for selected dissipation regimes. These stability and instability results are established in Chapters 3-5.
In the first part, we consider the inviscid ν = 0 and resistive κ > 0 case. We linearize around explicit low-frequency solutions of traveling waves to infer the main growth model. Small data in Gevrey 2 spaces are necessary and sufficient for this main growth model to ensure stability.
In the second part, we consider the MHD equations with viscosity and horizontal resistivity ν = κ_x > 0 and κ_y = 0. We show that small initial data in Sobolev spaces ensure stability. Furthermore, we show that for the viscous ν > 0 and non-resistive κ = 0 case, for the linearized MHD equations, there exists initial data such that the magnetic field grows unbounded as t → ∞. Thus,
global in time stability of the magnetic field in Sobolev spaces cannot hold for the nonlinear system.
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In the third part, we consider the case when resistivity is smaller than viscosity 0 < κ ≤ ν. We show that small initial data in Sobolev spaces yield stability. Furthermore, the stability properties qualitatively differ for the cases κ ≥ ν^3 and κ ≤ ν^3. For the case κ ≤ ν^3 we obtain norm inflation of order νκ^(− 1/3) due to growth in the magnetic field.