Part I: On the Stability threshold of Couette flow in a uniform magnetic field; Part II: Quantitative convergence to equilibrium for hypoelliptic stochastic differential equations with small noise

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This dissertation contains two parts. In Part I, We study the stability of the Couette flow (y,0,0) in the presence of a uniform magnetic field a*(b, 0, 1) on TxRxT using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, perfect conductor limit Re^(-1) = Rm^(-1) << 1 and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations that are O(Re^(-1)) in the Sobolev space H^N. More precisely, we establish the decay estimates predicted by a linear stability analysis and show that the perturbations u(t,x+yt,y,z) and b(t,x+yt,y,z) remain O(Re^(-1)) in H^M for some 1 << M(b) < N. In the Navier-Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption of O(Re^(-3/2)) data. The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier-Stokes equations linearized around Couette flow. In Part II, we study the convergence rate to equilibrium for a family of Markov semigroups (parametrized by epsilon>0) generated by a class of hypoelliptic stochastic differential equations on R^d, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter epsilon. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-epsilon upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a De Giorgi-type iteration (both uniform in epsilon). The spectral gap estimate on the semigroup is obtained by a weak Poincar'e inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.