Continuous data assimilation for the magnetohydrodynamic equations in 2D using one component of the velocity and magnetic fields

Abstract

We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in L 2 and H 1 norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.

Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm

“…Lastly, we emphasize that the 2D NSE is considered here only as a paradigm. Similar results can be obtained for other dissipative evolution equations, such as the 3D Navier-Stokes-α model [1], the 2D Bénard convection equations [2], and other models considered in [8], [17]-[22] and [55].…”

“…A rigorous treatment was given in [5] (see also [6]), where a general framework was introduced that can be applied to a large class of dissipative PDEs and various types of observables. Indeed, the broad applicability and complete analysis of this framework has been demonstrated in several works [1,7,8,9,10,17,18,19,20,21,22,29,36,45,51,55,56] for 2D and 3D dissipative systems that enjoy the global existence, uniqueness, and finite number of asymptotic (in time) determining parameters.…”