Seminar Analysis and Mathematical Physics: Wojciech Ozanski (Florida State University)

Exact boundary controllability of the 2D and 3D incompressible ideal MHD system

We are concerned with the ideal magneto-hydrodynamic system of PDEs posed on a domain in $\mathbb{R}^2$ or $\mathbb{R}^3$, such that a part $\Gamma $ of the boundary of the domain is controlled in the sense that $v\cdot n = k, b\cdot n =l$ on $\Gamma$, where $k,l$ are controls, and $v$ and $b$ denote the velocity field and the magnetic field of the system, respectively. The aim of the control is to bring the system from a given initial state $(v_0,b_0)$ into a given final state $(v_1,b_1)$ in finite time. Such a controllability problem was resolved for the 2D and 3D incompressible Euler equation in the 1990's by Coron and Glass. The case of the ideal MHD system is much harder due to the lack of the pressure function in the equation for $b$. Very recently, exact boundary controllability of the 2D MHD system was achieved in the case of a flat channel by Kukavica, Novack, Vicol. In the talk we will discuss the main difficulties of the problem and present a recent result (joint with Kukavica), which completely resolves the case of any domain in both 2D and 3D incompressible ideal MHD system.

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