Controllability of 3D incompressible Euler equations by a finite-dimensional external force

Abstract

Abstract.In this paper, we study the control system associated with the incompressible 3D Euler system. We show that the velocity field and pressure of the fluid are exactly controllable in projections by the same finite-dimensional control. Moreover, the velocity is approximately controllable. We also prove that 3D Euler system is not exactly controllable by a finite-dimensional external force.Mathematics Subject Classification. 35Q35, 93C20.

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

“…Furthermore, the methods used were rather ad hoc [HM06,HM11]. On the other hand, systematic results close to the setting of this paper were given previously in [AS05, AS06, Shi07, Shi08, Shi10, Ner10,Ner15]. While related, our results more directly extend the geometric control theory work of Jurdjevic and Kupka [JK81, JK85,Jur97].…”

“…Regarding existing literature concerning the controllability of the Euler equation, let us mention [Shi08,Ner10,Ner15] and also [Rom04,Shi07] for related work on the 3D Navier-Stokes equations. The reference [Ner10] treats the same control problem as below but using the Agrachev-Sarychev approach in the functional setting of H m for an arbitrary but fixed m ∈ N. Below we treat the dynamics on the space C ∞ = ∩ m≥0 H m using the methods of Section 3. In particular, because m ∈ N can be arbitrary the main result in [Ner10] implies the main control result for this dynamics (Theorem 5.112 below).…”

“…The reference [Ner10] treats the same control problem as below but using the Agrachev-Sarychev approach in the functional setting of H m for an arbitrary but fixed m ∈ N. Below we treat the dynamics on the space C ∞ = ∩ m≥0 H m using the methods of Section 3. In particular, because m ∈ N can be arbitrary the main result in [Ner10] implies the main control result for this dynamics (Theorem 5.112 below). For general background on the mathematical theory of inviscid, incompressible flow, see [MB02,MP12].…”

Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force

“…The following result is a version of Theorem 1.8 and Remark 1.9 in [Shi06] and Theorem 2.1 in [Ner10] in the case of the 3D NS system in the spaces H k , k ≥ 3. For the sake of completeness, we give all the details of the proof, even though it is very close to the proofs of the previous results.…”

Controllability of the 3D Compressible Euler System

“…It is straightforward to see that sup (û,ĝ)∈K Kf n (t,û,ĝ) C(JT ,H k+1 ) → 0, (e.g. see [11,Chapter 3] or [14,Section 6]). Thus v n − u 1 C(JT ,H k+1 ) → 0 as n → ∞.…”