Localized and degenerate controls for the incompressible Navier–Stokes system

IF 3.1 1区 数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2025-01-29 DOI:10.1002/cpa.22246
Vahagn Nersesyan, Manuel Rissel
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Abstract

We consider the global approximate controllability of the two‐dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier–Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well‐known open problem due to Agrachev.
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来源期刊
CiteScore
6.70
自引率
3.30%
发文量
59
审稿时长
>12 weeks
期刊介绍: Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.
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