Volume 21, Number 4, October-December 2015
|Page(s)||1002 - 1028|
|Published online||12 June 2015|
Some controllability results for linearized compressible navier−stokes system∗
Centre for Applicable Mathematics,
TIFR, Post Bag No. 6503, GKVK Post Office, 560065
Revised: 25 September 2014
In this article, we study the null controllability of linearized compressible Navier−Stokes system in one and two dimension. We first study the one-dimensional compressible Navier−Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around (ρ̅, 0, θ̅), with ρ̅ > 0, θ̅ > 0 is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier−Stokes system for non-barotropic fluid linearized around a constant steady state (ρ̅, v̅, θ̅), with ρ̅ > 0, v̅ > 0, θ̅ > 0 is not null controllable by localized interior control or by boundary control for small time T. Next we consider two-dimensional compressible Navier−Stokes system for barotropic fluid linearized around a constant steady state (ρ̅, 0). We prove that this system is also not null controllable by localized interior control.
Mathematics Subject Classification: 35Q30 / 93C20 / 93B05
Key words: Linearized compressible Navier−Stokes System / Null controllability / localized interior control / boundary control / Gaussian Beam
© EDP Sciences, SMAI 2015
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