33 research outputs found
Some Controllability Results for Linearized Compressible Navier-Stokes System
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 , with 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 , with is not
null controllable by localized interior control or by boundary control for
small time Next we consider two-dimensional compressible Navier-Stokes
system for barotropic fluid linearized around a constant steady state We prove that this system is also not null controllable by
localized interior control
- Maximal Regularity for some Operators Associated with Linearized Incompressible Fluid-Rigid Body Problems
We study an unbounded operator arising naturally after linearizing the system
modelling the motion of a rigid body in a viscous incompressible fluid. We show
that this operator is sectorial in for every , thus it has the maximal - regularity property. Moreover,
we show that the generated semigroup is exponentially stable with respect to
the norm. Finally, we use the results to prove the global existence for
small initial data, in an - setting, for the original nonlinear
problem
Lp theory for the interaction between the incompressible Navier-Stokes system and a damped beam
We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator
Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation
The article is devoted to the mathematical analysis of a fluid-structure
interaction system where the fluid is compressible and heat conducting and
where the structure is deformable and located on a part of the boundary of the
fluid domain. The fluid motion is modeled by the compressible
Navier-Stokes-Fourier system and the structure displacement is described by a
structurally damped plate equation. Our main results are the existence of
strong solutions in an setting for small time or for small data.
Through a change of variables and a fixed point argument, the proof of the main
results is mainly based on the maximal regularity property of the corresponding
linear systems. For small time existence, this property is obtained by
decoupling the linear system into several standard linear systems whereas for
global existence and for small data, the maximal regularity property is proved
by showing that the corresponding linear coupled {\em fluid-structure} operator
is sectorial
Controllability and positivity constraints in population dynamics with age structuring and diffusion
This Accepted Manuscript will be available for reuse under a CC BY-NC-ND licence after 24 months of embargo periodIn this article, we study the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka–McKendrick type). The control is localized in the space variable as well as with respect to the age. The first novelty we bring in is that the age interval in which the control needs to be active can be arbitrarily small and does not need to contain a neighbourhood of 0. The second one is that we prove that the whole population can be steered into zero in a uniform time, without, as in the existing literature, excluding some interval of low ages. Moreover, we improve the existing estimates of the controllability time and we show that our estimates are sharp, at least when the control is active for very low ages. Finally, we show that the system can be steered between two positive steady states by controls preserving the positivity of the state trajectory. The method of proof, combining final-state observability estimates with the use of characteristics and with L∞ estimates of the associated semigroup, avoids the explicit use of parabolic Carleman estimatesThe research of Enrique Zuazua was supported by the Advanced Grant DyCon (Dynamical Control) of the European Research Council Executive Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and the ICON project of the French ANR-16-ACHN-001