In this paper, we deal with the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a $m\times N$ state (corresponding to the velocities of the fluids) by $N-1$ distributed scalar controls. The proof of the controllability of the coupled Stokes system is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.
We consider the 3D Navier-Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearised around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity.
The goal of this article is to show a local exact controllability to smooth (C2) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the ow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the ow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.
The aim of this work is to show the local null controllability of a fluid-solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier-Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid-rigid body system with Navier boundary conditions. This work i…
We prove a null controllability result for the Vlasov-Navier-Stokes system, which describes the interaction of a large cloud of particles immersed in a fluid. We show that one can modify both the distribution of particles and the velocity field of the fluid from any initial state to the zero steady state, by means of an internal control. Indeed, we can modify the non-linear dynamics of the system in order to absorb the particles and let the fluid at rest. The proof is achieved thanks to the return method and a Leray-Schauder fixed-point argument.
In this paper we consider a cascade system in non divergence form which models the interaction between two different species, the first one can be seen as a predator and the other as a prey. Both of them depend on time, on age and on space. Moreover, the diffusion coefficients degenerate at the boundary of domain. We study, in particular, null controllability of the system via the observability inequality for the non homogeneous adjoint problem, which is deduced by Carleman estimates.
In the Eulerian approach, the motion of an incompressible fluid is usually described by the velocity field which is given by the Navier--Stokes system. The velocity field generates a flow in the space of volume-preserving diffeomorphisms. The latter plays a central role in the Lagrangian description of a fluid, since it allows to identify the trajectories of individual particles. In this paper, we show that the velocity field of the fluid and the corresponding flow of diffeomorphisms can be simultaneously approximately controlled using a finite-dimensional external force. The proof is based on…
In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in the density or stress equation is everywhere in the …
Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions
An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.
In this paper, we deal with the existence of insensitizing controls for the Navier-Stokes equations in a bounded domain with Dirichlet boundary conditions. We prove that there exist controls insensitizing the $L^2$ -norm of the observation of the solution in an open subset $\mathcal{O}$ of the domain, under suitable assumptions on the data. This problem is equivalent to an exact controllability result for a cascade system. First we prove a global Carleman inequality for the linearized Navier-Stokes system with right-hand side, which leads to the null controllability at any time $T>0$. Then, we…
An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier--Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.
In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation, both towards the null equilibrium state and towards weak trajectories. We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time glob…
This work is concerned with the necessary conditions of optimality for a minimal time control problem $(P)$ for the linearized Navier-Stokes periodic flow in a 2D-channel, subject to a boundary input which acts on the transversal component of the velocity. The objective in this problem is the reaching of the laminar regime in a minimum time, as well as its preservation after this time. The determination of the necessary conditions of optimality relies on the analysis of intermediate minimal time control problems $(P_{k})$ for the Fourier modes $"k"$ associated to the Navier-Stokes equations an…
We prove controllability results for abstract systems of weakly coupled $N$ evolution equations in cascade by a reduced number of boundary or locally distributed controls ranging from a single up to $N-1$ controls. We give applications to cascade coupled systems of $N$ multi-dimensional-hyperbolic, parabolic and diffusive equations.
The goal of this article is to present a local exact controllability result for the 2 and 3-dimensional compressible Navier-Stokes equations on a constant target trajectory when the controls act on the whole boundary. Our study is then based on the observability of the adjoint system of some linearized version of the system, which is analyzed thanks to a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces.
We study the local controllability properties of 2D and 3D bio-mimetic swimmers employing the change of their geometric shape to propel themselves in an incompressible fluid described by Navier-Stokes equations. It is assumed that swimmers' bodies consist of finitely many parts, identified with the fluid they occupy, that are subsequently linked by the rotational and elastic internal forces. These forces are explicitly described and serve as the means to affect the geometric configuration of swimmers' bodies. Similar models were previously investigated in [6]-[13].
In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time.
We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to …
In this paper we deal with the compressible Navier-Stokes equations with a friction term in one dimension on an interval. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. Inspired by the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger equation, we prove by choosing irrotational data and using the notion of effective velocity developed in \cite{cpde,cras} that the exact global controllability result does not hold for any time $T$.
This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixing in the dual-Lipschitz metric. This result is then applied to the 2D Navier-Stokes system driven by a random force acting through the boundary. A by-product of our analysis is the local exponential stabilisation of the boundary-driven Navier-Stokes system by a regular boundary co…
We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the $4/5$ law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness cri…
In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is small-time locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced by the first author that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L^2 parabolic Carleman estimat…
We provide explicit time-varying feedback laws that locally stabilize the two dimensional internal controlled incompressible Navier-Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as well as quantitative null controllability with explicit controls having $e^{C/T}$ costs.
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, …
This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.
Rigorous estimates for the total - (kinetic) energy plus pressure - flux in R^3 are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition - involving Taylor length scale and the size of the domain - sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in R^3 and no regularity or homogeneity/scaling assumptions are mad…
We consider a $n \times n$ nonlinear reaction-diffusion system posed on a smooth bounded domain $Ω$ of $\mathbb{R}^N$. This system models reversible chemical reactions. We act on the system through $m$ controls ($1 \leq m < n$), localized in some arbitrary nonempty open subset $ω$ of the domain $Ω$. We prove the local exact controllability to nonnegative (constant) stationary states in any time $T >0$. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space $L^\infty(Ω)^n$. The …
In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in…
We develop Ladyzhenskaya-Prodi-Serrin type spectral regularity criteria for 3D incompressible Navier-Stokes equations in a torus. Concretely, for any $N>0$, let $w_N$ be the sum of all spectral components of the velocity fields whose all three wave numbers are greater than $N$ absolutely. Then, we show that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of $w_N$ and corresponding energy spectru…
The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform about the probabilistic features of the associated stochastic cascades, and how some probabilistic features of the stochastic cascades can, in turn, inform about the existence and uniqueness (or the lack thereof) of solutions. Our method of incorporating the stochastic explosion gives a simpler and more natural method to construct the solution compared to the…
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the…
We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navierslip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We provethe small-time exact controllability of the position and velocity of the solid when the control takes the form of adistributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body.The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls forwhich the ``Navier-Stokes+rigid body'' sys…
The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise,…
Local analysis of the two dimensional Navier-Stokes equations is used to obtain estimates on the energy and enstrophy fluxes involving Taylor and Kraichnan length scales and the size of the domain. In the framework of zero driving force and non-increasing global energy, these bounds produce sufficient conditions for existence of the direct enstrophy and inverse energy cascades. Several manifestations of locality of the fluxes under these conditions are obtained. All the scales involved are actual physical scales in R^2 and no homogeneity assumptions are made.
By direct numerical simulation to the two-dimensional Navier-Stokes equations with small-scale forcing and large-scale damping, Xiao-Wan-Chen-Eyink (2009) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. On the other hand, Alexakis-Doering (2006) calculated upper bound of the bulk averaged enstrophy dissipation rate of the steady-state two dimensional turbulence. {In this paper, we show that vortex thinning induces enhanced dissipation with strictly slower vanishing order of the enstrophy dissipation than $Re^{-1}$.}
