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15
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 16 (8 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
2010: A simple framework to justify linear response theory
 Nonlinearity
"... The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced d ..."
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Cited by 5 (2 self)
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The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications. 1
Controllability of 3D incompressible Euler equations by a finitedimensional external force
, 810
"... 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 finitedimensional control. Moreover, the velocity is approximately controllable. We ..."
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Cited by 1 (0 self)
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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 finitedimensional control. Moreover, the velocity is approximately controllable. We also prove that 3D Euler system is not exactly controllable by a finitedimensional external force. 1
On the controllability of nonlinear partial differential equations
 In Proceedings of the International Congress of Mathematicians. Volume I
, 2010
"... A control system is a dynamical system on which one can act by using controls. A classical issue is the controllability problem: Is it possible to reach a desired target from a given starting point by using appropriate controls? We survey some methods to handle this problem when the control system i ..."
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Cited by 1 (1 self)
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A control system is a dynamical system on which one can act by using controls. A classical issue is the controllability problem: Is it possible to reach a desired target from a given starting point by using appropriate controls? We survey some methods to handle this problem when the control system is modeled by means of a nonlinear partial differential equation and when the nonlinearity plays a crucial role.
Fluid Simulation using Laplacian Eigenfunctions
"... We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity an ..."
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Cited by 1 (0 self)
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We present an algorithm for the simulation of incompressible fluid phenomena that is computationally efficient and leads to visually convincing simulations with far fewer degrees of freedom than existing approaches. Rather than using an Eulerian grid or Lagrangian elements, we represent vorticity and velocity using a basis of global functions defined over the entire simulation domain. We show that choosing Laplacian eigenfunctions for this basis provides benefits, including correspondence with spatial scales of vorticity and precise energy control at each scale. We perform Galerkin projection of the NavierStokes equations to derive a time evolution equation in the space of basis coefficients. Our method admits closed form solutions on simple domains but can also be implemented efficiently on arbitrary meshes.
ProjectTeam CORIDA Robust Control Of Infinite Dimensional Systems and Applications
"... c t i v it y e p o r t 2008 Table of contents ..."
On
, 2006
"... finitedimensional projections of distributions for solutions of randomly forced PDE’s ..."
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finitedimensional projections of distributions for solutions of randomly forced PDE’s
Contents
, 2005
"... The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finitedimensional projection. Our sufficient condition is verified for any torus in ..."
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The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finitedimensional projection. Our sufficient condition is verified for any torus in R 3. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D NavierStokes system has a unique strong solution for any initial function and a large class of external forces.
unknown title
, 2008
"... A probabilistic argument for the controllability of conservative systems ..."