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Reduced Basis Approximation and A Posteriori Error Estimation for the TimeDependent Viscous Burgers Equation
 CALCOLO
, 2008
"... In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive C ..."
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Cited by 43 (5 self)
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In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive Constraint Method. Numerical results indicate that the a posteriori error bounds are practicable for reasonably large times — many convective scales — and reasonably large Reynolds numbers — O(100) or larger.
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 40 (15 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
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
"... We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this ..."
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Cited by 32 (14 self)
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We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator Mt can be obtained. Informally, this bound can be read as “Fix any finitedimensional projection Π on a subspace of sufficiently regular functions. Then the eigenfunctions of Mt with small eigenvalues have only a very small component in the image of Π.” We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of “Wiener polynomials, ” where the coefficients are possibly nonadapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris ’ lemma, which is unavailable in the present context. We conclude by showing that the twodimensional stochastic NavierStokes equations and a large class of reactiondiffusion equations fit the framework of our theory. Contents 1
Markov solutions for the 3D stochastic NavierStokes equations with state dependent noise, available on the arXiv preprint archive at the web address http://www.arxiv.org/abs/math.AP/0512361
"... Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a ..."
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Cited by 31 (10 self)
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Abstract: We construct a Markov family of solutions for the 3D NavierStokes equation perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [3].
Some bound state problems in quantum mechanics, in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
 Proc. Sympos. Pure Math., 76.1, American Mathematical Society
, 2007
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Stationary statistical properties of RayleighBénard convection at large Prandtl
"... This is the third in a series of our study of RayleighBénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to those of the infinite Prandtl number model fo ..."
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Cited by 14 (11 self)
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This is the third in a series of our study of RayleighBénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for RayleighBénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for RayleighBénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form Ra1/3(ln Ra)1/3 + c Ra7/2 ln Ra
Global attractors and determining modes for the 3D NavierStokesVoight equations
, 2007
"... We investigate the longterm dynamics of the threedimensional NavierStokesVoight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D NavierStokesVoight equations and for the dimension of a global attractor of a semigro ..."
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Cited by 13 (4 self)
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We investigate the longterm dynamics of the threedimensional NavierStokesVoight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D NavierStokesVoight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the NavierStokesVoight model as a nonviscous (inviscid) regularization of the threedimensional NavierStokes equations. Furthermore, we also show that the weak solutions of the Navier StokesVoight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0.
Global wellposedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid voigtα regularization. Preprint arXiv:1010.5024v1 [math.AP
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Asymptotic behavior of global attractors to the Boussinesq system for RayleighBénard convection at large Prandtl number
 Communications on Pure and Applied Mathematics, vol 60, issue 9
"... We study asymptotic behavior of the global attractors to the Boussinesq system for RayleighBenard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for RayleighBenard convection converge to that of the innite Prandtl number model for c ..."
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Cited by 10 (8 self)
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We study asymptotic behavior of the global attractors to the Boussinesq system for RayleighBenard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for RayleighBenard convection converge to that of the innite Prandtl number model for convection as the Prandtl number approaches innity. This oers partial justication of the innite Prandtl number model for convection as a valid simplied model for convection at large Prandtl number even in the long time regime.
Global regularity of the NavierStokes equation on thin three dimensional domains with periodic boundary conditions
 Electronic J. Diff. Eqns
, 1999
"... Abstract: This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the NavierStokes equation on a thin three dimensional domain with periodic boundary conditions has global regularity, as long as ..."
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Cited by 9 (0 self)
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Abstract: This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the NavierStokes equation on a thin three dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger than that obtainable via “small data ” estimates. The approach taken is to consider the three dimensional equation as a perturbation of the equation when the vector field does not depend upon the coordinate in the thin direction. Keywords: NavierStokes equation, thin domain A.M.S. Classification (1991): Primary 35Q30, 76D05, Secondary 35B65. The celebrated NavierStokes equation is concerned with the velocity vector u on a domain Ω, describing the flow of an incompressible fluid. A famous unsolved problem is the following: if Ω is a nice enough 3 dimensional domain, and if the initial data is smooth, and the forcing term is uniformly smooth in time, then does it follow that the solution is smooth for all time? What is known is that a weak solution exists, although it is not