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Clément Mouhot - One of the best experts on this subject based on the ideXlab platform.

  • factorization of non symmetric operators and exponential h theorem
    2018
    Co-Authors: Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot
    Abstract:

    We present a factorization Method for estimating resolvents of non-symmetric operators in Banach or Hilbert spaces in terms of estimates in another (typically smaller) ``reference'' space. This applies to a class of operators writing as a ``regularizing'' part (in a broad sense) plus a dissipative part. Then in the Hilbert case we combine this factorization approach with an Abstract Plancherel identity on the resolvent into a Method for enlarging the functional space of decay estimates on semigroups. In the Banach case, we prove the same result however with some loss on the norm. We then apply these functional analysis approach to several PDEs: the Fokker-Planck and kinetic Fokker-Planck equations, the linear scattering Boltzmann equation in the torus, and, most importantly the linearized Boltzmann equation in the torus (at the price of extra specific work in the latter case). In addition to the Abstract Method in itself, the main outcome of the paper is indeed the first proof of exponential decay towards global equilibrium (e.g. in terms of the relative entropy) for the full Boltzmann equation for hard spheres, conditionnally to some smoothness and (polynomial) moment estimates. This improves on the result in [Desvillettes-Villani, Invent. Math., 2005] where the rate was ``almost exponential'', that is polynomial with exponent as high as wanted, and solves a long-standing conjecture about the rate of decay in the H-theorem for the nonlinear Boltzmann equation, see for instance [Cercignani, Arch. Mech, 1982] and [Rezakhanlou-Villani, Lecture Notes Springer, 2001].

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    Archive for Rational Mechanics and Analysis, 2016
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    The aim of the present paper is twofold: 1. We carry on with developing an Abstract Method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the Method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series. 2. We apply this Method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W1.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    The aim of the present paper is twofold:(1) We carry on with developing an Abstract Method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the Method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another-smaller or larger-Banach space under the condition that a certain iterate of the "mild perturba- tion" part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.(2) We apply this Method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap es- timates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W\_1.

  • Factorization for non-symmetric operators and exponential H-theorem
    2013
    Co-Authors: Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot
    Abstract:

    We present an Abstract Method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the Method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy Method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

  • kac s program in kinetic theory
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean \cite{Kac1956,McKean1967} giving rise to the Boltzmann equation. We solve the conjecture raised by Kac \cite{Kac1956}, motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates. Motivated by the inspirative paper by Gr\"unbaum \cite{Grunbaum}, we develop an Abstract Method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators ({\em consistency estimates}), along with differentiability estimates on the flow of the nonlinear limit equation ({\em stability estimates}). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac). Using this Method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of {\it entropic chaoticity}, as defined in \cite{CCLLV}. (3) Estimates on the time of relaxation to equilibrium, that are \emph{independent of the number of particles in the system}. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and \emph{true} Maxwell molecules interactions. The proof of the \emph{stability estimates} for these models requires significant analytic efforts and new estimates.

Stéphane Mischler - One of the best experts on this subject based on the ideXlab platform.

  • factorization of non symmetric operators and exponential h theorem
    2018
    Co-Authors: Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot
    Abstract:

    We present a factorization Method for estimating resolvents of non-symmetric operators in Banach or Hilbert spaces in terms of estimates in another (typically smaller) ``reference'' space. This applies to a class of operators writing as a ``regularizing'' part (in a broad sense) plus a dissipative part. Then in the Hilbert case we combine this factorization approach with an Abstract Plancherel identity on the resolvent into a Method for enlarging the functional space of decay estimates on semigroups. In the Banach case, we prove the same result however with some loss on the norm. We then apply these functional analysis approach to several PDEs: the Fokker-Planck and kinetic Fokker-Planck equations, the linear scattering Boltzmann equation in the torus, and, most importantly the linearized Boltzmann equation in the torus (at the price of extra specific work in the latter case). In addition to the Abstract Method in itself, the main outcome of the paper is indeed the first proof of exponential decay towards global equilibrium (e.g. in terms of the relative entropy) for the full Boltzmann equation for hard spheres, conditionnally to some smoothness and (polynomial) moment estimates. This improves on the result in [Desvillettes-Villani, Invent. Math., 2005] where the rate was ``almost exponential'', that is polynomial with exponent as high as wanted, and solves a long-standing conjecture about the rate of decay in the H-theorem for the nonlinear Boltzmann equation, see for instance [Cercignani, Arch. Mech, 1982] and [Rezakhanlou-Villani, Lecture Notes Springer, 2001].

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    Archive for Rational Mechanics and Analysis, 2016
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    The aim of the present paper is twofold: 1. We carry on with developing an Abstract Method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the Method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series. 2. We apply this Method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W1.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    The aim of the present paper is twofold:(1) We carry on with developing an Abstract Method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the Method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another-smaller or larger-Banach space under the condition that a certain iterate of the "mild perturba- tion" part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.(2) We apply this Method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap es- timates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W\_1.

  • Factorization for non-symmetric operators and exponential H-theorem
    2013
    Co-Authors: Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot
    Abstract:

    We present an Abstract Method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the Method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy Method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

  • kac s program in kinetic theory
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Stéphane Mischler, Clément Mouhot
    Abstract:

    This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean \cite{Kac1956,McKean1967} giving rise to the Boltzmann equation. We solve the conjecture raised by Kac \cite{Kac1956}, motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates. Motivated by the inspirative paper by Gr\"unbaum \cite{Grunbaum}, we develop an Abstract Method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators ({\em consistency estimates}), along with differentiability estimates on the flow of the nonlinear limit equation ({\em stability estimates}). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac). Using this Method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of {\it entropic chaoticity}, as defined in \cite{CCLLV}. (3) Estimates on the time of relaxation to equilibrium, that are \emph{independent of the number of particles in the system}. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and \emph{true} Maxwell molecules interactions. The proof of the \emph{stability estimates} for these models requires significant analytic efforts and new estimates.

Praveen Kumar - One of the best experts on this subject based on the ideXlab platform.

  • does lycra sleeve alter biomechanics of shoulder in people with post stroke hemiplegia a mechanistic study
    2018
    Co-Authors: Praveen Kumar, L Macleod, P. Mohan, W Wheeler
    Abstract:

    Abstract Introduction Glenohumeral subluxation (GHS) is reported in up to 81% of patients with stroke. Our previous studies found that a Lycra sleeve can reduce acromion-greater tuberosity distance (used for assessment of GHS) in people with chronic stroke (n=5). In a recent study on healthy people (n=31), we report reduction in AGT, change in scapula measurements and change in muscle activity after the application of Lycra sleeve. The aim of this study was to investigate the effect of Lycra sleeves on the acromion-greater tuberosity (AGT) distance, muscle activity around the shoulder region and scapular position in people with stroke. Abstract Method People with stroke who gave informed consent were recruited. Measurements were taken before and immediately after application of the sleeve. Portable diagnostic ultrasound, electromyography and a tape measure were used to measure AGT distance, muscle activity (biceps, triceps, deltoid, and supraspinatus) and position of the scapula respectively. Abstract Results Six participants with mean age 53±8 years were recruited. Mean±SD and 95% confidence intervals for AGT distances on the affected side before and after the application of sleeve were 2.1±0.3 (1.8-2.5cm) and 2.0±0.4 (1.6-2.4cm) respectively. There was a very slight increase in muscle activity after the application of Lycra sleeve in all muscles tested. Likewise there was reduction in scapula position (posterior tilt) (Mean difference 0.75±0.2cm after the application of sleeve. Abstract Discussion Findings from this study are in agreement with the previous research. Further research is required to establish the effectiveness of the Lycra sleeve using a well-designed randomised controlled trial.

  • the effects of lycra sleeves on acromion greater tuberosity distance muscle activity and scapula position in people with post stroke hemiplegia
    International Journal of Stroke, 2017
    Co-Authors: Praveen Kumar, L Macleod, P. Mohan, Catherine Wheeler
    Abstract:

    Abstract Introduction Glenohumeral subluxation (GHS) is reported in up to 81% of patients with stroke. Our previous studies found that a Lycra sleeve can reduce acromion-greater tuberosity distance (used for assessment of GHS) in people with chronic stroke (n=5). In a recent study on healthy people (n=31), we report reduction in AGT, change in scapula measurements and change in muscle activity after the application of Lycra sleeve. The aim of this study was to investigate the effect of Lycra sleeves on the acromion-greater tuberosity (AGT) distance, muscle activity around the shoulder region and scapular position in people with stroke. Abstract Method People with stroke who gave informed consent were recruited. Measurements were taken before and immediately after application of the sleeve. Portable diagnostic ultrasound, electromyography and a tape measure were used to measure AGT distance, muscle activity (biceps, triceps, deltoid, and supraspinatus) and position of the scapula respectively. Abstract Results Six participants with mean age 53±8 years were recruited. Mean±SD and 95% confidence intervals for AGT distances on the affected side before and after the application of sleeve were 2.1±0.3 (1.8-2.5cm) and 2.0±0.4 (1.6-2.4cm) respectively. There was a very slight increase in muscle activity after the application of Lycra sleeve in all muscles tested. Likewise there was reduction in scapula position (posterior tilt) (Mean difference 0.75±0.2cm after the application of sleeve. Abstract Discussion Findings from this study are in agreement with the previous research. Further research is required to establish the effectiveness of the Lycra sleeve using a well-designed randomised controlled trial.

Catherine Wheeler - One of the best experts on this subject based on the ideXlab platform.

  • the effects of lycra sleeves on acromion greater tuberosity distance muscle activity and scapula position in people with post stroke hemiplegia
    International Journal of Stroke, 2017
    Co-Authors: Praveen Kumar, L Macleod, P. Mohan, Catherine Wheeler
    Abstract:

    Abstract Introduction Glenohumeral subluxation (GHS) is reported in up to 81% of patients with stroke. Our previous studies found that a Lycra sleeve can reduce acromion-greater tuberosity distance (used for assessment of GHS) in people with chronic stroke (n=5). In a recent study on healthy people (n=31), we report reduction in AGT, change in scapula measurements and change in muscle activity after the application of Lycra sleeve. The aim of this study was to investigate the effect of Lycra sleeves on the acromion-greater tuberosity (AGT) distance, muscle activity around the shoulder region and scapular position in people with stroke. Abstract Method People with stroke who gave informed consent were recruited. Measurements were taken before and immediately after application of the sleeve. Portable diagnostic ultrasound, electromyography and a tape measure were used to measure AGT distance, muscle activity (biceps, triceps, deltoid, and supraspinatus) and position of the scapula respectively. Abstract Results Six participants with mean age 53±8 years were recruited. Mean±SD and 95% confidence intervals for AGT distances on the affected side before and after the application of sleeve were 2.1±0.3 (1.8-2.5cm) and 2.0±0.4 (1.6-2.4cm) respectively. There was a very slight increase in muscle activity after the application of Lycra sleeve in all muscles tested. Likewise there was reduction in scapula position (posterior tilt) (Mean difference 0.75±0.2cm after the application of sleeve. Abstract Discussion Findings from this study are in agreement with the previous research. Further research is required to establish the effectiveness of the Lycra sleeve using a well-designed randomised controlled trial.

P. Mohan - One of the best experts on this subject based on the ideXlab platform.

  • does lycra sleeve alter biomechanics of shoulder in people with post stroke hemiplegia a mechanistic study
    2018
    Co-Authors: Praveen Kumar, L Macleod, P. Mohan, W Wheeler
    Abstract:

    Abstract Introduction Glenohumeral subluxation (GHS) is reported in up to 81% of patients with stroke. Our previous studies found that a Lycra sleeve can reduce acromion-greater tuberosity distance (used for assessment of GHS) in people with chronic stroke (n=5). In a recent study on healthy people (n=31), we report reduction in AGT, change in scapula measurements and change in muscle activity after the application of Lycra sleeve. The aim of this study was to investigate the effect of Lycra sleeves on the acromion-greater tuberosity (AGT) distance, muscle activity around the shoulder region and scapular position in people with stroke. Abstract Method People with stroke who gave informed consent were recruited. Measurements were taken before and immediately after application of the sleeve. Portable diagnostic ultrasound, electromyography and a tape measure were used to measure AGT distance, muscle activity (biceps, triceps, deltoid, and supraspinatus) and position of the scapula respectively. Abstract Results Six participants with mean age 53±8 years were recruited. Mean±SD and 95% confidence intervals for AGT distances on the affected side before and after the application of sleeve were 2.1±0.3 (1.8-2.5cm) and 2.0±0.4 (1.6-2.4cm) respectively. There was a very slight increase in muscle activity after the application of Lycra sleeve in all muscles tested. Likewise there was reduction in scapula position (posterior tilt) (Mean difference 0.75±0.2cm after the application of sleeve. Abstract Discussion Findings from this study are in agreement with the previous research. Further research is required to establish the effectiveness of the Lycra sleeve using a well-designed randomised controlled trial.

  • the effects of lycra sleeves on acromion greater tuberosity distance muscle activity and scapula position in people with post stroke hemiplegia
    International Journal of Stroke, 2017
    Co-Authors: Praveen Kumar, L Macleod, P. Mohan, Catherine Wheeler
    Abstract:

    Abstract Introduction Glenohumeral subluxation (GHS) is reported in up to 81% of patients with stroke. Our previous studies found that a Lycra sleeve can reduce acromion-greater tuberosity distance (used for assessment of GHS) in people with chronic stroke (n=5). In a recent study on healthy people (n=31), we report reduction in AGT, change in scapula measurements and change in muscle activity after the application of Lycra sleeve. The aim of this study was to investigate the effect of Lycra sleeves on the acromion-greater tuberosity (AGT) distance, muscle activity around the shoulder region and scapular position in people with stroke. Abstract Method People with stroke who gave informed consent were recruited. Measurements were taken before and immediately after application of the sleeve. Portable diagnostic ultrasound, electromyography and a tape measure were used to measure AGT distance, muscle activity (biceps, triceps, deltoid, and supraspinatus) and position of the scapula respectively. Abstract Results Six participants with mean age 53±8 years were recruited. Mean±SD and 95% confidence intervals for AGT distances on the affected side before and after the application of sleeve were 2.1±0.3 (1.8-2.5cm) and 2.0±0.4 (1.6-2.4cm) respectively. There was a very slight increase in muscle activity after the application of Lycra sleeve in all muscles tested. Likewise there was reduction in scapula position (posterior tilt) (Mean difference 0.75±0.2cm after the application of sleeve. Abstract Discussion Findings from this study are in agreement with the previous research. Further research is required to establish the effectiveness of the Lycra sleeve using a well-designed randomised controlled trial.