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21
ENTROPYENERGY INEQUALITIES AND IMPROVED CONVERGENCE RATES FOR NONLINEAR PARABOLIC EQUATIONS
, 2005
"... In this paper, we prove new functional inequalities of Poincaré type on the onedimensional torus S 1 and explore their implications for the longtime asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global ..."
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Cited by 16 (5 self)
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In this paper, we prove new functional inequalities of Poincaré type on the onedimensional torus S 1 and explore their implications for the longtime asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropyentropy production methods and based on appropriate Poincaré type inequalities.
On the BakryEmery criterion for linear diffusions and weighted porous media equations
 Comm. Math. Sci
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EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEADSPRING CHAINS
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2010
"... We show the existence of globalintime weak solutions to a general class of coupled FENEtype beadspring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stok ..."
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Cited by 13 (5 self)
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We show the existence of globalintime weak solutions to a general class of coupled FENEtype beadspring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in R d, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extrastress tensor appearing on the righthand side in the momentum equation. The extrastress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Plancktype parabolic equation, a crucial feature of which is the presence of a centreofmass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a squareintegrable and divergencefree initial velocity datum u ∼ 0 for the Navier–Stokes equation and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove
Stochastic stokes’ drift, homogenized functional inequalities, and large time behavior of brownian ratchets, preprint hal00270521
, 2008
"... Abstract. A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincare ́ and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized cons ..."
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Cited by 11 (2 self)
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Abstract. A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincare ́ and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates towards equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes ’ drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behaviour of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in selfsimilar variables attached to the center of mass to a stationary solution of a FokkerPlanck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes ’ drift with given potential flow.
A LOGARITHMIC HARDY INEQUALITY
, 2010
"... We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with superquadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev ..."
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Cited by 11 (8 self)
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We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with superquadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue’s measure nor a probability measure. All terms are scale invariant. After an EmdenFowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved among radial functions, in some range of the parameters.
A qualitative study of linear driftdiffusion equations with timedependent or vanishing coefficients
, 2005
"... This paper is concerned with entropy methods for linear driftdiffusion equations with explicitly timedependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the ..."
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Cited by 8 (2 self)
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This paper is concerned with entropy methods for linear driftdiffusion equations with explicitly timedependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the socalled Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions. We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some eventually timedependent large time asymptotic solutions. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(x  α ∇·), we prove that the inequality relating the entropy with the entropy production term is a HardyPoincaré type inequality, that we establish. Here we assume that α ∈ (0, 2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of timeperiodic coefficients, we prove the existence of a unique timeperiodic solution which attracts all other solutions. The case of a degenerate diffusion coefficient taking the form x  α with α> 2 is also studied. The Gibbs state exhibits a non integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional timedependence restores the smoothness of the asymptotic solution.
EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO FINITELY EXTENSIBLE NONLINEAR BEADSPRING CHAIN MODELS FOR DILUTE POLYMERS
, 2010
"... We show the existence of globalintime weak solutions to a general class of coupled FENEtype beadspring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stok ..."
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Cited by 6 (1 self)
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We show the existence of globalintime weak solutions to a general class of coupled FENEtype beadspring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in Rd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extrastress tensor appearing on the righthand side in the momentum equation. The extrastress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Plancktype parabolic equation, a crucial feature of which is the presence of a centerofmass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a squareintegrable and divergencefree initial velocity datum u ∼ 0 for the Navier–Stokes equation and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove the existence of a globalintime weak solution t ↦ → (u(t), ψ(t)) to the coupled Navier– Stokes–Fokker–Planck system, satisfying the initial condition (u(0), ψ(0)) = (u ∼ ∼0, ψ0), such that t ↦ → u(t) belongs to the classical Leray space and t ↦ → ψ(t) has bounded relative entropy and square integrable Fisher information over any time interval. It is also shown that in the absence of a body force, t ↦ → (u(t), ψ(t)) decays exponentially in time to (0, M) in the L2 × L1 norm, at a rate that is independent of the choice of (u∼ 0, ψ0) and of the centreofmass diffusion coefficient.
The entropy dissipation method for spatially inhomogeneous reactiondiffusiontype systems
, 2008
"... We study the large–time asymptotics of reaction–diffusion type systems, which feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expr ..."
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Cited by 5 (0 self)
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We study the large–time asymptotics of reaction–diffusion type systems, which feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimising) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion–convection equations and the main goal of this paper is to study its generalisation to systems of partial differential equations, which contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of reaction–diffusion–convection system arising in solid state physics as a paradigm for general nonlinear systems.
From Poincaré to logarithmic Sobolev inequalities: A gradient flow approach
 SIAM Journal on Mathematical Analysis
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Nonlinear flows and rigidity results on compact manifolds
"... Abstract. This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is ..."
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Cited by 5 (3 self)
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Abstract. This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality. Our approach relies in a nonlinear flow of porous medium / fast diffusion type which gives a clearcut interpretation of technical choices of exponents done in earlier works. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constant in the interpolation inequalities. hal00784887, version 1 4 Feb 2013 1. Introduction and