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INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES
"... Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1. ..."
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Cited by 21 (9 self)
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Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1.
On the BakryEmery criterion for linear diffusions and weighted porous media equations
 Comm. Math. Sci
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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 15 (4 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.
Phientropy inequalities for diffusion semigroups. Prépublication, 2009. [BGL01] [CE02
 J. Math. Pures Appl
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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 6 (1 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.
A quantitative logSobolev inequality for a two parameter family of functions
 Int. Math. Res. Not. IMRN
, 1093
"... Abstract. We prove a sharp, dimensionfree stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C1,1 functions. Moreover, we show how to enlarge this space at the expense of the dime ..."
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Cited by 6 (1 self)
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Abstract. We prove a sharp, dimensionfree stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C1,1 functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy. 1.
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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SHARP INTERPOLATION INEQUALITIES ON THE SPHERE: NEW METHODS AND CONSEQUENCES
, 2012
"... Abstract. These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion betwe ..."
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Cited by 3 (3 self)
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Abstract. These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the LaplaceBeltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.