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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 9 (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
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 3 (0 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.
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 3 (1 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.
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 ..."
Abstract

Cited by 3 (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.
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 2 (1 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
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 2 (2 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.
EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO HOOKEANTYPE BEADSPRING CHAIN MODELS FOR DILUTE POLYMERS
"... We show the existence of globalintime weak solutions to a general class of coupled Hookeantype 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–S ..."
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Cited by 1 (1 self)
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We show the existence of globalintime weak solutions to a general class of coupled Hookeantype 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 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 with respect to M and t ↦ → ψ(t)/M has integrable Fisher information (w.r.t. the measure dν: = M(q) dq dx) over any time interval [0, T], T> 0. If the density of body forces f on the righthand side of the Navier–Stokes momentum equation vanishes, then t ↦ → (u(t), ψ(t)) decays exponentially in time to (0, M) in
Mathematical Models and Methods in Applied Sciences c ○ World Scientific Publishing Company EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEADSPRING CHAINS
, 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 ..."
Abstract
 Add to MetaCart
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
SOLUTIONS TO FINITELY EXTENSIBLE NONLINEAR BEADSPRING CHAIN MODELS FOR DILUTE POLYMERS
"... Report no. OxPDE10/05 Existence and equilibration of global weak solutions to finitely extensible nonlinear ..."
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Report no. OxPDE10/05 Existence and equilibration of global weak solutions to finitely extensible nonlinear
Improved intermediate asymptotics for the heat equation ✩,✩✩
, 908
"... This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in selfsimilar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using ..."
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This letter is devoted to results on intermediate asymptotics for the heat equation. We study the convergence towards a stationary solution in selfsimilar variables. By assuming the equality of some moments of the initial data and of the stationary solution, we get improved convergence rates using entropy / entropyproduction methods. We establish the equivalence of the exponential decay of the entropies with new, improved functional inequalities in restricted classes of functions. This letter is the counterpart in a linear framework of a recent work on fast diffusion equations, see [8]. Results extend to the case of a FokkerPlanck equation with a general confining potential. Key words: Heat equation, FokkerPlanck equation, OrnsteinUhlenbeck equation, intermediate asymptotics, selfsimilar variables, stationary solutions, large time behavior, rate of convergence, entropy, Poincaré inequality,