Logarithmic Sobolev Inequality Research Articles

Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat equations

The main goal of this work is to study the initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity in a bounded domain. By using the logarithmic Sobolev inequality and potential wells method, we obtain the decay, blow-up and non-extinction of solutions under some conditions, and the results extend the results of a recent paper Lijun Yan and Zuodong Yang (2018).

The Poincaré inequality and quadratic transportation-variance inequalities

It is known that the Poincare inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_{V}$ than before, which equals the double of Poincare constant. Applying the same argument leads to more characterizations of the Poincare inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_{2}^{2}(f\mu ,\mu )$ and its centralization $W_{2}^{2}(f_{c}\mu ,\mu )$ for $f_{c} = \frac{|\sqrt {f} - \mu (\sqrt {f})|^{2}} {\mathrm{Var} _{\mu }(\sqrt{f} )}$, which may be viewed as some special counterpart of the Rothaus’ lemma for relative entropy. Then it yields some new bound of $W_{2}^{2}(f\mu ,\mu )$ associated to the variance of $\sqrt{f} $ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze’s characterization of the Talagrand’s inequality.

Multiscale functional inequalities in probability: Concentration properties

In a companion article we have introduced a notion of multiscale functional inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. These inequalities are multiscale weighted versions of standard Poincare, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields $A$ arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results.

On Fokker-Planck equations with In- and Outflow of Mass

Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.

Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups

In this paper, we prove the fractional logarithmic Sobolev inequality on homogeneous groups. Also, we establish fractional logarithmic Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups. In addition, we show blow-up results for the fractional heat equation with logarithmic nonlinearity on homogeneous groups.

Behavior of Solutions to a Fourth-Order Nonlinear Parabolic Equation with Logarithmic Nonlinearity

This paper deals with a class of fourth-order parabolic equations modeling the evolution of the epitaxial growth of nanoscale thin films. By exploiting the variational structure of the equation, the logarithmic Sobolev inequality and a family of potential wells, the conditions on existence of global and infinite time blow-up solutions are obtained. Moreover, the finite time extinction and exponential decay of the solutions are also discussed.

Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C^*-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

Second order concentration via logarithmic Sobolev inequalities

We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.

Global regularity and convergence to equilibrium of reaction\u2013diffusion systems with nonlinear diffusion

We study the boundedness and convergence to equilibrium of weak solutions to reaction–diffusion systems with nonlinear diffusion. The nonlinear diffusion is of porous medium type, and the nonlinear reaction terms are assumed to grow polynomially and to dissipate (or conserve) the total mass. By utilising duality estimates, the dissipation of the total mass and the smoothing effect of the porous medium equation, we prove that if the exponents of the nonlinear diffusion terms are high enough, then weak solutions are bounded, locally Hölder continuous and their L^{infty }(Omega )-norm grows in time at most polynomially. In order to show convergence to equilibrium, we consider a specific class of nonlinear reaction–diffusion models, which describe a single reversible reaction with arbitrarily many chemical substances. By exploiting a generalised logarithmic Sobolev inequality, an indirect diffusion effect and the polynomial in time growth of the L^{infty }(Omega )-norm, we show an entropy–entropy production inequality which implies exponential convergence to equilibrium in L^p(Omega )-norm, for any 1le p < infty , with explicit rates and constants.

Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

This paper concerns with a semilinear heat equation with singular potential and logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, the existence of global solutions and infinite time blow-up solutions are obtained. The results of this paper indicate that the polynomial nonlinearity is a critical condition of existence of finite time blow-up solutions to semilinear heat equation with singular potential.

Multiplicity of weak solutions to a fourth-order elliptic equation with sign-changing weight function

ABSTRACT In this paper, a fourth-order elliptic equation with a sign-changing weight function is investigated. By the modified logarithmic Sobolev inequality and Nehari manifold it is shown that the problem admits at least two nontrivial weak solutions, which shows how the sign-changing weight function affect the existence and multiplicity of weak solutions to the problem.

Statistical mechanics of the periodic Benjamin–Ono equation

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({\mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $n\rightarrow\infty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.

Wright–Fisher–type equations for opinion formation, large time behavior and weighted logarithmic-Sobolev inequalities

We study the rate of convergence to equilibrium of the solution of a Fokker–Planck type equation introduced in [19] to describe opinion formation in a multi-agent system. The main feature of this Fokker–Planck equation is the presence of a variable diffusion coefficient and boundaries, which introduce new challenging mathematical problems in the study of its long-time behavior.

A Class of Fourth-order Parabolic Equations with Logarithmic Nonlinearity

In this paper, we apply the modified potential well method and the logarithmic Sobolev inequality to study the fourth-order parabolic equation with $p$-Laplacian and logarithmic nonlinearity. Some results are obtained under the different initial data conditions. More precisely, we give the global existence of weak solution by combining the classical Galerkin's method with the modified potential well method, decay estimates, and blow-up in finite time when the initial energy is subcritical and critical, respectively. In addition, sufficient conditions for the global existence and blow-up of the weak solution are also provided for supercritical initial energy. These results extend and improve many results in the literature.

Logarithmic-Sobolev and multilinear Hölder’s inequalities via heat flow monotonicity formulas

Heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriving functional and geometric inequalities which are in turn useful in mathematical analysis and applications. This paper aims mainly at proving Logarithmic Sobolev and multilinear Hölder’s inequalities through the heat flow method. Precisely, two entropy monotonicity formulas are constructed via the heat flow. It is shown that the first entropy monotonicity formula is intimately related to the concavity of the power of Shannon entropy and Fisher Information, from which the associated logarithmic Sobolev inequality for probability measure in Euclidean setting is recovered. The second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to establish the proof of the multilinear Hölder inequalities.

Multiple solutions for a class of infinitely degenerate elliptic equations with a sign‐changing weight function

In this paper, we study the Dirichlet problem for a class of degenerate elliptic equations with a sign‐changing weight function. Under some conditions on the weight function, we obtain at least two nontrivial solutions by the method of Nehari manifold and the logarithmic Sobolev inequality.

Entropic multipliers method for Langevin diffusion and weighted log Sobolev inequalities

In his work about hypocoercivity, Villani [20] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.

The Nehari manifold for fractional [formula omitted]-Laplacian equation with logarithmic nonlinearity on whole space

We study a fractional p-Laplacian equation in the whole space with the sign-changing logarithmic nonlinearity. By using fibrering maps and Nehari manifold we obtain the existence of at least two nontrivial solutions. Our result extends a recent result by Tian (2017). The main difficulty is the lack of logarithmic Sobolev inequality concerning to fractional p-Laplacian.

A Generalization of the Logarithmic Gross–Sobolev Inequality

A sharp integral inequality is proved that is used to derive a Sobolev interpolation inequality. A generalization of the logarithmic Sobolev inequality is proposed based on the Sobolev interpolation inequality.

Equality in the logarithmic Sobolev inequality

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems.