Logarithmic Sobolev Inequality Research Articles

On the variational interpretation of local logarithmic Sobolev inequalities

On the variational interpretation of local logarithmic Sobolev inequalities

Blow-up at infinity of solutions to a class of pseudo-parabolic equations with logarithmic nonlinearity and singular potential

We study the initial–boundary value problem for a class of pseudo-parabolic equations in a bounded domain Ω with smooth boundary ∂Ω and 0∈Ω⊂BR(0)⊂Rn(n≥1), where BR(0) is a ball centered at 0 with radius R. First, we simplify the scheme of potential well method, which helps us obtain the invariant set in an easier way. Combining logarithmic Sobolev inequality and Hardy inequality, we prove that the solutions to the pseudo-parabolic equations with logarithmic nonlinearity ulog|u| blow up at infinity when R≤1. However, there is no restriction on R if the singular potential is 1. Second, when the singular potential is 1 and the pseudo-parabolic term vanishes, the equations model the epitaxial growth of thin films. We also show that the solutions have better growth rate under weaker conditions, which improves the result in Han et al. (2019).

Corrigendum to “Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results”

Corrigendum to “Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results”

Sobolev trace-type inequalities via time-space fractional heat equations

Abstract This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$ . Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$ logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$ . As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges

Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges

Discrete logarithmic Sobolev inequalities in Banach spaces

AbstractLet be the discrete hypercube equipped with the uniform probability measure . We prove that if is a Banach space of finite cotype and , then every function satisfies the dimension‐free vector‐valued logarithmic Sobolev inequality The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector‐valued version of a deep inequality of Talagrand (1993). As an application, we use such vector‐valued logarithmic Sobolev inequalities to derive new lower bounds for the bi‐Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.

Sum-of-Squares Proofs of Logarithmic Sobolev Inequalities on Finite Markov Chains

Sum-of-Squares Proofs of Logarithmic Sobolev Inequalities on Finite Markov Chains

Existence and stability results of nonlinear swelling equations with logarithmic source terms

<abstract><p>We considered a swelling porous-elastic system characterized by two nonlinear variable exponent damping and logarithmic source terms. Employing the Faedo-Galerkin method, we established the local existence of weak solutions under suitable assumptions on the variable exponents functions. Furthermore, we proved the global existence utilizing the well-depth method. Finally, we established several decay results by employing the multiplier method and the Logarithmic Sobolev inequality. To the best of our knowledge, this represents the first study addressing swelling systems with logarithmic source terms.</p></abstract>

On Behavior of Solutions for Nonlinear Klein–Gordon Wave Type Models with a Logarithmic Nonlinearity and Multiple Time-Varying Delays

In this paper, we study the existence and exponential stability of solutions to a class of nonlinear delay Klein–Gordon wave type models on a bounded domain. Such models include multiple time-varying delays, frictional damping, and nonlinear logarithmic source terms. After showing the local existence result of the solutions using Faedo–Galerkin’s method and logarithmic Sobolev inequality, the global existence is analyzed. Then, under some appropriate conditions, energy decay estimates and exponential stability results of the global solutions are investigated.

Large-time asymptotics for degenerate cross-diffusion population models with volume filling

The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including “iterated” degenerated functions.

Long time behavior of weak solutions for the nonlinear damped beam equation

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data.

The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature

The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature

Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results

Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results

Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker–Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner's inequality.

Well‐posedness and asymptotic behavior for a p‐biharmonic pseudo‐parabolic equation with logarithmic nonlinearity of the gradient type

AbstractThis paper is concerned with the well‐posedness and asymptotic behavior for an initial boundary value problem of a pseudo‐parabolic equation with p‐biharmonic operator and logarithmic nonlinearity of the gradient type. The existence of the global weak solution is established by combining the technique of potential‐well and the method of Faedo–Galerkin approximation. Meantime, by virtue of the improved logarithmic Sobolev inequality and modified differential inequality, we obtain the results on infinite and finite time blow‐up and derive the lifespan of blow‐up solutions in various energy levels. Furthermore, the extinction phenomenon with extinction time is presented.

Gaussian upper bounds for the heat kernel on evolving manifolds

AbstractIn this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method that was previously used by Davies (Amer. J. Math. 109 (1987) 319–334) in the case of a non‐evolving manifold. This result directly implies Gaussian‐type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.

Dynamics of a spatially homogeneous Vicsek model for oriented particles on a plane

We consider a spatially homogeneous Kolmogorov–Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov–Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869–896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality.

Some Geometric Inequalities by the ABP Method

Abstract In this paper, we apply the so-called Alexandrov–Bakelman–Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $\Sigma $ of $\mathbb S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb S^{n}| \leq |\Sigma |$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for positive symmetric two-tensors on smooth domains in $\mathbb R^{n}$, which was established by D. Serre when the domain is convex. In the last application of the ABP method, we formulate and prove an inequality related to quermassintegrals of closed hypersurfaces of the Euclidean space.

Sharp Stability for LSI

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier–Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability inequality, the remainder is in terms of the entropy, not a metric. Recently, a stability result for H1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an Lp norm. Afterward, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H1 convergence is identified in this paper. Moreover, an explicit H1 bound via a moment assumption is shown. Additionally, the Lp stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.