Logarithmic Sobolev Inequality Research Articles

The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that ). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).

Logarithmic Sobolev inequalities for bounded domains and applications to drift-diffusion equations

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal drift-diffusion models and prove the exponential decay of their solutions to constant steady states.

Revised logarithmic Sobolev inequalities of fractional order

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on Rn with an explicit expression for the constant. Namely, we show that for all 0<s<n2 and a>0 we have the inequality∫Rn|f(x)|2log⁡(|f(x)|2‖f‖L2(Rn)2)dx+ns(1+log⁡a)‖f‖L2(Rn)2≤C(n,s,a)‖(−Δ)s/2f‖L2(Rn)2 with an explicit C(n,s,a) depending on a, the order s, and the dimension n, and investigate the behaviour of C(n,s,a) for large n. Notably, for large n and when s=1, the constant C(n,1,a) is asymptotically the same as the sharp constant of Lieb and Loss that was computed in [13]. Moreover, we prove a similar type inequality for functions f∈Lq(Rn)∩W1,p(Rn) whenever 1<p<n and p<q≤p(n−1)n−p.

Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.

The logarithmic Sobolev inequality on the Heisenberg group and applications to the uncertainty inequality and heat equation

We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

Stability for the logarithmic Sobolev inequality

This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.

Logarithmic Sobolev Inequalities, Gaussian Upper Bounds for the Heat Kernel, and the $$\textrm{G}_{2}$$-Laplacian Flow

Logarithmic Sobolev Inequalities, Gaussian Upper Bounds for the Heat Kernel, and the $$\textrm{G}_{2}$$-Laplacian Flow

Coercive inequalities on Carnot groups: taming singularities

In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson–Ornstein–Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators.

On the variational interpretation of local logarithmic Sobolev inequalities

On the variational interpretation of local logarithmic Sobolev inequalities

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

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”

This corrigendum corrects several mistakes in Appendix B.3 of [Ann. Inst. H. Poincaré Anal. Non Linéaire (2023). .

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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.