Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

Improved Poincaré inequalities

Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be built, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy–Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.     

Nonlinear flows and rigidity results on compact manifolds

This paper is devoted to rigidity results for some elliptic PDEs and to optimal constants in 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. The largest value of this parameter provides an estimate for the optimal constant in the corresponding interpolation inequality. Our approach relies on a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works on rigidity. 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 constants in the interpolation inequalities.     

Nonlinear degenerate elliptic equation with Hardy potential and critical parameter

Some embedding inequalities in Hardy–Sobolev spaces with general weight functions were proved, and a positive answer to an open problem raised by Brezis–Vázquez was given. In the weighted Hardy–Sobolev spaces, the existence of nontrivial (many) solutions to the corresponding nonlinear degenerated elliptic equations with Hardy potential and critical parameter under conditions weaker than Ambrosetti–Rabinowitz condition, was obtained.     

Optimal lower bounds on asymptotic support propagation rates for the thin-film equation

We derive lower bounds on asymptotic support propagation rates for strong solutions of the Cauchy problem for the thin-film equation. The bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp. Our results hold in case of at most three spatial dimensions and n∈(1,2.92)$n\in (1,2.92)$. The result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality; combined with some entropy production estimates, the optimal rate of propagation is obtained. To the best of our knowledge, these are the first lower bounds on asymptotic support propagation rates for higher-order nonnegativity-preserving parabolic equations.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

直线上函数的Banach空间的Poincaré型不等式的变分公式

基于研究对数Sobolev,Nash和其它泛函不等式的需要,将Poincare不等式 的变分公式拓广到一大类直线上函数的Banach(Orlicz)空间.给出了这些不等式成立 与否的显式判准和显式估计. 作为典型应用,仔细考察了对数Sobolev常数.     

On the Two-Dimensional Navier—Stokes Equations with the Free Boundary Condition

In this article we consider the two-dimensional Navier—Stokes equations with free boundary condition (open surface), and derive a number of different results: a new orthogonal property for the nonlinear term, improved a priori estimates on the solution, an upper bound on the dimension of the attractor which agrees with the conventional theory of turbulence; finally, for elongated rectangular domains, an improved Lieb—Thirring (collective Sobolev) inequality leads to an upper bound on the dimension of the attractor which might be optimal. Accepted 11 July 1996     

The optimal Euclidean L-Sobolev logarithmic inequality

We prove an optimal logarithmic Sobolev inequality in . Explicit minimizers are given. This result is connected with best constants of a special class of Gagliardo-Nirenberg-type inequalities.     

Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift–Hohenberg Equation in one and two space dimensions

In this work we have obtained explicit and accurate estimates of the sup-norm for solutions of the Swift–Hohenberg Equation (SHE) in one and two space dimensions. By using the best (so far) available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations, we have evaluated in an explicit manner the values of the sup-norm of the solutions of the SHE. In addition we have calculated the so-called time-averaged dissipative length scale associated to the above solutions.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002     

Dirichlet and Neumann problems to critical Emden–Fowler type equations

We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations.      

Some remarks on Sobolev type inequalities

We extend two Sobolev type inequalities for balls to arbitrary smooth bounded domains. In the case of balls, one inequality is due to Brezis and Lieb and another is due to Escobar. The extension has been achieved by analyzing the asymptotic behaviour of solutions of certain semilinear Neumann problems.     

Capacitary Criteria for Poincaré-Type Inequalities

The Poincaré-type inequality is a unification of various inequalities including the F-Sobolev inequalities, Sobolev-type inequalities, logarithmic Sobolev inequalities, and so on. The aim of this paper is to deduce some unified upper and lower bounds of the optimal constants in Poincaré-type inequalities for a large class of normed linear (Banach, Orlicz) spaces in terms of capacity. The lower and upper bounds differ only by a multiplicative constant, and so the capacitary criteria for the inequalities are also established. Both the transient and the ergodic cases are treated. Besides, the explicit lower and upper estimates in dimension one are computed. Mathematics Subject Classifications (2000) 60J55, 31C25, 60J35, 47D07.Research supported in part NSFC (No. 10121101) and 973 Project.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. (Received 11 October 2000; in revised form 13 March 2001)     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.     

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

Logarithmic Sobolev,isoperimetry and transport inequalities on graphs

In this paper,we study some functional inequalities(such as Poincaré inequality,logarithmic Sobolev inequality,generalized Cheeger isoperimetric inequality,transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of(random) path method.We provide estimates of the involved constants.