Duality Estimates Research Articles

Admissible inertial manifolds for neutral equations and applications

We study the existence of admissible inertial manifolds for parabolic neutral functional differential equations of the form where the linear differential operator A is positive definite and self-adjoint with a discrete spectrum, the difference operator F is a bounded linear operator, and the delay nonlinear operator f is φ-Lipschitz for φ belonging to an admissible function space defined on . Our method is based on Lyapunov–Perron's equations, duality estimates in admissible spaces and F-induced trajectories. An application to heat transfer with delays in materials with memory is also given to illustrate our results.

Delayed blow‐up for chemotaxis models with local sensing

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the $(H^1)'$-norm of the solutions, which can only grow with a square-root law in time. This additional $(H^1)'$-bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.

Parabolic evolution equations in interpolation spaces: boundedness, stability, and applications

We develop a general approach for the study of the boundedness and stability of solution to general parabolic semi-linear evolution equations in real-interpolation spaces. Our approach yields a unified method to handle problems in Newtonian viscous incompressible fluid flows. Moreover, our method will be applied to derive new results for the existence and polynomial stability of bounded solutions to the Ornstein–Uhlenbeck semi-linear equations and various diffusion equations with rough coefficients. The method is based on a combination of interpolation functors, duality estimates, smoothing properties of the linearized equation, and fixed point arguments.

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part II, below the critical regime

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size $K$ placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The reservoirs have slow dynamics, in the sense that they intervene at speed $N^{-\theta}$ w.r.t. the bulk dynamics. In the first part of this article, we treated the case $\theta>1$ for which the entropy method can be adapted. We treat here the case where the boundary dynamics is too fast for the Entropy Method to apply. We prove, using duality estimates inspired by previous work by Erignoux, Landim and Xu, that the hydrodynamic limit is given by the heat equation with Dirichlet boundary conditions, where the density at the boundaries is fixed by the parameters of the model.

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.

Admissible Inertial Manifolds for Delay Equations and Applications to Fisher-Kolmogorov Model

For solutions to the delay equations of the form $\dot{u}(t) + Au(t) = f(t,u_{t}), t\in\mathbb{R}$ , we prove the existence of an admissible inertial manifold. Here, the linear differential operator $A$ is positive definite and self-adjoint with a discrete spectrum, and the nonlinear part $f$ is $\varphi$ -Lipschitz for $\varphi$ belonging to an admissible space of functions defined on the whole line. An application to Fisher-Kolmogorov model with time-dependent environmental capacity and finite delay is also given to illustrate our results. Our main method is based on Lyapunov-Perron’s equation in combination with admissibility and duality estimates.

Light-by-light forward scattering sum rules for charmonium states

We apply three forward light-by-light scattering sum rules to charmonium states. We show that these sum rules imply a cancellation between charmonium bound state contributions, which are mostly known from the $\gamma \gamma$ decay widths of these states, and continuum contributions above $D \bar D$ threshold, for which we provide a duality estimate. We also show that two of these sum rules allow to predict the yet unmeasured $\gamma^\ast \gamma$ coupling of the $\chi_{c1}(1P)$ state, which can be tested at present high-luminosity $e^+ e^-$ colliders.

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.

Entropic structure and duality for multiple species cross-diffusion systems

This paper deals with the existence of global weak solutions for a wide class of (multiple species) cross-diffusions systems. The existence is based on two different ingredients: an entropy estimate giving some gradient control and a duality estimate that gives naturally L2 control. The heart of our proof is a semi-implicit scheme tailored for cross-diffusion systems firstly defined in Desvillettes et al. (2015) and a (nonlinear Aubin–Lions type) compactness result developed in Moussa (2016) and Andreianov et al. (2015) that turns the (potentially weak) gradient estimates into almost everywhere convergence. We apply our results to models having an entropy relying on the detailed balance condition exhibited by Chen et. al. in Chen et al. (2016).

Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems

We consider a system of reaction–diffusion equations describing the reversible reaction of two species $${\mathcal{U}}$$ , $${\mathcal{V}}$$ forming a third species $${\mathcal{W}}$$ and vice versa according to mass action law kinetics with arbitrary stoichiometric coefficients (equal or larger than one). Firstly, we prove existence of global classical solutions via improved duality estimates under the assumption that one of the diffusion coefficients of $${\mathcal{U}}$$ or $${\mathcal{V}}$$ is sufficiently close to the diffusion coefficient of $${\mathcal{W}}$$ . Secondly, we derive an entropy entropy-dissipation estimate, that is a functional inequality, which applied to global solutions of these reaction–diffusion systems proves exponential convergence to equilibrium with explicit rates and constants.

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.

The duality between Morrey spaces and its pre-dual spaces characterized by heat kernel

Let L be an elliptic operator, we give arguments about the duality estimate between Morrey spaces and its pre-dual spaces characterized by the heat kernel associated to L that is given in [6],[8].

Duality Estimates and Multigrid Analysis for Saddle Point Problems Arising from Mortar Discretizations

We present an abstract framework for the analysis of multigrid methods for a saddle point problem arising from mortar finite element discretizations. In contrast to other approaches, the iterates do not have to be in the positive definite subspace. Moreover, our approach also covers the case of nonnested Lagrange multiplier spaces. We apply the multigrid method to different mortar settings including dual Lagrange multipliers, linear elasticity, and a rotating geometry. Numerical results in two dimensions and three dimensions demonstrate the flexibility, efficiency, and reliability of our multigrid method.

Duality estimates for the numerical solution of integral equations

We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.

Semi-local duality estimate of the triple-Regge couplings of three-meson trajectories

Semi-local duality in Eeggeon-partiole scattering is used to establish the magnitude and small-t dependence of the triple-Regge couplings of three (nonpomeron) meson trajectories. Simple analytic formulae in terms of a few known parameters are given.

Improved duality estimates: time discrete case and applications to a class of cross-diffusion systems

We adapt the improved duality estimates for bounded coefficients derived by Canizo et al. to the framework of cross diffusion. Since the estimates can not be directly applied we need to derive a time discrete version of their results and apply it to an implicit semi-discretization in time of the cross diffusion systems. This leads to new global existence results for cross diffusion systems with bounded cross diffusion pressures and potentially superquadratic reaction.