Boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term

AbstractIn this paper, we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing termx″+f(x)g(x′)+γ|x|α−1x=p(t), where f(x), p(t) are odd functions, p(t) is smooth 1-periodic function and γ≠0 is a constant. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, we show the existence of Aubry–Mather sets as well

Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems

We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions.Comment: 35 pages, to appear in Archive Rational Mech. Ana

Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $\epsilon$ is proportional to $N^{-1/\theta}$ for $\theta>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Fa\`a di Bruno's formula.Comment: 28 pages, no figure

Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory

In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular, we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single-porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of plant-soil interaction when trying to understand solute losses from the soil

Entropy, Duality and Cross Diffusion

This paper is devoted to the use of the entropy and duality methods for the existence theory of reaction-cross diffusion systems consisting of two equations, in any dimension of space. Those systems appear in population dynamics when the diffusion rates of individuals of two species depend on the concentration of individuals of the same species (self-diffusion), or of the other species (cross diffusion)

Asymptotic analysis for Hamilton-Jacobi equations associated with sub-riemannian control systems

The long-time average behaviour of the value function in the calculus of variations, where both the Lagrangian and Hamiltonian are Tonelli, is known to be connected to the existence of the limit of the corresponding Abel means as the discount factor goes to zero. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or, ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when the Hamiltonian fails to be Tonelli: in particular, for control systems that can be associated with a family of vector fields which satisfies the Lie Algebra rank condition. First, following a dynamical approach we characterise the unique constant for which the ergodic equation admits solutions. Then, we construct a critical solution which coincides with its Lax-Oleinik evolution.Comment: 29 page