We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–Landau-type potential.

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of $orthodox$ crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical...

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodox crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical...

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

We investigate the steady transport equation $$\lambda z+w·\nabla z+az=f,\lambda >0$$ in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions $w,a$ are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular stress is put onto the problem to extend the results to as less regular vector fields $w,a$, as possible (conserving the requirement of...

We study the discrete Schrödinger operator $H$ in ${𝐙}^d$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left(x_1\right)tan\pi (\alpha ·x_2+\omega )$, where $x=(x_1,x_2),x_1\in {𝐙}^{d_1},x_2\in {𝐙}^{d_2},\alpha \in {𝐑}^{d_2},\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha$ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${𝐙}^{d_1}$ waves, the form...

Žemlička, Jan: Structure of steady rings. Zemek, Martin: On some aspects of subdifferentiality of functions on Banach spaces. Hlubinka, Daniel: Construction of Markov kernels with application for moment problem solution. Somberg, Petr: Properties of the BGG resolution on the spheres. Krump, Lukáš: Construction of Bernstein-Gelfand-Gelfand for almost hermitian symmetric structures. Kolář, Jan: Simultaneous extension operators. Porosity.

We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...

We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time,...

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $logt$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.