Related papers: The nonlinear heat equation on W-random graphs
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
We consider the long-term evolution of an inhomogeneous long-range interacting $N$-body system. Placing ourselves in the dynamically hot limit, i.e. neglecting collective effects, we derive a large deviation principle for the system's…
We derive an energy-based continuum limit for $\varepsilon$-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most $O(\varepsilon)$; the prefactor involves…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics,…
We study the steady states and dynamics of a thin film-type equation with non-conserved mass in one dimension. The evolution equation is a nonlinear fourth-order degenerate parabolic PDE motivated by a model of volatile viscous fluid films…
We derive a variational expression for the correlation time of physical observables in steady-state diffusive systems. As a consequence of this variational expression, we obtain lower bounds on the correlation time, which provide speed…
Gas transport and other complex real-world challenges often require solving and controlling partial differential equations (PDEs) defined on graph structures, which typically demand substantial memory and computational resources. The Random…
A large class of evolutionary processes can be modeled by a rule which involves self-replication of some physical quantity with a non local rescaling. I show that a class of such models are exactly solvable -- in the discrete as well as…
We consider some interacting particle processes with long-range dynamics: the zero-range and exclusion processes with long jumps. We prove that the hydrodynamic limit of these processes corresponds to a (possibly non-linear) fractional heat…
It is proposed to consider the fast thermalization of gluons in relativistic heavy-ion collisions as a diffusion process in momentum space. Closed-form analytical solutions of a nonlinear boson diffusion equation (NBDE) with constant drift…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
We consider the monomer-dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We…
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of…
We consider an evolution equation of parabolic type in R having a travelling wave solution. We perform an appropriate change of variables which transforms the equation into a non local evolution one having a travelling wave solution with…
We study a random resistors network model on a euclidean geometry $\bt{Z}^d$. We formulate the model in terms of a variational principle and show that, under appropriate boundary conditions, the thermodynamic limit of the dissipation per…
We introduce the idea of weakly coherent collisional models, where the elements of an environment interacting with a system of interest are prepared in states that are approximately thermal, but have an amount of coherence proportional to a…
The combinatorial heat and wave equations on all finite Cayley and coset graphs with discrete time variable was solved by Lal {\it et al}. In this paper, the results of the above paper are extended for infinite Cayley and coset graphs,…
In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to…
Microwave electromagnetic heating are widely used in many industrial processes. The mathematics involved is based on the Maxwell's equations coupled with the heat equation. The thermal conductivity is strongly dependent on the temperature,…