Showing papers in "Nonlinearity in 2011"
TL;DR: In this article, a family of interaction potentials for which the equilibria are of finite density and compact support is studied, and global well-posedness of solutions and their global stability are investigated analytically and numerically.
Abstract: We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR n , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of R n and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density ¯ ρ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which ¯ ρ is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.
196 citations
TL;DR: In this paper, the authors generalized the Wasserstein metric to reaction-diffusion systems with reversible mass-action kinetic and showed that this gradient structure can be generalized to systems including electrostatic interactions and correct energy balance via coupling to the heat equation.
Abstract: In recent years the theory of the Wasserstein metric has opened up new treatments of diffusion equations as gradient systems, where the free energy or entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction–diffusion systems with reversible mass-action kinetic. The metric is constructed using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential.The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in Albinus et al (2002 Nonlinearity 15 367–83), which stimulated this work, have a gradient structure.
154 citations
TL;DR: In this article, it was shown that sub-critical problems are globally well-posed for PKS with degenerate diffusion, and for a fairly general class of problems, the existence of a critical mass which sharply divides the possibility of finite time blow-up and global existence was shown.
Abstract: Recently, there has been a wide interest in the study of aggregation equations and Patlak–Keller–Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions d ≥ 2 and in all of space for d ≥ 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow-up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow-up is possible for initial data of arbitrary mass.
128 citations
TL;DR: In this article, an entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is established by proving convergence of a difference-quadrature scheme, and it turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one.
Abstract: The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid–fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.
105 citations
TL;DR: A sufficient and necessary condition for the existence of monotone travelling waves in the non-local Fisher-KPP equation is established in this article, and the uniqueness of travelling wavefronts (up to translation) is also proved.
Abstract: A sufficient and necessary condition for the existence of monotone travelling waves in the nonlocal Fisher–KPP equation is established, and the uniqueness of travelling wavefronts (up to translation) is also proved.
100 citations
TL;DR: In this paper, the authors provide an explicit rigorous derivation of a diffusion limit from a deterministic skew-product flow, which is assumed to exhibit time-scale separation and has the form of a slowly evolving system driven by a fast chaotic flow.
Abstract: We provide an explicit rigorous derivation of a diffusion limit—a stochastic differential equation (SDE) with additive noise—from a deterministic skew-product flow. This flow is assumed to exhibit time-scale separation and has the form of a slowly evolving system driven by a fast chaotic flow. Under mild assumptions on the fast flow, we prove convergence to a SDE as the time-scale separation grows. In contrast to existing work, we do not require the flow to have good mixing properties. As a consequence, our results incorporate a large class of fast flows, including the classical Lorenz equations.
97 citations
TL;DR: In this paper, a Lyapunov-like functional decreasing along the travelling wave is constructed, and the authors prove an existence and non-existence result for travelling wave solutions.
Abstract: The aim of this paper is to investigate the spatial invasion of some infectious disease. The contamination process is described by the age since infection. Compared with the classical Kermack and McKendrick’s model, the vital dynamic is not omitted, and we allow some constant input flux into the population. This problem is rather natural in the context of epidemic problems and it has not been studied. Here we prove an existence and non-existence result for travelling wave solutions. We also describe the minimal wave speed. We are able to construct a suitable Lyapunov like functional decreasing along the travelling wave allowing to derive some qualitative properties, namely their convergence towards equilibrium points at x =± ∞.
93 citations
TL;DR: In this article, it is shown that the stable fixed points of such dynamical systems are the index-1 saddle points, and generalizations to high index saddle points are discussed.
Abstract: Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.
92 citations
TL;DR: For Axiom A flows on basic sets satisfying certain additional conditions, this article proved strong spectral estimates for Ruelle transfer operators similar to those of Dolgopyat (1998 Ann. Math. 147 357-90) for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on a compact manifold with C1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential).
Abstract: For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to those of Dolgopyat (1998 Ann. Math. 147 357–90) for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C1 jointly non-integrable horocycle foliations (for the Sinai–Bowen–Ruelle potential). Here we deal with general potentials and on spaces of arbitrary dimension, although under some geometric and regularity conditions. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, and in other areas.
86 citations
TL;DR: In this paper, a KAM theorem for an infinite dimensional reversible system with unbounded perturbation is established, by which quasi-periodic solutions of a class of nonlinear Schodinger equations with a derivative in nonlinear terms are obtained subject to Dirichlet boundary conditions.
Abstract: In this paper, we establish a KAM theorem for an infinite dimensional reversible system with unbounded perturbation, by which we obtain quasi-periodic solutions of a class of nonlinear Schodinger equations with a derivative in nonlinear terms, subject to Dirichlet boundary conditions.
85 citations
Abstract: The nonlinear dynamics of biochemical reactions in a small-sized system on the order of a cell are stochastic. Assuming spatial homogeneity, the populations of n molecular species follow a multi-dimensional birth-and-death process on . We introduce the Delbruck–Gillespie process, a continuous-time Markov jump process, whose Kolmogorov forward equation has been known as the chemical master equation, and whose stochastic trajectories can be computed via the Gillespie algorithm. Using simple models, we illustrate that a system of nonlinear ordinary differential equations on emerges in the infinite system size limit. For finite system size, transitions among multiple attractors of the nonlinear dynamical system are rare events with exponentially long transit times. There is a separation of time scales between the deterministic ODEs and the stochastic Markov jumps between attractors. No diffusion process can provide a global representation that is accurate on both short and long time scales for the nonlinear, stochastic population dynamics. On the short time scale and near deterministic stable fixed points, Ornstein–Uhlenbeck Gaussian processes give linear stochastic dynamics that exhibit time-irreversible circular motion for open, driven chemical systems. Extending this individual stochastic behaviour-based nonlinear population theory of molecular species to other biological systems is discussed.
TL;DR: In this article, the formation of rogue waves in nonlinear hyperbolic systems with an application to nonlinear shallow-water waves is studied in the framework of nonlinear hypersphere.
Abstract: The formation of rogue waves is studied in the framework of nonlinear hyperbolic systems with an application to nonlinear shallow-water waves. It is shown that the nonlinearity in the random Riemann (travelling) wave, which manifests in the steeping of the face-front of the wave, does not lead to extreme wave formation. At the same time, the strongly nonlinear Riemann wave cannot be described by the Gaussian statistics for all components of the wave field. It is shown that rogue waves can appear in nonlinear hyperbolic systems only in the result of nonlinear wave–wave or/and wave–bottom interaction. Two special cases of wave interaction with a vertical wall (interaction of two Riemann waves propagating in opposite directions) and wave transformation in the basin of variable depth are studied in detail. Open problems of the rogue wave occurrence in nonlinear hyperbolic systems are discussed.
TL;DR: In this paper, the authors consider a class of quasi-Markovian GLEs, similar to the model that was introduced in Eckmann J-P et al. 1999 Commun. Math. Soc. Phys. 202 iv, 141) and make use of a hypoelliptic operator which is the generator of an auxiliary Markov process.
Abstract: Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in Eckmann J-P et al 1999 Commun. Math. Phys. 201 657–97. Ergodicity, exponentially fast convergence to equilibrium, short time asymptotics, a homogenization theorem (invariance principle) and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity (Villani C 2009 Mem. Am. Math. Soc. 202 iv, 141) is made.
TL;DR: In this article, the structural stability of the generalized Forchheimer equations for slightly compressible fluids in porous media is established with respect to either the boundary data or the coefficients of the forchheimer polynomials and a weighted Poincare-Sobolev inequality related to the nonlinearity of the equation is used to study the asymptotic behaviour of the solutions.
Abstract: We study the generalized Forchheimer equations for slightly compressible fluids in porous media. The structural stability is established with respect to either the boundary data or the coefficients of the Forchheimer polynomials. A weighted Poincare–Sobolev inequality related to the nonlinearity of the equation is used to study the asymptotic behaviour of the solutions. Moreover, we prove a perturbed monotonicity property of the vector field associated with the resulting non-Darcy equation, where the correction is explicit and Lipschitz continuous in the coefficients of the Forchheimer polynomials.
TL;DR: In this article, a new Poisson-Boltzmann type (PB_n) equation with a small dielectric parameter 2 and non-local nonlinearity was proposed to model the equilibrium of bulk ionic species in different media and solvents.
Abstract: The Poisson–Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents In this paper we study a new Poisson–Boltzmann type (PB_n) equation with a small dielectric parameter 2 and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion This equation can be derived from the original Poisson–Nernst–Planck system Under Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB_n equations as the parameter approaches zero In particular, we show that in case of electroneutrality, ie α = β, solutions of 1D PB_n equations have a similar asymptotic behaviour as those of 1D PB equations However, as α ≠ β (non-electroneutrality), solutions of 1D PB_n equations may have blow-up behaviour which cannot be found in 1D PB equations Such a difference between 1D PB and PB_n equations can also be verified by numerical simulations
TL;DR: In this article, a model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature, the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain.
Abstract: A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature , the velocity field u and the director field d, representing preferred orientation of molecules in a neighbourhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier–Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field d, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg–Landau type, with a suitable penalization term to relax the constraint |d| = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field d. The proposed model is shown to be compatible with first and second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.
TL;DR: In this paper, the dynamics of Bose-Einstein condensates in optical lattices were analyzed in the presence of residual harmonic trapping and in interferometry configurations suitable to investigate discrete breathers' interactions.
Abstract: Discrete breathers, originally introduced in the context of biopolymers and coupled nonlinear oscillators, are also localized modes of excitation of Bose–Einstein condensates (BEC) in periodic potentials such as those generated by counter-propagating laser beams in an optical lattice. Static and dynamical properties of breather states are analysed in the discrete nonlinear Schrodinger equation that is derived in the limit of deep potential wells, tight-binding and the superfluid regime of the condensate. Static and mobile breathers can be formed by progressive re-shaping of initial Gaussian wave-packets or by transporting atomic density towards dissipative boundaries of the lattice. Static breathers generated via boundary dissipations are determined via a transfer-matrix approach and discussed in the two analytic limits of highly localized and very broad profiles. Mobile breathers that move across the lattice are well approximated by modified analytical expressions derived from integrable models with two independent parameters: the core-phase gradient and the peak amplitude. Finally, possible experimental realizations of discrete breathers in BEC in optical lattices are discussed in the presence of residual harmonic trapping and in interferometry configurations suitable to investigate discrete breathers' interactions.
TL;DR: In this article, the authors developed a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms, where the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator were considered when the spatial domain takes the whole space or torus and when there is a confining force or not.
Abstract: In this paper we develop a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker–Planck operator and linearized Boltzmann operator are considered when the spatial domain takes the whole space or torus and when there is a confining force or not. The key part of the developed approach is to construct some equivalent temporal energy functionals for obtaining time rates of the solution trending towards equilibrium in some Hilbert spaces. The result in the case of the linear Boltzmann equation with confining forces is new. The proof mainly makes use of the macro–micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic–parabolic system. At the end, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.
TL;DR: In this article, an integrable generalization of the nonlinear Schrodinger equation proposed by Fokas and Lenells is proposed and the relationship between this equation and other integrably general models is discussed.
Abstract: This work is devoted to an integrable generalization of the nonlinear Schrodinger equation proposed by Fokas and Lenells I discuss the relationships between this equation and other integrable models Using the reduction of the Fokas–Lenells equation to the already known ones I obtain the N-dark soliton solutions
TL;DR: In this paper, the authors start from density models for streamers, i.e. from reaction-drift-diffusion equations coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models.
Abstract: Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.
TL;DR: In this article, the authors define a stability index σ(x) of a point x X that characterizes the local extent of the basin and show that this index is constant along trajectories, and relate this orbit invariant to other notions of stability such as Milnor attraction.
Abstract: Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set with a basin of attraction N, we define a stability index σ(x) of a point x X that characterizes the local extent of the basin. Let B denote a ball of radius about x. If σ(x) > 0, then the measure of B N relative the measure of the ball is O(|σ(x)|), while if σ(x) < 0, then the measure of B ∩ N relative the measure of the ball is of this order. We show that this index is constant along trajectories, and we relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where N is defined as the set of initial conditions that are in the basin and whose trajectories remain local to X).This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995 Ergod. Theory Dyn. Syst. 15 121–48; 2004 Proc. R. Soc. Edinb. A 134 1177–97) and give criteria for simple heteroclinic cycles in to be Milnor attractors.
TL;DR: In this paper, lower-edge spectral and dynamical localization for a multi-particle Anderson model in a Euclidean space was established in the presence of a non-trivial short-range interaction and an alloy-type random external potential.
Abstract: We establish lower-edge spectral and dynamical localization for a multi-particle Anderson model in a Euclidean space , d ≥ 1, in the presence of a non-trivial short-range interaction and an alloy-type random external potential.
TL;DR: In this article, the authors deal with the problem of estimating the topological entropy of non-autonomous systems of differential equations via the second Lyapunov method, and show that the results of this method are applicable to the Lorenz system and the Duffing oscillator.
Abstract: This paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.
TL;DR: In this article, a general, nonperturbative result about finite-time blowup solutions for the L2-critical Boson star equation in d = 3 space dimensions was proved.
Abstract: We prove a general, non-perturbative result about finite-time blowup solutions for the L2-critical Boson star equation in d = 3 space dimensions. Under the sole assumption that u = u(t, x) blows up in the energy space H1/2 at finite time 0 0. For radial solutions, this result establishes a large data blowup conjecture for the L2-critical Boson star equation.We also discuss some extensions of our results to other L2-critical theories of gravitational collapse, in particular to critical Hartree-type equations.
TL;DR: In this article, it was shown that the particular BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP for the elliptic version of the Ernst equation.
Abstract: The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analysing boundary-value problems (BVPs) for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is not yet effective. (b) Characterize the unknown boundary values by analysing a certain equation called the global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain BVPs called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain BVPs for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurrence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP. In this way the remarkable results of Neugebauer and Meinel are recovered.
TL;DR: In this article, the authors present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrodinger operators of scattering systems, in cases where the corresponding classical dynamics is chaotic; more precisely, the classical Hamiltonian flow admits a fractal set of trapped trajectories, which hosts chaotic (hyperbolic) dynamics.
Abstract: We present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrodinger (or wave) operators of scattering systems, in cases where the corresponding classical dynamics is chaotic; more precisely, we assume that in some energy range, the classical Hamiltonian flow admits a fractal set of trapped trajectories, which hosts chaotic (hyperbolic) dynamics. The aim is then to connect the information on this trapped set with the distribution of quantum resonances in the semiclassical limit.Our study encompasses several models sharing these dynamical characteristics: free motion outside a union of convex hard obstacles, scattering by certain families of compactly supported potentials, geometric scattering on manifolds with (constant or variable) negative curvature. We also consider the toy model of open quantum maps, and sketch the construction of quantum monodromy operators associated with a Poincare section for a scattering flow.The semiclassical density of long-living resonances exhibits a fractal Weyl law, related to the fact that the corresponding metastable states are 'supported' by the fractal trapped set (and its outgoing tail). We also describe a classical condition for the presence of a gap in the resonance spectrum, equivalently a uniform lower bound on the quantum decay rates, and present a proof of this gap in a rather general situation, using quantum monodromy operators.
TL;DR: In this article, the Euler equations in a three-dimensional Gevrey-class bounded domain were considered and the persistence of the solution up to the boundary with an explicit estimate on the rate of decay in terms of Sobolev norms.
Abstract: We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius, in terms of Sobolev norms.
TL;DR: In this article, the authors describe a general approach to the transient and steady state fluctuation theorems of non-equilibrium statistical mechanics within the abstract framework of dynamical system theory, and discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.
Abstract: Within the abstract framework of dynamical system theory we describe a general approach to the transient (or Evans–Searles) and steady state (or Gallavotti–Cohen) fluctuation theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. In addition to its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.
TL;DR: In this paper, the dyadic model is used to test well-posedness and blow-up for the Navier-Stokes and Euler equations in the Dyadic model.
Abstract: We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.
TL;DR: In this paper, a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force, is investigated, where the dry friction between the two solids is modelled phenomenologically as being proportional to the sign of the object's velocity relative to the surface, and therefore shows a discontinuity at zero velocity.
Abstract: We investigate a model for the dynamics of a solid object, which moves over a randomly vibrating solid surface and is subject to a constant external force. The dry friction between the two solids is modelled phenomenologically as being proportional to the sign of the object's velocity relative to the surface, and therefore shows a discontinuity at zero velocity. Using a path integral approach, we derive analytical expressions for the transition probability of the object's velocity and the stationary distribution of the work done on the object due to the external force. From the latter distribution, we also derive a fluctuation relation for the mechanical work fluctuations, which incorporates the effect of the dry friction.