Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Ergodicity in infinite Hamiltonian systems with conservative noise

Summary. We study the stationary measures of an infinite Hamiltonian system of interacting particles in ℝ ³ subject to a stochastic local perturbation conserving energy and momentum. We prove that the translation invariant measures that are stationary for the deterministic Hamiltonian dynamics, reversible for the stochastic dynamics, and with finite entropy density, are convex combination of “Gibbs” states. This result implies hydrodynamic behavior for the systems under consideration. Received: 17 December 1994/In revised form: 12 April 1996     

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBV _loc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].     

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

We study Langevin dynamics of N particles on ℝ^d interacting through a singular repulsive potential, e.g., the well-known Lennard-Jones type, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof of the main result relies on an explicit construction of a Lyapunov function. In contrast to previous results for such systems, our result implies geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. © 2019 Wiley Periodicals, Inc.     

具有非线性记忆项的耗散波动方程的整体吸引子

LetΩRn be a bounded domain with a smooth boundary.We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term utt+αut-△u-∫0t 0μ(t-s)|u(s)| βu(s)ds + g(u)=f.Based on a time-uniform priori estimate method,the existence of the compact global attractor is proved for this model in the phase space H10(Ω)×L2(Ω).     

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤ ^d at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L ⁻². Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β _c of the pure system. Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001     

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction x = \fracpq,\xi=\frac{p}{q}, where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.     

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ₀ of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ₀. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.     

An exponential decay estimate for the stationary axisymmetric perturbation of Poiseuille flow in a circular pipe

We prove a decay estimate for the steady state incompressible Navier-Stokes equations. The estimate describes the exponential decay, in the axial direction of a semi-infinite circular tube, for an energy-type functional in terms of the axisymmetric perturbation of Poiseuille flow, provided that the Reynolds number does not exceed a critical value, for which we exhibit a lower and an upper bound. Since the motion is considered axisymmetric we use a stream function formulation, and the results are similar to those obtained by Horgan [8], for a two-dimensional channel flow problem. For the Stokes problem our estimate for the rate of decay is a lower bound to the actual rate of decay which is obtained from an asymptotic solution to the Stokes equations. Finally we describe a numerical approach to computing bounds to the energy functionalE(0).     

Emergent dynamics of the first‐order stochastic Cucker‐Smale model and application to finance

In this paper, we study stochastic aggregation properties of the financial model for the N‐asset price process whose dynamics is modeled by the weakly geometric Brownian motions with stochastic drifts. For the temporal evolution of stochastic components of drift coefficients, we employ a stochastic first‐order Cucker‐Smale model with additive noises. The asset price processes are weakly interacting via the stochastic components of drift coefficients. For the aggregation estimates, we use the macro‐micro decomposition of the fluctuations around the average process and show that the fluctuations around the average value satisfies a practical aggregation estimate over a time‐independent symmetric network topology so that we can control the differences of drift coefficients by tuning the coupling strength. We provide numerical examples and compare them with our analytical results. We also discuss some financial implications of our proposed model.     

The Vlasov-Maxwell-Boltzmann system near Maxwellians

Perhaps the most fundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field. Despite its importance, no global in time solutions, weak or strong, have been constructed so far. It is shown in this article that any initially smooth, periodic small perturbation of a given global Maxwellian, which preserves the same mass, total momentum and reduced total energy (22), leads to a unique global in time classical solution for such a master system. The construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation: the linear collision operator L, not its time integration, is positive definite for any solutionf(t,x,v) with small amplitude to the Vlasov-Maxwell-Boltzmann system (8) and (12). As a by-product, such an estimate also yields an exponential decay for the simpler Vlasov-Poisson-Boltzmann system (24).     

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi‐discrete piecewise optimal control problem and the desired states in L² and H¹ norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

On dispersive stability of Hamiltonian systems on lattices

We study the long-time dynamics of oscillations in lattices of infinitely many particles interacting via certain non-linear potentials. The aim is to proof dispersive stability of such Hamiltonian systems analogously to results known for PDEs. To do so we first recapitulate the dynamics of linear Hamiltonian systems on an infinite chain and give optimal decay rates based on the dispersion relation. Based on this we proof that if the non-linearity is weak enough, the non-linear system shows a similar behaviour like its linearization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The dynamics of structures on an elastic base

The problem of the dynamics of structures interacting with an elastic homogeneous half-plane through a rigid unyielding plate is reduced to the construction of the matrix of transmission momentum functions (the Green's matrix), which establishes the dependence between the generalized coordinates of the oscillations of the plate and the force characteristics. The elements obtained for this matrix are represented by graphs as the result of numerical solution of the nonsteady-state dynamic contact problem. We conclude that approximating a Green's matrix of exponential type on an infinite time interval is unjustified.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 60–63.     

On <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">1</Emphasis></Superscript>-summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain

The exterior nonstationary problem is studied for the 3D Navier-Stokes equations. The L ¹ -summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions. The result is then applied to show that such solutions decay in time more rapidly than observed in general. Furthermore, an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time. Mathematics Subject Classifications (1991): 35Q30, 76D05.On leave of absence from Institute of Applied Mathematics, Academy of Mathematics and System Sciences. Academia Sinica, Beijing 100080, Peoples Republic of China. Supported by JSPS     

Polynomial stability of solutions for a system of non-linear viscoelastic equations

In this article, we consider an interacting system consisting of two weakly-coupled viscoelastic equations. This work proves a uniform stabilization result without assuming any decay rate of the relaxation function provided that the kernels have some flat zones, the size of the flat zones are sufficiently small.     

Study of continuity of random sequences by constructing proximity points

A method of estimation of time-uniform continuity of random sequences is considered that makes it possible to interpret previous methods from a single point of view.Translated from Problemy Ustroichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 52–57, 1980.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Scaling Limits of Additive Functionals of Interacting Particle Systems

Using the renormalization method introduced by the authors, we prove what we call the local Boltzmann‐Gibbs principle for conservative, stationary interacting particle systems in dimension d = 1. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by‐product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation. © 2013 Wiley Periodicals, Inc.