Time Asymptotic Behavior Research Articles

Sufficient condition for Gaussian departure in turbulence

The interaction of two isotropic turbulent fields of equal integral scale but different kinetic energy generates the simplest kind of inhomogeneous turbulent field. In this paper we present a numerical experiment where two time decaying isotropic fields of kinetic energies E1 and E2 initially match over a narrow region. Within this region the kinetic energy varies as a hyperbolic tangent. The following temporal evolution produces a shearless mixing. The anisotropy and intermittency of velocity and velocity derivative statistics is observed. In particular the asymptotic behavior in time and as a function of the energy ratio E_{1}E_{2}-->infinity is discussed. This limit corresponds to the maximum observable turbulent energy gradient for a given E1 and is obtained through the limit E_{2}-->0 . A field with E_{1}E_{2}-->infinity represents a mixing which could be observed near a surface subject to a very small velocity gradient separating two turbulent fields, one of which is nearly quiescent. In this condition the turbulent penetration is maximum and reaches a value equal to 1.2 times the nominal mixing layer width. The experiment shows that the presence of a turbulent energy gradient is sufficient for the appearance of intermittency and that during the mixing process the pressure transport is not negligible with respect to the turbulent velocity transport. These findings may open the way to the hypothesis that the presence of a gradient of turbulent energy is the minimal requirement for Gaussian departure in turbulence.

Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation u t = Δ u + V ( x ) | u | p − 1 u in Ω × ( 0 , T ) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u ( x , 0 ) = M φ ( x ) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].

Time decay of solution for the KdV equation with multiplicative space-time noise

We consider the asymptotic behavior in large time of solutions for the KdV equation with multiplicative space-time noise. Under certain assumptions on the noise, we prove that the $L^2$ norm of solutions almost surely decays to zero as time goes to infinity. This phenomena is called stabilization by noise.

The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients

This paper concerns with Cauchy problems for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. Two cases are considered here, first, the initial density is assumed to be integrable on the whole real line. Second, the deviation of the initial density from a positive constant density is integrable on the whole real line. It is proved that for both cases, weak solutions for the Cauchy problem exist globally in time and the large time asymptotic behavior of such weak solutions are studied. In particular, for the second case, the phenomena of vanishing of vacuum and blow-up of the solutions are presented, and it is also shown that after the vanishing of vacuum states, the globally weak solution becomes a unique strong one. The initial vacuum is permitted and the results apply to the one-dimensional Saint-Venant model for shallow water.

Time-asymptotic interactions of Boltzmann shock layers in the presence of boundary

We study the time-asymptotic behavior of the Boltzmann shock layers with a given physical boundary in a half-space. As boundary conditions, we prescribe a Maxwellian at the far field and require a specular reflection at the wall x=0 . When the macroscopic velocity at the far field is negative, we prove that if the initial data are suitably chosen, then a solution exists globally in time and tends toward the corresponding outgoing Boltzmann shock profile as time goes to infinity. The proof is essentially based on the macro-micro decomposition of solutions and the elementary energy methods.

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

The aim of this paper is to study the existence and uniqueness of solutions of an initial-boundary value problem for a viscoelastic two-phase material with capillarity in one space dimension. Therein, the capillarity is modelled via a nonlocal interaction potential. The proof relies on uniform energy estimates for a family of difference approximations: with these estimates at hand we show the existence of a global weak solution. Then, by means of a nontrivial variant of the arguments in [2], uniqueness and optimal regularity are proven. The results of this paper also apply to interaction potentials with non-vanishing negative part and constitute a base for an analysis of the time-asymptotic behaviour.

Spin Dynamics in a One-Dimensional Ferromagnetic Bose Gas

We investigate the propagation of spin excitations in a one-dimensional ferromagnetic Bose gas. While the spectrum of longitudinal spin waves in this system is soundlike, the dispersion of transverse spin excitations is quadratic, making a direct application of the Luttinger liquid theory impossible. By using a combination of different analytic methods we derive the large time asymptotic behavior of the spin-spin dynamical correlation function for strong interparticle repulsion. The result has an unusual structure associated with a crossover from the regime of trapped spin wave to an open regime and does not have analogues in known low-energy universality classes of quantum 1D systems.

The Broadwell model with a subsonic boundary

In this paper, we focus on the time-asymptotic behavior of an initial boundary value problem (IBVP) for the Broadwell model with a subsonic physical boundary. By using the Green’s function for the initial problem established in [C.-Y. Lan, H.-E. Lin, S.-H. Yu, The Green’s functions for the Broadwell model in half space problem, Netw. Heterog. Media 1 (1) (2006)] and the weighted energy estimates, we construct the Green’s function for IBVP and show that the solution converges pointwise to the equilibrium state when the perturbations are sufficiently small.

ON A MICROSCOPIC MODEL OF VISCOUS FRICTION

We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V0 be the initial velocity of the body and V∞ its asymptotic velocity, then for |V0 - V∞| small enough it results |V(t) - V∞| ≈ C t-d-2 for t large, where V(t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,3 the case of E constant and positive, with 0 < V0 < V∞, for a disk orthogonal to the x-axis has been discussed. Here we complete the analysis in the cases E > 0 with V0 > V∞ and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.

Asymptotic Behavior and Synchronizability Characteristics of a Class of Recurrent Neural Networks

We propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model. The model we introduce is a variation of the Zhang model. We investigate the time-asymptotic behavior of the corresponding dynamical system (in particular, the conditions for the existence of an invariant compact asymptotic set) and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with Poisson connectivity distribution.

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L1-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C0-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ | u | p − 1 u , where 1 < p < 1 + 2 / n , n is the space dimension and λ is a complex constant satisfying Im λ < 0 . We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n ⩽ 3 , p is sufficiently close to 1 + 2 / n and the initial data is sufficiently small.

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation u t = − ( u u x x x ) x . It was shown in previous work of Carrillo and Toscani that for non-negative initial data u 0 that belongs to H 1 ( R ) and also has a finite mass and second moment, the strong solutions relax in the L 1 ( R ) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H 1 ( R ) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H 1 ( R ) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.

Nonlinear equilibria, stability and generation of zonal structures in toroidal plasmas

The long-lived saturated zonal flow (ZF) structures, spontaneously generated by drift-wave (DW) turbulence, are viewed as generators of nonlinear equilibria. Here, we derive a nonlinear evolution equation for the zonal response, valid for arbitrary wavelengths, whose time-asymptotic behaviour corresponds to the nonlinear equilibrium. Its corresponding stability, meanwhile, determines the nature of the ZF instability and the nonlinear up-shift of turbulent transport thresholds. On a shorter-time scale, the temporal evolution of the zonal response describes the DW–ZF generation and the regulation of the DW intensity by the ZFs. As an application, we present a kinetic stability analysis of zonal structures in the ion temperature gradient driven DW turbulence case as well as the corresponding collisionless zonal dissipation rates.

Long Range Scattering and Modified Wave Operators for the Maxwell–Schrödinger System II. The General Case

We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr"odinger equation, which then becomes both nonlinear and nonlocal in time. The Schr"odinger function is then parametrized in terms of an amplitude and a phase satisfying a suitable auxiliary system, and the Cauchy problem for that system, with prescribed asymptotic behaviour determined by the asymptotic data, is solved by an energy method, thereby leading to solutions of the original system with prescribed asymptotic behaviour in time. This paper is the generalization of a previous paper with the same title. However it is entirely selfcontained and can be read without any previous knowledge of the latter.

Asymptotics of flat, radiation universes in quadratic gravity

We consider the asymptotics of flat, radiation-dominated isotropic universes in four-dimensional theories with quadratic curvature corrections which may arise when contributions related to the string parameter α′ are switched on. We show that all such universes are singular initially, and calculate all early time asymptotics using the method of asymptotic splittings. We also examine the late time asymptotic behaviour of these models and show that there are no solutions which diverge as et2, the known radiation solution of general relativity is essentially the only late time asymptotic possibility in these models.

Unstable sixth-order thin film equation: II. Global similarity patterns

We continue the study begun in part I (Evans J D, Galaktionov V A and King J R 2007 Unstable sixth-order thin film equation: I. Blow-up of similarity solutions Nonlinearity 20 1799–841) of asymptotic large time behaviour of global solutions of the sixth-order thin film equation with bounded integrable initial data. We show that for the first critical exponent, where N is the space dimension, the free-boundary problem with zero-contact-angle, zero-moment and zero-flux conditions at the interface admits continuous families (branches) of radially symmetric self-similar solutions defined for all t > 0, We also study the Cauchy problem, for which we construct global similarity solutions of the maximal regularity, these being oscillatory near the interfaces for n ∊ (0, nh), where is a ‘heteroclinic bifurcation’ point for a related nonlinear ordinary differential equation. We use various concepts based on the branching of sufficiently small solutions from the known eigenfunctions of the linear rescaled operator corresponding to n = 0.

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

AbstractWe study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta‐Kawashima condition. We show that these solutions approach a constant equilibrium state in the Lp‐norm at a rate O(t− (m/2)(1 − 1/p)) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman‐Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. © 2007 Wiley Periodicals, Inc.

On the focusing critical semi-linear wave equation

The wave equation ∂ttψ − Δψ − ψ5 = 0 in ℝ3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions φ(x, a) = (3a) ¼ (1 + a|x|2)−½ for all a &gt; 0 which are linearly unstable. We view these functions as a curve in the energy space ˙H1 ×L2. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold M of small diameter in a suitable topology. Loosely speaking, M acts as a center-stable manifold with the curve Φ(·, a) as an attractor in M.

ABOUT DECAY OF SOLUTION OF THE WAVE EQUATION WITH DISSIPATION

In this work, we consider the problem of existence of global solutions for a scalar wave equation with dissipation. We also study the asymptotic behaviour in time of the solutions. The method used here is based in nonlinear techniques. Key words: wave equation, evolution model, decay of solution, asymptotic behaviour.