Large-time Asymptotic Behaviour Research Articles

Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as .

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces RN−k×(0,∞)k, k≥0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L2(RN−k×(0,∞)k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k=0, to any k≥0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

Periodic entropy solution to a conservation law with nonlocal source arising in radiative gas

We establish the global well-posedness and large-time asymptotic behaviour of spatially periodic entropy solutions to a scalar conservation law with nonlocal source arising in radiative gas. The global existence is established by the L1-contraction and the comparison principle for the vanishing viscosity approximation. Moreover, we show that if the initial data is periodic, the source term induces the solution to decay in L2-norm at an exponential rate to the mean value of initial data over one space period.

Large time asymptotics for the higher-order nonlinear nonlocal Schrödinger equation

We consider the Cauchy problem for the higher-order nonlinear nonlocal Schrödinger equation in one space dimension i∂tu−Λu=λu3u,t>0,x∈R,u0,x=u0x,x∈R,where λ>0, and the linear operator Λu=∑j=351j∂xju. Our purpose in this paper is to prove the large time asymptotic behaviour of solutions for the defocusing case λ>0 with a logarithmic correction under the nonzero mass condition ∫Ru0xdx≠0.

Long-distance and large-time asymptotic behaviour of dynamic correlation functions in the massless regime of the XXZ spin-1/2 chain

Starting from the massless form factor expansion for the two-point dynamical correlation functions obtained recently, I extract the long-distance and large-time asymptotics of these correlators. The analysis yields the critical exponents and associated amplitudes characterizing the asymptotics. The results are obtained on the basis of exact and first principle based considerations: they do not rely, at any stage, on some hypothetical correspondence with a field theory or the use of any other phenomenological approach. Being based on form factor expansion, the method allows one to clearly identify which contributions to the asymptotics issue from which class of excited states. All this permits to settle the long-standing question of the contribution of bound states to the asymptotics of two-point functions. For instance, when considering the long-distance m behavior of equal-time correlators, the analysis shows that while, in fine, the bound states only produce contributions that are exponentially small in m, they also play a key role in canceling out certain power-law contributions which, should they be present, would break explicitly the universality structure of the long-distance behavior.

Asymptotics of solutions of a modified Whitham equation with surface tension

We study the large-time behaviour of solutions of the Cauchy problem for a modified Whitham equation, where the pseudodifferential operator is given by the symbol with a parameter . This symbol corresponds to the total dispersion relation for water waves taking surface tension into account. Assuming that the total mass of the initial data is equal to zero () and the initial data are small in the norm of , , we prove the existence of a global-in-time solution and describe its large-time asymptotic behaviour.

On a cell division equation with a linear growth rate

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Lax Integrability and the Peakon Problem for the Modified Camassa–Holm Equation

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes' continued fractions and multi-point Pad\'{e} approximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons which share asymptotic speeds, as well as Toda-like sorting property.

ON A CELL DIVISION EQUATION WITH A LINEAR GROWTH RATE

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws

In this article, we study the interaction of delta shock waves for the one-dimensional strictly hyperbolic system of conservation laws with split delta function. We prove that Riemann solutions are stable under local small perturbations of the Riemann initial data. The global structure and large time asymptotic behaviour of the perturbed Riemann solutions are constructed and analyzed case by case.

Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

Comparing methods for modelling spreading cell fronts

Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher–Kolmogorov equation; however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of individual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and the asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performance of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.

Hybrid competitive Lotka–Volterra ecosystems with a hidden Markov chain

This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching. Distinct from the existing formulations, the Markov chain that models random environments is unobservable. For such partially observed systems, we use Wonham’s filter to estimate the Markov chain from the observable evolution of the population, and convert the original system to a completely observable one. We then show that the positive solution of our model does not explode in finite time with probability 1. Several properties including stochastic boundedness, finite moments, sample path continuity and large-time asymptotic behaviour are also obtained. Moreover, stochastic permanence, extinction and feedback controls are also investigated.

Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.

Asymptotic behaviour of solutions to the Keller–Segel model for chemotaxis with prevention of overcrowding

This paper deals with the Keller–Segel chemotaxis model of parabolic–elliptic type with the volume-filling effect studied by Burger et al (2006 The Keller–Segel model for chemotaxis with prevention of overcrowding: linear versus nonlinear diffusion SIAM J. Math. Anal. 38 1288–315). In their discussion on the large time asymptotic behaviour of solutions, the diffusion rate of ρ (the density of cells) had to be assumed to be large with . While for the nonlinear diffusion model, it was proved that the asymptotic behaviour of solutions is fully determined by the diffusion constant being larger or smaller than the threshold value ε = 1. The same ‘large ε-restriction’ was also made for studying the parallel parabolic–parabolic model in Di Francesco and Rosado (2008 Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding Nonlinearity 21 2715–30), where it was pointed out that ‘Whether this condition is necessary to have large time decay (and consequently a self-similar behaviour) for ρ is still an open problem even in the (simpler) parabolic–elliptic case’. The aim of the paper is to answer this problem for the parabolic–elliptic model. We prove the mentioned time decay estimate without the restriction . The main technique used in this paper is the Lp-Lq estimate method.

EXACT PRICING AND LARGE-TIME ASYMPTOTICS FOR THE MODIFIED SABR MODEL AND THE BROWNIAN EXPONENTIAL FUNCTIONAL

We derive a closed-form expression for the stock price density under the modified SABR model [see section 2.4 in Islah (2009)] with zero correlation, for β = 1 and β < 1, using the known density for the Brownian exponential functional for μ = 0 given in Matsumoto and Yor (2005), and then reversing the order of integration using Fubini's theorem. We then derive a large-time asymptotic expansion for the Brownian exponential functional for μ = 0, and we use this to characterize the large-time behaviour of the stock price distribution for the modified SABR model; the asymptotic stock price "density" is just the transition density p(t, S0, S) for the CEV process, integrated over the large-time asymptotic "density" [Formula: see text] associated with the Brownian exponential functional (re-scaled), as we might expect. We also compute the large-time asymptotic behaviour for the price of a call option, and we show precisely how the implied volatility tends to zero as the maturity tends to infinity, for β = 1 and β < 1. These results are shown to be consistent with the general large-time asymptotic estimate for implied variance given in Tehranchi (2009). The modified SABR model is significantly more tractable than the standard SABR model. Moreover, the integrated variance for the modified model is infinite a.s. as t → ∞, in contrast to the standard SABR model, so in this sense the modified model is also more realistic.

Schur flows and orthogonal polynomials on the unit circle

Connections between Toda lattices (Toda chains) and similar non-linear chains and the theory of orthogonal polynomials on the real axis have been studied in detail during the last decades. Another system of difference differential equations, known as the Schur flow, is considered in this paper in the framework of the theory of orthogonal polynomials on the unit circle. A Lax pair for this system is found and the dynamics of the corresponding spectral measure is described. The general result is illustrated by an example of Bessel modified measures on the unit circle: the large-time asymptotic behaviour of their reflection coefficients is determined.

The Cauchy problem for an equation of Sobolev type with power non-linearity

This paper deals with the study of the large-time asymptotic behaviour of solutions of the Cauchy problem for a non-linear non-local equation of Sobolev type with dissipation. In the case when the initial data are small our approach is based on a detailed study of the Green's function of the linear problem and the use of the contraction-mapping method. We also consider the case when the initial data are large. In the supercritical case the asymptotics is quasilinear. The asymptotic behaviour of solutions in the critical case differs from the behaviour of solutions of the corresponding linear equation by a logarithmic correction. In the subcritical case we prove that the principal term of the large-time asymptotics of the solution can be represented by a self-similar solution if the initial data have non-zero total mass.

Cauchy problem for non-linear systems of equations in the critical case

The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations where is a linear pseudodifferential operator and the non-linearity is a quadratic pseudodifferential operator where is the Fourier transform. Under the assumptions that the initial data , are sufficiently small, where is a Sobolev weighted space, and that the total mass vector is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector of the initial data.

Asymptotics of solutions of non-linear dissipative equations

We study the large-time asymptotic behaviour of solutions of the Cauchy problem for a system of non-linear evolution equations with dissipation. In the case when the initial data are small, we construct solutions using the contraction-mapping principle. When the initial data are large, we obtain the large-time asymptotics of solutions by taking into account a certain symmetry of the non-linear terms.