Large Time Asymptotics Research Articles

System of degenerate parabolic p-Laplacian

Abstract In this article, we study the mathematical properties of the solution u = ( u 1 , … , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) to the degenerate parabolic system u t = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) , ( p > 2 ) . {{\bf{u}}}_{t}=\nabla \hspace{0.25em}\cdot \hspace{0.25em}({| \nabla {\bf{u}}| }^{p-2}\nabla {\bf{u}}),\hspace{1.0em}(p\gt 2). More precisely, we show the existence and uniqueness of solution u {\bf{u}} and investigate a priori L ∞ {L}^{\infty } boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component u l {u}^{l} , ( 1 ≤ l ≤ k ) (1\le l\le k) , converges to the function c l ℬ {c}^{l}{\mathcal{ {\mathcal B} }} in space as t → ∞ t\to \infty . Here, the function ℬ {\mathcal{ {\mathcal B} }} is the fundamental or Barenblatt solution of p p -Laplacian equation, and the constant c l {c}^{l} is determined by the L 1 {L}^{1} -mass of u l {u}^{l} . The proof is based on the existence of entropy functional. As an application of the asymptotic large-time behaviour, we establish a Harnack-type inequality, which makes the size of the spatial average controlled by the value of the solution at one point.

Edge currents for the time-fractional, half-plane, Schrödinger equation with constant magnetic field

We study the large-time asymptotics of the edge current for a family of time-fractional Schrödinger equations with a constant, transverse magnetic field on a half-plane (x,y)∈Rx+×Ry. The time-fractional Schrödinger equation is parameterized by two constants (α, β) in (0, 1], where α is the fractional order of the time derivative, and β is the power of i in the Schrödinger equation. We prove that for fixed α, there is a transition in the transport properties as β varies in (0, 1]: For 0 < β < α, the edge current grows exponentially in time, for α = β, the edge current is asymptotically constant, and for β > α, the edge current decays in time. We prove that the mean square displacement in the y∈R-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin [Phys. Rev. E 62, 3135 (2000)] that the latter two cases, α = β and α < β, are the physically relevant ones.

Sharp decay characterization for the compressible Navier-Stokes equations

The low-frequency L1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical Lp framework. Precisely, it is proved that the Besov space B˙2,∞σ1-boundedness condition (with d2−2dp≤σ1<d2−1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B˙2,∞σ1. To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids.

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Asymptotics for viscous Burgers equation under impulsive forcing

An initial value problem to the viscous Burgers equation under the influence of an external forcing of distribution type is considered. The existence and uniqueness of weak solutions for it are investigated. To do this, we linearize the viscous Burgers equation and show the existence and uniqueness of common boundary function for the associated initial-boundary problems. Jump discontinuity of the spatial derivative of the solution makes a key role in determining large time asymptotics to the solutions. The explicit representation of the common boundary function and its asymptotic behavior are discussed for a specific case of the jump discontinuity. Here, the common boundary is obtained in terms of Associated Legendre functions of the first and second kind. For the general case, asymptotic behavior of the common boundary of the associated initial-boundary problems is used to determine the large time asymptotics for the Cauchy problem to the viscous Burgers equation.

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped $p$-system

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta–Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows us to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform, which we avoid here, developing the “physical space version” of the hyperbolic hypocoercivity approach introduced in Beauchard and Zuazua [Arch. Ration. Mech. Anal. 199 (2011), 177–227], to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows us to recover the natural heat-like time decay of solutions under sharp rank conditions, without employing Fourier analysis or L^{1} assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped p -system.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

A Two-Component Sasa–Satsuma Equation: Large-Time Asymptotics on the Line

A Two-Component Sasa–Satsuma Equation: Large-Time Asymptotics on the Line

Asymptotic analysis of time-fractional quantum diffusion

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrödinger equation in Rd. We define the time-fractional derivative by the Caputo derivative. We consider the initial-value problem for the free evolution of wave packets in Rd governed by the time-fractional Schrödinger equation iβ∂tαu=−Δu, with initial condition u(t=0)=u0, parameterized by two indices α,β∈(0,1]. We show distinctly different long-time evolution of the mean square displacement according to the relation between α and β. In particular, asymptotically ballistic motion occurs only for α=β.

Large-time asymptotics for degenerate cross-diffusion population models with volume filling

The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including “iterated” degenerated functions.

Long-Term Behaviour in an Exactly Solvable Model of Pure Decoherence and the Problem of Markovian Embedding

We consider a well-known, exactly solvable model of an open quantum system with pure decoherence. The aim of this paper is twofold. Firstly, decoherence is a property of open quantum systems important for both quantum technologies and the fundamental question of the quantum–classical transition. It is worth studying how the long-term rate of decoherence depends on the spectral density characterising the system–bath interaction in this exactly solvable model. Secondly, we address a more general problem of the Markovian embedding of non-Markovian open system dynamics. It is often assumed that a non-Markovian open quantum system can be embedded into a larger Markovian system. However, we show that such embedding is possible only for Ohmic spectral densities (for the case of a positive bath temperature) and is impossible for both sub- and super-Ohmic spectral densities. On the other hand, for Ohmic spectral densities, an asymptotic large-time Markovianity (in terms of the quantum regression formula) takes place.

Population growth in discrete time: a renewal equation oriented survey

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

The complex MKDV equation with step-like initial data: Large time asymptotic analysis

In this paper, we study large-time asymptotics for the complex modified Korteveg–de Vries equation with step-like initial data. It is shown that the step-like initial problem can be described by a matrix Riemann–Hilbert problem. Further we apply the steepest descent method to obtain different large-time asymptotics in the Zakharov–Manakov region, a plane wave region, and a slow decay region.

Asymptotics of Solutions for Periodic Problem for the Korteweg-de Vries Equation with Landau Damping, Pumping and Higher Order Convective Non Linearity

We study the periodic problem for the Korteweg–de Vries equation with Landau damping, linear pumping and a higher-order convective nonlinearity wt+wxxx-αwxx=βw+λwx2wxx,x∈Ω,t>0,w(0,x)=ψx,x∈Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{c} w_{t}+w_{xxx}-\\alpha w_{xx}=\\beta w+\\lambda w_{x}^{2}w_{xx},\ ext { }x\\in \\Omega ,t>0,\\\\ w(0,x)=\\psi \\left( x\\right) ,\ ext { }x\\in \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where, alpha ,beta >0,lambda in mathbb {R},Omega =left[ -pi ,pi right] . We assume that the initial data psi left( xright) are 2pi - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.

Electromagnetic field generated by tsunamigenic seabed deformation

We derive a mathematical model of an electromagnetic (EM) field generated by tsunamigenic seabed deformation over an ocean of constant depth. We solve the governing Maxwell equations for the EM field, coupled with a potential flow model of Cauchy–Poisson type for the transient fluid motion forced by seabed deformation. Our new model advances previous studies, where simplified formulae without direct forcing were assumed for the wave field. Using complex integration and large-time asymptotics, we obtain a novel analytical solution for the magnetic field propagating at large distance from the seabed deformation in two dimensions. We show that this magnetic field is made of two terms, one proportional to an Airy function, and thus propagating similarly to the surface gravity wave, and one proportional to a Scorer function, which exhibits a phase lag with respect to the surface gravity wave. Such a phase lag explains the time difference between the arrival of the EM field and the surface gravity wave generated by seabed deformation, which were observed in recent measurements and numerical results. Finally, we discuss the opportunity to detect EM fields as precursors of surface gravity waves in tsunami early warning systems. We introduce a novel non-dimensional parameter to identify the propagation regime of the magnetic field, i.e. self-induction versus diffusion-dominated. We show that tsunami early warning via EM field is possible for diffusion-dominated regimes when the water depth is less than 2 km. Our findings provide a rigorous analytical explanation of existing observations and numerical results.

Analysis of large time asymptotics of the fourth‐order parabolic system involving variable coefficients and mixed nonlinearities

In a general domain, we discuss an initial value problem for the symmetry generalized ‐coupled system of fourth‐order parabolic equations with space–time‐dependent diffusion coefficients and mixed nonlinearities. For the power‐type nonlinearities containing the sub‐critical Sobolev exponents, we achieve the local existence, a unique continuation, global existence, or finite‐time blow‐up of solutions to the system in time‐weighted Sobolev space. For the logarithmic nonlinearities containing the critical Sobolev exponents, we investigate the local existence in time‐weighted Sobolev space.

Self-similar solution for Hardy operator

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.

Modified Scattering for the One-Dimensional Schrödinger Equation with a Subcritical Dissipative Nonlinearity

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $$\lambda |u|^\alpha u$$ , where $$0<\alpha <2$$ , and $$\lambda $$ is a complex constant satisfying $$\text {Im} \lambda >\frac{\alpha |\text {Re} \lambda |}{2\sqrt{ \alpha +1}}$$ . For arbitrary large initial data, we present the uniform time decay estimates when $$4/3\le \alpha <2$$ , and the large time asymptotics of the solution when $$\frac{7+\sqrt{145}}{12}<\alpha <2$$ . The proof is based on the vector fields method and a semiclassical analysis method.

On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial differential operators

We consider certain constant-coefficient differential operators on Rd that have positive-definite symbols. Each such operator Λ with symbol P defines a semigroup of operators e−tΛ, t>0, admitting a continuous convolution kernel HPt for which the large-time behavior of HPt(0) cannot be deduced by basic scaling arguments. The simplest example has symbol P(ξ)=(η+ζ2)2+η4, ξ=(η,ζ)∈R2. We devise a method that allows us to determine the large-time behavior of HPt(0) for several classes of examples of this type and we show that these large-time asymptotics are preserved by perturbations of Λ by certain higher-order differential operators. For the P just given, it turns out that HPt(0)∼cPt−5/8 when t tends to infinity. We show how such results are relevant to understand the iterated convolution powers of certain finitely-supported complex functions on Zd. We also discuss how these techniques provide precise small-time asymptotics for HPt(0) in some cases when the operator Λ is not hypoelliptic. The simplest such example Λ has symbol P(ξ)=η2+(η−ξ2)4 and we show that HPt(0)∼cPt−1/2 as t tends to 0 in this case. Our work represents a first basic step towards a good understanding of the semigroups associated with these differential operators. Obtaining meaningful off-diagonal upper bounds for the convolution kernels of these semigroups remains an interesting challenge.

The matrix nonlinear Schrödinger equation with a potential

This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schrödinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with a potential and a general point interaction, in the whole supercritical regime. We prove that the small solutions are scattering solutions that asymptotically in time, t→±∞, behave as solutions to the associated linear matrix Schrödinger equation with the potential identically zero. The potential can be either generic or exceptional. Our approach is based on detailed results on the spectral and scattering theory for the associated linear matrix Schrödinger equation with a potential, and in a factorization technique that allows us to control the large-time behavior of the solutions in appropriate norms.