Time Asymptotic Behavior Research Articles

Stability and receptivity analyses of the heated flat-plate boundary layer with variable viscosity

This article presents the modal, non-modal, and resolvent analyses of the boundary layer developed over a heated flat-plate with temperature-dependent viscosity using linear stability theory. The governing equations are derived in the normal velocity–vorticity form by imposing small infinitesimal disturbances on the base flow with the Oberbeck-Boussinesq (OB) approximation. The spectral method is employed to discretize the governing stability equations in the wall-normal direction. The base flow comes from the similarity solution using R-K4th order method along with shooting techniques. The effects of viscosity stratification, inertia, shearing, and buoyancy on the stability of the boundary layer are investigated by varying the sensitivity parameter (ϵ), Reynolds (Re), Prandtl (Pr), and Richardson numbers (Ri). The modal analysis shows the time-asymptotic behavior of the disturbances and onset of instability is mainly caused by amplification of Tollmien–Schlichting (T-S) waves similar to as observe in the traditional Blasius case. The modal stability increases with increase in sensitivity parameter and stabilizing effects become more pronounced for liquid than gas. However, the thermal effects lead to destabilize the flow and strong destabilizing effects produced for higher Prandtl number. On the other hand, non-modal analysis displays an early transient growth of disturbances and an existence of these non-normality effects identified from the pseudospectra via. resolvent analysis. The non-modal growth increases with increase in ϵ even though T-S modes shift towards the damped region, which indicates the continuous modes in the eigenspectrum are more dominated than the discrete T-S modes due to the thermal effects. To clarify this qualitative change, a component-wise input–output analysis is performed to measure the receptivity to particular external disturbances. The results show the thermal energy of the disturbances is converted into kinetic energy due to thermal effects, resulting in strong receptivity amplification at the continuous mode due to the non-normality of the linear operator. Thus, the boundary layer under the influence of viscosity stratification, heating, and shearing effects is vulnerable to free-stream disturbances that could significantly affect bypass transition

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

In this paper, we study the heat content for the Schrödinger operator related to a subordinate Brownian motion and we also establish its small time asymptotic behavior for suitable potentials V. The case V=c1Ω for c>0 and Ω a Borel set of finite measure is investigated in detail.

Integrability and Inverse Scattering Transform of the Modified Benjamin-Ono Equation

Abstract This paper presents a B{"a}cklund transformation, the Lax representation, and conserved quantities for the modified Benjamin--Ono equation. The initial problem of the modified Benjamin--Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann--Hilbert problem in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.

On the Cauchy problem for acoustic waves in hereditary fluids: Decay properties and inviscid limits

This paper considers the viscous/inviscid Moore–Gibson–Thompson (MGT) equations with memory of type I in the whole space . For one thing, associating with a new condition on initial data, we derive the optimal estimates and the optimal leading term of the acoustic velocity potential for large time, where we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, by using the multiscale analysis and energy methods in the Fourier space, we demonstrate the inviscid limits (i.e., as the diffusivity of sound tends to zero), which match our Wentzel‐Kramers‐Brillouin (WKB) expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan‐MGT equation in viscous hereditary fluids.

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

In order to analyze the wave propagation in three-dimensional isotropic and viscoelastic body, the Cauchy initial value problem on unbounded domain is considered for the wave equation written as a system of fractional partial differential equations consisting of equation of motion of three-dimensional solid body, equation of strain, as well as of the constitutive equation, obtained by generalizing the classical Hooke’s law of three-dimensional isotropic and elastic body by replacing Lamé coefficients with the relaxation moduli to account for different memory kernels corresponding to the propagation of compressive and shear waves. By the use of the method of integral transforms, namely Laplace and Fourier transforms, the displacement field, as a solution of the Cauchy initial value problem, is expressed through Green’s functions corresponding to both compressive and shear wave propagation. Two approaches are adopted in solving the Cauchy problem: in the first one the displacement field is expressed through the scalar and vector fields, obtained as solutions of wave equations which are consequences of decoupling the wave equation for displacement field, while in the second approach the displacement field is obtained by the action of the resolvent tensor on the initial conditions. Fractional anti-Zener and Zener model I+ID.ID, as well as fractional Burgers model VII, as representatives of models yielding infinite and finite wave propagation speed, are used in numerical calculations and graphical representations of Green’s functions and its short and large time asymptotic behavior.

On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup

<p>This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.</p>

Time-asymptotic behavior of solution for NLS with large-size initial value

Time-asymptotic behavior of solution for NLS with large-size initial value

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 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.

Dynamics of a spatially homogeneous Vicsek model for oriented particles on a plane

We consider a spatially homogeneous Kolmogorov–Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov–Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869–896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality.

On echoes in magnetohydrodynamics with magnetic dissipation

We study the long time asymptotic behavior of the inviscid magnetohydrodynamic equations with magnetic dissipation near a combination of Couette flow and a constant magnetic field. Here we show that there exist nearby explicit global in time low frequency solutions, which we call waves. Moreover, the linearized problem around these waves exhibits resonances under high frequency perturbations, called echoes, which result in norm inflation in Gevrey regularity and infinite time blow-up in Sobolev regularity.

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.

A stochastic predator–prey model with Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function

This paper concerns with a stochastic system modeling the population dynamical behavior of one prey and two predators. In this paper, we adopt a special method to simulate the effect of the environmental interference to the system instead of using the linear functions of white noise, i.e., the growth rate of the prey and the death rates of the two predators are all governed by Ornstein–Uhlenbeck processes, which is more practical and interesting. The dynamical behavior among the three species in the model is discussed in detail. The existence and uniqueness, moment boundedness and long time asymptotic behavior of the unique global solution are investigated. Through rigorous proof and theoretical derivation, some sharp sufficient conditions for the persistence and extinction of the three species have been established. Furthermore, we obtain that when one or two predators die out, the remaining species can be stable in time average under same conditions. Moreover, by constructing some suitable Lyapunov functions, the sufficient conditions for the existence of the stationary distribution are explicitly determined. Under the same parametric restrictions, it is worth noting that we obtain the six-dimensional probability density function of the stochastic one-prey two-predator model around the quasi-equilibrium E∗, which can reflect most statistical properties. Finally, several numerical simulations are carried out to illustrate the theoretical results.

A small time large deviation principle for stochastic differential delay equations

In this paper, we prove a small time large deviation principle for a type of stochastic differential delay equations. The key point is to translate small time asymptotic behavior problems into Freidlin–Wentzell type large deviation problems. The weak convergence method plays an important role in obtaining the large deviation principle.

Modified scattering for the derivative fractional nonlinear Schrödinger equation

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional derivative nonlinear Schrödinger equation in one space dimension{i∂tu−1α|∂x|αu=iλ∂x(|u|2u), t>0,x∈R,u(0,x)=u0(x),x∈R,where α∈(0,1)∪(1,2)∪(2,3), and λ∈R. The fractional derivative |∂x|α=F−1|ξ|αF, where F stands for the Fourier transformation ϕˆ(ξ)=12π∫Re−ixξϕ(x)dx, and F−1 is the inverse Fourier transformation. Equation (1.1) with α=2 is the derivative NLS equation and studied extensively. The case of α=1 corresponds to the so-called derivative half-wave equation. We prove that the modified scattering of solutions occurs when 0<α<3 without the exceptional point α=1.

Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.

Time evolution of a decaying quantum state: Evaluation and onset of nonexponential decay

The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.

Existence, uniqueness, and long-time behavior of linearized field dislocation dynamics

This paper examines a system of partial differential equations describing dislocation dynamics in a crystalline solid. In particular we consider dynamics linearized about a state of zero stress and use linear semigroup theory to establish existence, uniqueness, and time-asymptotic behavior of the linear system.

Long Time Asymptotic Behavior for the Nonlocal mKdV Equation in Solitonic Space–Time Regions

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\ \ \lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\xi<6$ with $\xi=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\xi<-6$ and $\xi>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\bar{\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $\xi<-6$, phase function $\theta(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(\Lambda)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\rm Im}\nu(\zeta_i)$. In the region $\xi>6$, phase function $\theta(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(\Lambda)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.

Global well-posedeness and large time asymptotic behavior of 2D density-dependent Boussinesq equations of Korteweg type with vacuum

In this paper, we are devoted to study the Cauchy problem of the incompressible density-dependent Boussinesq equations of Korteweg type with vacuum. We establish some key spatial weighted estimates and a priori decay-in-time rate of the strong solutions by using the energy method. Furthermore, we prove that there is a unique global strong solution for the 2D Cauchy problem when the spatial weighted norm of the initial density is suitably small. Finally, we also obtain the large time decay rates for the gradients of velocity, temperature and pressure.