Time Decay Rates Research Articles

Well-posedness and decay of solutions for three-dimensional generalized Navier–Stokes equations

In this paper, we study the generalized incompressible Navier–Stokes equations in R3. Based on the energy estimates and regularization of the initial data with the heat semigroup, we prove the well-posedness of solutions in H5−4α2(R3) provided that the H5−4α2(R3)-norm of initial data is sufficiently small. In addition, in contrast to the generalized heat equation, the upper bound of the time decay rate of solutions to the generalized Navier–Stokes equations is also established.

Global existence and large time behavior for a two-dimensional chemotaxis–shallow water system

This paper concerns the chemotaxis–shallow water system in two dimensional whole spaces. We first establish the global existence of strong solution under the condition that the initial data are close to the constant equilibrium state in HN(R2)(N≥3)-framework. Then, the optimal time decay rates of the global solution are obtained by the method of energy estimates, if the initial data belong to H˙−s(R2)(0<s<1) additionally.

Er3+-ions doped lithium‑bismuth-boro-phosphate glass for 1532 nm emission and efficient red emission up conversion for telecommunication and lasing applications

Er3+-ions doped lithium‑bismuth-boro-phosphate glass samples were prepared by conventional melt quenching method and their structural, thermal, fluorescence, and decay times of the glasses were investigated. Prepared glass samples exhibits amorphous nature and shows very good thermal stability in the temperature range of RT to 500 °C. Judd-Ofelt (JO) analysis was carried out to obtain the parameters which will evidence the glass system for fibre and lasing applications. The intensity parameters Ωλ(2,4 and 6) and stimulated-emission cross-sections were estimated for present glass system. The magnitude of Ω2 confirms the covalency nature and compared with available other host. The near infrared emission spectra were measured by 980 nm excitation in which the emission intensity is found to be at 1532 nm for the 4I11/2–4I15/2 transition. The FWHM (78 nm), σ(λp) (10.11 × 10−21 cm2), ΔG (7.93 × 10−28 cm3) and G (9.1 × 10−24 cm2 s), for 4I11/2–4I15/2 transition of present glasses are compared to some of the glass systems. Furthermore, the time decay rate found to be slight decreased from 1.04 ms to 0.90 ms when the concentration increased from 0.1 to 0.5 mol% of Er3+ ions and also all follow the single exponential behaviour which is attributed to the self-quenching effect due to the cross-relaxation channels. McCumber's theory reveals peak emission cross-section value is also good agreement with σ(λp) which is 10.11×10−21cm2. The obtained gain coefficient population inversion 40% with broad band of emission proves to consider the present system for C band amplifiers. The up conversion of luminescence spectra upon 980 nm excitation exhibits red emission at 665 nm due to 4F9/2 - 4I15/2 transitions also two photon absorption was obtained for power dependence up conversion.

Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier–Stokes equations with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier–Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier–Stokes equations on the whole space admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure, which are the same as those of the homogeneous case.

Decay rates of the compressible viscoelastic flows with electric potential

We study the global existence and decay rates of strong solutions to the three dimensional compressible viscoelastic flows with an electric potential. The energy method for high frequency parts combined with a spectral analysis for low frequency parts is used to obtain the global solution and its decay rate (in time). We show that the density of the solution converges to its equilibrium state at the L2-decay rate (1+t)−34−12 when the initial perturbation admits some small assumptions. Compared to the compressible viscoelastic flows Hu and Wu (2013) [13], our results imply that the rotating effect of the electric field affects the dispersion of fluids and enhances the time decay rate of the density.

Asymptotic Behavior of Solutions to a New Hall-MHD System

In this paper, by using Fourier splitting method and the properties of decay character $r^{*}$ , we consider the algebraic time decay rate of solutions to a new Hall-MHD system in Sobolev space $H^{m}(\mathbb{R} ^{3})\times H^{m+1}(\mathbb{R}^{3})$ for $m\geq 0$ .

Long-time behavior of solutions for full compressible quantum model in R3

In this paper, long-time behavior of solutions for full compressible quantum flows in three-dimensional whole space is studied. We establish the optimal time decay rates for higher-order spatial derivatives of density, velocity and temperature, which will improve the work of Pu and Guo (2016).

Decay Rates of the Compressible Hall-MHD Equations for Quantum Plasmas

In this paper, we consider the global existence and decay rates of strong solutions to the three-dimensional compressible quantum Hall-magneto-hydrodynamics equations. By combing the Lp-Lq estimates for the linearized equations and a standard energy method, the global existence and its convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the stationary solution is small in some Sobolev norms. More precisely, the decay rates in time of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

In this paper, we consider the large time behavior of the strong solutions to the three dimensional compressible viscoelastic flows with damping. Based on the energy method and spectral analysis, we analyze the influences of the damping on the global existence and decay rates of compressible viscoelastic flows under some small assumptions in H3-framework. Compared with the time decay rates of solutions to the compressible viscoelastic flows in [1], our results imply that the friction of the damping is stronger than the dissipation effect of the viscosities.

Optimal Time Decay of Navier–Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not to be small). Finally, combining the recent results in \cite{zhang2014} with our methods, we can only need the initial data to be small in homogeneous Besov space $\dot{B}^{N/2-2, N/2} \cap \dot{B}^{N/2-1}$ to get the optimal time decay rate in space $L^{2}$.

Nd3+-doped lanthanum lead boro-tellurite glass for lasing and amplification applications

Nd3+-doped lanthanum lead boro-tellurite glass samples were prepared by conventional melt quenching method and their structural, thermal, fluorescence, and decay times of the glasses were investigated. Prepared glass samples exhibits amorphous nature and shows good thermal stability in the temperature range of 100–800 °C. Judd-Ofelt (JO) analysis was carried out and the intensity parameters (Ωλ = 2, 4, 6) also spontaneous radiative probability and stimulated-.emission cross-sections were estimated. The magnitude of Ωλ confirms the covalency nature. The near infrared emission spectra were measured by 808 nm excitation in which the emission intensity is found to be high at 1060 nm for the 4F3/2 → 4F11/2 transition. The stimulated cross section, effective band width and branching ratios are found to be 8.910 × 10−20 cm2, 21.57 nm and 53.72 % respectively, for 4F3/2 → 4F11/2 transition. The derived gain bandwidth, figure of merit, threshold and saturation intensity found to be comparable to some of the glass systems. Furthermore, the time decay rate found to decrease from 100 μs to 27 μs when the concentration increased from 0.1 to 3.0 mol% of Nd3+ ions and also all follow the single exponential behaviour which is attributed to the self quenching effect due to the cross-relaxation channels.

A stability analysis of the compressible boundary layer flow over indented surfaces

This contribution presents a stability analysis for compressible boundary layer flows over indented surfaces. Specifically, the effects of increasing depth D/δ* and Ma∞ number on perturbation time-decay rates and spatial amplification factors are quantified and compared with those of an unindented configuration.The indented surfaces represent aeronautical lifting surfaces endowed with the smooth gap resulting when a filler material applied at the junction of leading-edge and wing-box components retracts upon its curing process. Since the configuration considered is such that the parallel/weakly-parallel assumptions are necessarily compromised, a global temporal stability analysis is considered in this study. Our analysis does not require a parallel flow constrain, and hence it is believed to be valid when two dimensional effects are relevant.We find that small surface modifications enhance certain flow instabilities. An increase in Ma∞ enhances further this behaviour: for the D/δ*=1.5,Ma∞=0.5 case, amplification factors at a given location can be up to 20 times larger than those corresponding to the unindented case.

Optimal decay rate for the compressible Navier–Stokes–Poisson system in the critical Lp framework

In this paper, we consider the large time behavior of global solutions to the initial value problem for the compressible Navier–Stokes–Poisson system in theLpcritical framework and in any dimensionN≥3. We obtain the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The proof is mainly based on the Littlewood–Paley theory and refined time weighted inequalities in Fourier space.

Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

We consider the global existence and large time behavior of solutions near a constant equilibrium state to the bipolar non-isentropic compressible Euler–Maxwell system in , where the background magnetic field could be non-zero. The global existence is established under the assumption that the H3 norm of the initial data is small, but its higher order derivatives could be large. Combining the negative Sobolev (or Besov) estimates with the interpolation estimates, we prove the optimal time decay rates of the solution and its higher order spatial derivatives. In this sense, our results improve the similar ones in Wang et al (2012 SIAM J. Math. Anal. 44 3429–57).

Quantum preservation of the measurements precision using ultra-short strong pulses in exact analytical solution

Various schemes have been proposed to improve the parameter-estimation precision. In the present work, we suggest an alternative method to preserve the estimation precision by considering a model that closely describes a realistic experimental scenario. We explore this active way to control and enhance the measurements precision for a two-level quantum system interacting with classical electromagnetic field using ultra-short strong pulses with an exact analytical solution, i.e. beyond the rotating wave approximation. In particular, we investigate the variation of the precision with a few cycles pulse and a smooth phase jump over a finite time interval. We show that by acting on the shape of the phase transient and other parameters of the considered system, the amount of information may be increased and has smaller decay rate in the long time. These features make two-level systems incorporated in ultra-short, of-resonant and gradually changing phase good candidates for implementation of schemes for the quantum computation and the coherent information processing.

Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity

In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with the Ostwald–de Waele type viscosity (p-Laplacian type degenerate viscosity) ∂tu+∂x(f(u))=μ∂x(|∂xu|p−1∂xu) where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. When the corresponding Riemann problem admits a multiwave pattern which consists of the rarefaction waves and the contact discontinuity, it has already been proved by Yoshida that the solution to the Cauchy problem tends toward the linear combination of the rarefaction waves and contact wave for the Ostwald–de Waele type viscosity as time goes to infinity. We investigate the decay rate in time of the solution toward the multiwave pattern. Furthermore, we investigate the decay rate in time of the solution for the derivative. The proof is given by L1, L2-energy and time-weighted Lq-energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves.

Nonlinear stability of traveling wavefronts for competitive-cooperative Lotka–Volterra systems of three species

This paper is concerned with nonlinear stability of traveling wavefronts for competitive-cooperative Lotka–Volterra systems of three species. It is proved that all traveling wavefronts with speed c > c* are exponentially stable in L∞ space by the weighted energy method and comparison principle. Moreover, the time decay rates are obtained by weighted energy estimates.

Global well-posedness of the non-isothermal model for incompressible nematic liquid crystals

In this paper, a macromolecular non-isothermal model for the incompressible hydrodynamics flow of nematic liquid crystals on mathbb {T}^{3} is considered. By a Galerkin approximation, we prove the local existence of a unique strong solution if the initial data u_{0}, d_{0} and theta_{0} satisfy some natural conditions and provided that the viscosity coefficients μ and the heat conductivity κ, h, which are temperature dependent, are properly differentiable and bounded. Moreover, with small initial data, we can extend the local strong solution to be a global one by the argument of contradiction. In this case, the exponential time decay rate is also established.

The Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space

AbstractIn this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space. The global solutions to this system near the relativistic Maxwellian are constructed and the optimal time decay rate of global solutions are also obtained by an approach by combining the compensating function and energy method.

Large time decay estimates for the Muskat equation

ABSTRACTWe prove time decay of solutions to the Muskat equation in 2D and in 3D. In the previous work, the following norms were utilized to prove global existence of solutions to the Muskat problem: In this paper, for the 3D Muskat problem, given initial data for some l≥3 such that for a constant k0≈1∕5, we prove uniform in time bounds of ∥f∥s(t) for −2<s<l−1 and assuming we prove time decay estimates of the form for 0≤s≤l−1 and −2≤ν<s. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.