Decay Estimates Research Articles

Stability and decay estimates of the 2D incompressible magneto-micropolar fluid system with partial viscosity on a flat strip

Stability and decay estimates of the 2D incompressible magneto-micropolar fluid system with partial viscosity on a flat strip

Decay estimates and extinction properties for some parabolic equations with fractional time derivatives

Decay estimates and extinction properties for some parabolic equations with fractional time derivatives

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Decay estimates in time for fractional evolution equations

We consider the homogeneous time-fractional diffusion equation \partial^{\alpha} (u-u_0)(t)+A u(t)=0, \quad t\text{-a.e.} Here, \partial^{\alpha} denotes the Riemann–Liouville fractional derivative of order \alpha \in (0,1) with respect to time and A is a uniformly elliptic operator on \mathbb{R}^d . We prove decay estimates in time and other regularity properties for the solution of the above equation.

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 .

On the Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Coupled Petrovsky-Wave System

In this paper, we consider the initial-boundary value problem for a class of nonlinear coupled wave equation and Petrovesky system in a bounded domain. The strong damping is nonlinear. First, we prove the existence of global weak solutions by using the energy method combined with Faedo-Galarkin method and the multiplier method. In addition, under suitable conditions on functions gi(⋅), i = 1, 2 and a(⋅), we obtain both exponential and polynomial decay estimates. The method of proofs is direct and based on the energy method combined with the multipliers technique, on some integral inequalities due to Haraux and Komornik.

Time decay estimates of solutions to a two-phase flow model in the whole space

Time decay estimates of solutions to a two-phase flow model in the whole space

On Behavior of Solutions for Nonlinear Klein–Gordon Wave Type Models with a Logarithmic Nonlinearity and Multiple Time-Varying Delays

In this paper, we study the existence and exponential stability of solutions to a class of nonlinear delay Klein–Gordon wave type models on a bounded domain. Such models include multiple time-varying delays, frictional damping, and nonlinear logarithmic source terms. After showing the local existence result of the solutions using Faedo–Galerkin’s method and logarithmic Sobolev inequality, the global existence is analyzed. Then, under some appropriate conditions, energy decay estimates and exponential stability results of the global solutions are investigated.

On the Cauchy problem for semilinear σ‐evolution equations with time‐dependent damping

In this paper, we would like to consider the Cauchy problem for semilinear ‐evolution equations with time‐dependent damping for any . Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from to and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.

Long time behavior of weak solutions for the nonlinear damped beam equation

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data.

Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation ρ(x)utt+μ(x)ut+(r(x)uxx)xx=0, subject to the clamped boundary conditions u(0,t)=ux(0,t)=0 at x=0, and the boundary conditions (−r(x)uxx)x=ℓ=krux(ℓ,t)+kauxt(ℓ,t), (−(r(x)uxx)x)x=ℓ=−kdu(ℓ,t)−kvut(ℓ,t) at x=ℓ, is analyzed. The boundary conditions at x=ℓ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient μ(x)≥0, and the boundary springs kr,kd≥0 and dampers ka,kv≥0. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.

Multiway Decomposition Followed by Reconvolution of Fluorescence Time Decay Data.

The use of time-resolved emission spectroscopy (TRES) is increasing as these instruments become more available, and the prices are decreasing, especially with the development of cheaper LED light sources. In this article, we propose a new methodology for analyzing TRES data. It combines two existing methods: PARAFAC and reconvolution. PARAFAC is a soft modeling curve resolution technique which has been extensively applied to steady-state fluorescence data, and reconvolution is the most common method for fitting TRES data. The proposed method is compared to two well-established methods of analyzing these data, namely, global reconvolution and tail fitting. In addition, we compare our approach with the SLICING method proposed in 2021 by Devos et al. which is also based on a soft model, but does not include the reconvolution step. All of these methods follow the assumption that the measured fluorescence signal is a linear combination of the underlying fluorophores. The comparison is based on a measured TRES data set with a mixture of three fluorophores and two sets of simulated data sets with up to four fluorophores. The results show that global fitting works well as long as the signal-to-noise ratio (SNR) is high (more than 15 dB), independent of the spacing between the emission peak maxima. SLICING does not give as good estimates of the time decay, mainly due to the challenge of defining the tail. Our proposed method gives robust and accurate results, outperforming the other techniques in cases with broad instrument response functions and high noise levels with SNRs down to 5 dB.

Decay rates of the highest order spatial derivatives for solutions of compressible Hall MHD system with or without Coulomb force

In this thesis, we study the decay rates of the highest order (M-th order) spatial derivatives for solutions of both compressible Hall MHD system and that with Coulomb force by a pure energy method inspired by Guo and Wang (2012)[12] and Gao, Li and Yao (2023)[9]. When the norm of H3 for the initial perturbation is small enough, and the norms of both HM(M≥3) and H˙−α(α∈[0,32)) for that only need to be bounded, we first prove the decay rates of compressible Hall MHD system in L2 are (1+t)−(M+α)2. And then affected by the Coulomb force, for compressible Hall MHD system with Coulomb force, we obtain the decay rates are (1+t)−(M+α)2 for α∈[12,32) and (1+t)12[(14−12α)−(M+α)] for α∈[0,12), which are consistent with the decay estimates for compressible MHD system with Coulomb force. In particular, the results improve the original decay rates (1+t)−(M−1+α)2 established by Xu et al. (2016)[46] and Tan, Tong and Wang (2015)[31].

On the global wellposedness of free boundary problem for the Navier-Stokes system with surface tension

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ωt⊂R3, t>0, to a problem in the lower half-space R−3. We then establish some time-weighted estimate of solutions, in an Lp-in-time and Lq-in-space setting, for the linearized problem around the trivial steady state with the help of Lr-Ls time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R−3 admits a global-in-time solution in the Lp-Lq setting and that the solution decays polynomially as time t tends to infinity under the assumption that p, q satisfy the conditions: 2<p<∞, 3<q<16/5, and (2/p)+(3/q)<1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R−3 to prove our main results mentioned above for the original problem in Ωt. Here we want to emphasize that it is not allowed to take p=q in the above assumption about p, q, which means that the different exponents p, q of Lp-Lq setting play an essential role in our approach.

Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schrödinger operator with smooth coefficients PA (x, hD x ) = (hD x − µA(x))2 + V (x) on L 2(R d ). We shall essentially consider the case where µ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon [1], also in the case of a Riemannian manifold M. Agmon estimates hold true for any h, but are particularly useful in the limit h → 0 when studying tunneling.

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

Sharp decay estimate for solutions of general Choquard equations

Consider the general Choquard equation{Δpu+(∫RN|u(y)|pα⁎|x−y|αdy)|u|pα⁎−2u=0,inRN,u∈D1,p(RN), where 1<p<N,0<α<N,p<pα⁎=(N−α2)pN−p and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. By using the Wolff potential theory, we prove that for any solution u of the equation, there exists a constant c such that|u(x)|≤c(1+|x|)−N−pp−1, for x∈RN. The decay estimate is sharp in the sense that the positive solution u satisfiesu(x)≥c(1+|x|)−N−pp−1.

Decay estimates of solution to the two‐dimensional fractional quasi‐geostrophic equation

This paper studies the fractional quasi‐geostrophic equation with modified dissipation term. We first prove the existence and uniqueness of smooth solution of the fractional quasi‐geostrophic equation by using the regularization method. Then, utilizing the Fourier‐splitting method, we obtain the long‐time behavior of the solution. More precisely, we show that under suitable assumptions on the initial data, for any multi‐index and , the solution satisfies and

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordström Spacetime

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, |Q|=M,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ |Q|=M, $$\\end{document} Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}. In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component α̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\underline{\\alpha }} $$\\end{document} not decaying asymptotically along the event horizon H+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+, $$\\end{document} a result previously unknown in the literature.