Decay Estimates Research Articles

Rapid biphasic decay of intact and defective HIV DNA reservoir during acute treated HIV disease

AbstractDespite antiretroviral therapy (ART), HIV persists in latently-infected cells (the HIV reservoir) which decay slowly over time. Here, leveraging >500 longitudinal samples from 67 people living with HIV (PLWH) treated during acute infection, we developed a mathematical model to predict reservoir decay from peripheral CD4 + T cells. Nonlinear generalized additive models demonstrated rapid biphasic decay of intact DNA (week 0-5: t1/2 ~ 2.83 weeks; week 5-24: t1/2 ~ 15.4 weeks) that extended out to 1 year. These estimates were ~5-fold faster than prior decay estimates among chronic treated PLWH. Defective DNA had a similar biphasic pattern, but data were more variable. Predicted intact and defective decay rates were faster for PLWH with earlier timing of ART initiation, higher initial CD4 + T cell count, and lower pre-ART viral load. In this study, we advanced our limited understanding of HIV reservoir decay at the time of ART initiation, informing future curative strategies targeting this critical time.

Global Kato smoothing and Strichartz estimates for Schrödinger type equations with rough decay potentials

Let [Formula: see text] be a higher-order elliptic operator on [Formula: see text], where [Formula: see text] is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with [Formula: see text]. In particular, we first establish sharp global Kato smoothing estimates for [Formula: see text], based on uniform resolvent estimates of Kato–Yajima type for the absolutely continuous part of [Formula: see text]. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig–Ruiz–Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

Dispersive Decay Estimates for Dirac Equations with a Domain Wall

Dispersive Decay Estimates for Dirac Equations with a Domain Wall

Visco‐elastic damped wave models with time‐dependent coefficient

AbstractIn this paper, we study the following Cauchy problem for linear visco‐elastic damped wave models with a general time‐dependent coefficient : We are interested to study the influence of the damping term on qualitative properties of solutions to () as decay estimates for energies of higher order and the parabolic effect. The main tools are related to WKB‐analysis. We apply elliptic as well as hyperbolic WKB‐analysis in different parts of the extended phase space.

Conversion of a piston–cylinder dimensional dataset to the effective area of a mechanical pressure generator

Recent developments in diameter metrology at NIST have improved the dimensional characterization of piston-cylinder assemblies (PCAs) to unprecedented precision. For the newest generation of PCAs, the standard uncertainty in the measurement of the outer diameter is 12 nm, while uncertainty in the measurement of the inner diameter is 14 nm. With a high-accuracy dimensional dataset in hand, the task of determining the pressure generated by a specific PCA is reduced to converting the diameter (and straightness and roundness) to an effective area (and distortion coefficient). The details on how this was performed for the artifact PCA2062 are described. PCA2062 was dimensioned in 2017 and 2020; the area repeated within . The calculation produced estimates of fall rate and rotation decay that agreed with experimental observations within 12 %. The fall rate is proportional to the square of the gap width; therefore, the agreement between calculation and measurement validates the dimensional estimate of the gap width within (36 ± 42) nm, where the 42 nm standard uncertainty is governed by the present state of flow theory. The piston gage model is buttressed by three comparison tests against a laser barometer, which support a view that PCA2062 is linear and reproducible within 0.2 μPa Pa-1. Finally, an estimate of uncertainty in the effective area of a dimensioned artifact is provided: as expected, the diameter measurement is the main culprit, but there are open questions regarding the flow model that preclude an accurate evaluation of the distortion coefficient. For the 530 kPa operating range of PCA2062, distortion is not a significant problem, but the effect would be dominant in assemblies operating at 1 MPa and above.

Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation

This paper focuses on the stability and decay estimates of the 3D MHD equations with only one direction dissipation near a background magnetic field. The global stability and decay estimates of the 3D Navier-Stokes equations with only one direction dissipation remain an outstanding open problem. However, when coupled with the magnetic field in the MHD system involved here, the velocity field is shown to stabilize and decay in time. This result reveals that the magnetic field stabilizes and damps the electrically conducting fluids. By utilizing time-weighted methods and the method of bootstrapping argument, we establish that any perturbation near a magnetic field is globally stable in H3(R3). Furthermore, explicit decay rates in H2(R3) are obtained.

Decay estimates for Beam equations with potential in dimension three

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potentialutt+(Δ2+V)u=0,u(0,x)=f(x),ut(0,x)=g(x) in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of H=Δ2+V, we first prove the following optimal time decay estimates of the solution operators‖cos⁡(tH)Pac(H)‖L1→L∞≲|t|−32and‖sin⁡(tH)HPac(H)‖L1→L∞≲|t|−12. Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators cos⁡(tH) and sin⁡(tH)H, but their decay will be significantly changed for the second and third-kind resonances.

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

Rapid Biphasic Decay of Intact and Defective HIV DNA Reservoir During Acute Treated HIV Disease.

Despite antiretroviral therapy (ART), HIV persists in latently-infected cells ("the reservoir") which decay slowly over time. Here, leveraging >500 longitudinal samples from 67 people with HIV (PWH) treated during acute infection, we developed a novel mathematical model to predict reservoir decay from peripheral CD4+ T cells. Nonlinear generalized additive models demonstrated rapid biphasic decay of intact DNA (week 0-5: t1/2~2.83 weeks; week 5-24: t1/2~15.4 weeks) that extended out to 1 year. These estimates were ~5-fold faster than prior decay estimates among chronic treated PWH. Defective DNA had a similar biphasic pattern, but data were more variable. Predicted intact and defective decay rates were faster for PWH with earlier timing of ART initiation, higher initial CD4+ T cell count, and lower pre-ART viral load. These data add to our limited understanding of HIV reservoir decay at the time of ART initiation, informing future curative strategies targeting this critical time.

Spectral structure of the Neumann–Poincaré operator on axially symmetric functions

We consider the Neumann–Poincaré operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when it is restricted to axially symmetric functions. If the boundary of the domain is smooth, we show that there are infinitely many axially symmetric eigenfunctions and derive Weyl-type asymptotics of the corresponding eigenvalues. We also derive the leading order terms of the asymptotic limits of positive and negative eigenvalues. The coefficients of the leading order terms are related to the convexity and concavity of the domain. If the boundary of the domain is less regular, we derive decay estimates of the eigenvalues. The decay rate depends on the regularity of the boundary. We also consider the domains with corners and prove that the essential spectrum of the Neumann–Poincaré operator on the axially symmetric three-dimensional domain is non-trivial and contains that of the planar domain.

Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.

Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity

In this paper, a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The purpose of this study is twofold. First, it is shown that the system is locally well-posed. For that purpose, the existence of classical solutions which are perturbation of constant states is established. Second, it is proved that in the particular case of subsonic equilibrium states, sufficiently small perturbations decay globally in time. In order to prove this stability property, the linearized system around the subsonic state is examined. Using an appropriately constructed compensating matrix symbol in the Fourier space, it is proved that solutions to the linear system decay globally in time, underlying a dissipative mechanism of regularity gain type. These linear decay estimates, together with the local existence result, imply the global existence and the decay of perturbations to constant subsonic equilibrium states as solutions to the full nonlinear system.

Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation

This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.

Global solution for semilinear σ$$ \sigma $$‐evolution models with memory term

In this paper, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

Asymptotic estimates of solution to damped fractional wave equation

It is known that the damped fractional wave equation has the diffusive structure as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t\\rightarrow \\infty $\\end{document}. Let u(t,x)=e−tcosh(tL)f(x)+e−tsinh(tL)L(f(x)+g(x))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)=e^{-t}\\cosh (t\\sqrt{L})f(x)+e^{-t} \\frac{\\sinh (t\\sqrt{L})}{\\sqrt{L}}(f(x)+g(x))$\\end{document} be the solution of the Cauchy problem for the damped fractional wave equation, where L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sqrt{L}$\\end{document} involves the fractional Laplacian (−△)α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\ riangle )^{\\alpha}$\\end{document} on the space variable. We can study the decay estimate of the solution u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)$\\end{document} over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\alpha <1$\\end{document}, the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha =1$\\end{document}, the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.

Improved energy decay estimate for Dir-stationary Q-valued functions and its applications

In this paper, we establish an improved decay estimate for the Dirichlet energy of Dir-stationary Q-valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary Q-valued functions defined on Rm. Additionally, in an attempt to establish the continuity of Dir-stationary Q-valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain.

Asymptotic spatial behaviour in linearised thermoelasticity for non-compact regions

A quasi-static approximation is studied of linearised nonhomogeneous anisotropic compressible thermoelasticity on a non-compact region. Differential inequalities are constructed which under appropriate conditions lead to algebraic spatial growth and decay estimates for various cross-sectional and volume measures of the temperature, displacement and their spatial gradients.