Decay Estimates Research Articles

Global well-posedness of three-dimensional incompressible Boussinesq system with temperature-dependent viscosity

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space Ḃ3,10. The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary.

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

A pseudo‐parabolic equation with logarithmic nonlinearity: Global existence and blowup of solutions

This article deals with a fourth‐order pseudo‐parabolic partial differential equation with logarithmic nonlinearity. We adopt the Faedo–Galerkin method to analyze the global existence of weak solutions and discuss the existence of a solution in subcritical and critical initial energy situations. Further, applying the concavity approach, we have shown that the solution blows up when the initial energy is subcritical, critical, and supercritical. Moreover, we have provided an upper and lower bound for blow‐up time and a decay estimate for the global solutions.

Optimal decay rates and space–time analyticity of solutions to the Patlak-Keller–Segel equations

Based on some new elementary estimates for the space–time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the Lq(Rd) (1≤q≤∞, d∈N) norm of the space–time derivatives of solutions to the (modified) Patlak-Keller–Segel equations with initial data in L1(Rd), which implies the joint space–time analyticity of solutions. When the L1(Rd) norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space–time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in L1(Rd). The decay estimates and space–time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.

Weak Gurov–Reshetnyak class in metric measure spaces

We introduce a weak Gurov–Reshetnyak class and discuss its connections to a weak Muckenhoupt A ∞ A_\infty condition and a weak reverse Hölder inequality in the setting of metric measure spaces with a doubling measure. A John–Nirenberg type lemma is shown for the weak Gurov–Reshetnyak class which gives a specific decay estimate for the oscillation of a function. It implies that a function in the weak Gurov–Reshetnyak class satisfies the weak reverse Hölder inequality. This comes with an upper bound for the reverse Hölder exponent depending on the Gurov–Reshetnyak parameter which allows the study of the asymptotic behavior of the exponent.

Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen–Cahn system

We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.

Global well-posedness for Ericksen-Leslie system with zero viscosity

We study the global well-posedness of the simplified Ericksen-Leslie system with zero viscosity on the periodic domain T 2 \mathbb {T}^2 . Precisely, we prove the global existence and uniqueness of smooth solution to Ericksen-Leslie system if the initial data ( u 0 , ∇ d 0 ) (u_0,\nabla d_0) is small in H 4 ( T 2 ) × H 4 ( T 2 ) H^4(\mathbb {T}^2)\times H^4(\mathbb {T}^2) . Furthermore, we derive the time decay estimate of ∇ d \nabla d in H 1 ( T 2 ) H^1(\mathbb {T}^2) .

Decay estimate for Yang-Mills fields on ALE spaces and applications

We study the decay estimates at the infinity for Yang-Mills connections on Ricci flat ALE manifolds and explore some applications.

Dissipative structures for the system of Moore–Gibson–Thompson thermoelasticity in the whole space

We investigate the dissipative structure for the system of Moore–Gibson–Thompson (MGT) thermoelasticity in the whole space. To analyze the dissipative structure, it is very useful to rewrite the equations into a symmetric hyperbolic system and apply the so‐called stability condition. When we rewrite our system into the symmetric hyperbolic form in the multidimensional case, it is important to take the constraint conditions into account. Indeed, the stability condition with the constraint conditions guarantees the dissipative property for our system in some cases. In this paper, we introduce the stability condition with constraints for the general problem and apply this argument to the system of MGT thermoelasticity. Furthermore, we discuss the optimality of the decay estimates we obtain, together with the no regularity‐loss phenomenon.

Adiabatic Evolution of Low-Temperature Many-Body Systems

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].

Spherically symmetric evolution of self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3

Abstract The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution ( ρ s ( x ) , 0 , 0 ) T {\left({\rho }_{s}\left(x),0,0)}^{T} is in a small neighborhood of the constant state ( ρ ¯ , 0 , 0 ) T {\left(\bar{\rho },0,0)}^{T} at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear L p {L}^{p} - L q {L}^{q} decay estimates.

Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Abstract This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) r\left(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω \omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{1,p}({\\mathbb R}^N)$$\\end{document}

We prove decay estimates for solutions of quasilinear elliptic equations in the whole RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} of the form u∈X:-divA(x,∇u)=a(x)f(x,u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\ ext{ div }\\,A(x,\ abla u)=a(x) f(x,u), \\end{aligned}$$\\end{document}where X=D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=D^{1,p}({\\mathbb R}^N)$$\\end{document} is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional Φ:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi : X\\rightarrow {\\mathbb R}$$\\end{document} related to the p-Laplacian equation in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} in the form u∈X:-Δpu=a(x)g(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\\Delta _p u=a(x) g(u), \\end{aligned}$$\\end{document}saying that for the subspace V of bounded continuous functions with weight 1+|x|N-pp,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+|x|^{\\frac{N-p}{p}},$$\\end{document} a local minimizer of Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} in the finer V topology is also a local minimizer in the X-topology. Global L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.

The higher regularity and decay estimates for positive solutions of fractional Choquard equations

AbstractIn the paper, we study the higher regularity and decay estimates for positive solutions of the following fractional Choquard equations: where , is a constant, , , , . Let be a positive solution of (). We obtain an optimal decay estimate for and reveal the relation between and the decay rate of for all permitting . We skillfully use the properties of Bessel kernel to deduce our main result by iteration. Moreover, by the decay estimates, the stronger regular estimates for are obtained.

The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting

abstract: The current work considers solutions to the wave equation on asymptotically flat, stationary, Lorentzian spacetimes in $(1+3)$ dimensions. We investigate the relationship between the rate at which the geometry tends to flat and the pointwise decay rate of solutions. The case where the spacetime tends toward flat at a rate of $|x|^{-1}$ was studied by Tataru (2013), where a $t^{-3}$ pointwise decay rate was established. Here we extend the result to geometries tending toward flat at a rate of $|x|^{-\kappa}$ and establish a pointwise decay rate of $t^{-\kappa-2}$ for $\kappa\in\Bbb{N}$ with $\kappa\ge 2$. We assume a weak local energy decay estimate holds, which restricts the geodesic trapping allowed on the underlying geometry. We use the resolvent to connect the time Fourier Transform of a solution to the Cauchy data. Ultimately the rate of pointwise wave decay depends on the low frequency behavior of the resolvent, which is sensitive to the rate at which the background geometry tends to flat.

The Compressible Euler and Acoustic Limits from Quantum Boltzmann Equation with Fermi–Dirac Statistics

This paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi–Dirac statistics rigorously. By employing Hilbert expansion, in particular analyzing the nonlinear implicit transformation between the classical form of compressible Euler equations and the one obtained directly from BFD, and some new type of Grad–Caflisch type decay estimate of the linearized collision operator, we establish the compressible Euler limit from scaled BFD equation, which was formally derived by Zakrevskiy in (Kinetic models in the near-equilibrium regime. Thesis at Polytechnique, 2015) by moment method. Consequently, the acoustic limit is obtained in optimal scaling with respect to Knudsen number.

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