Optimal Decay Research Articles

Whole space case for solution formula of Korteweg type fluid motion in R3

In this paper we consider the solution formula of linearized diffusive capillary model of Korteweg type without surface tension in three-dimensional Euclidean space ℝ3 using Fourier transform. Firstly, we construct the matrix of differential operators from the model problem. Then, we apply Fourier transform to the matrix. In the third step, we consider the resolvent problem of model problem. Finally, we find the solution formula of velocity and density by using inverse Fourier transform. For the further research we can consider not only estimating the solution operator families of the Korteweg theory of capillarity but also estimating the optimal decay for solution to the non-linear problem.

Optimal decay rate to the contact discontinuity for Navier–Stokes equations under generic perturbations

Optimal decay rate to the contact discontinuity for Navier–Stokes equations under generic perturbations

Global existence and optimal time decay rate of the solutions to the incompressible biaxial nematic liquid crystals flows in $$\mathbb {R}^3$$

Global existence and optimal time decay rate of the solutions to the incompressible biaxial nematic liquid crystals flows in $$\mathbb {R}^3$$

Large Time Behavior for Solutions to the Anisotropic Navier–Stokes Equations in a 3D Half-space

Abstract We consider the large time behavior of the solution to the anisotropic Navier–Stokes equations in a $3$D half-space. Investigating the precise anisotropic nature of linearized solutions, we obtain the optimal decay estimates for the nonlinear global solutions in anisotropic Lebesgue norms. In particular, we reveal the enhanced dissipation mechanism for the third component of velocity field. We notice that, in contrast to the whole space case, some difficulties arises on the $L^{1}(\mathbb{R}^{3}_{+})$-estimates of the solution due to the nonlocal operators appearing in the linear solution formula. To overcome this, we introduce suitable Besov-type spaces and employ the Littlewood–Paley analysis on the tangential space.

On the isoperimetric Riemannian Penrose inequality

AbstractWe prove that the Riemannian Penrose inequality holds for asymptotically flat 3‐manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the mass being a well‐defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential‐theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3‐manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well‐posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.

Structural modification of MgO/Au thin films by aluminum Co-doping and related studies on secondary electron emission

The secondary electron emission (SEE) has the potential applications in electron multiplication, mass spectrometer, scanning electron microscope, photomultiplier tubes and so on. Among these, the electron multiplier (EM) based on MgO/Au thin film has attracted great attention over the past few years. Here, we design a new kind of MgO thin film by co-doping of gold and aluminum elements based on the structural modification through reactive magnetron sputtering method. The doping of Al into MgO/Au film gives a rise to the maximum SEE coefficient (δmax) between 9.02 and 10.23 while the optimal decay rate is around 10.4 % after 2 h of consistent electron bombardment. Herein, we also improved the stability of the film stored in air by sputtering Al2O3 layer on it. For the sample sputtered with the protecting layer Al2O3, its SEE coefficient (δ) at the incident electron energy of 200 eV still kept at a relatively high level that exceed 4.5 after stored in air for 72 h. The film properties after Al and Au co-doping are investigated in detail by the SEM, AFM, XPS and UV-VIS spectroscopy. Our results provide a new candidate for SEE layer in the application of long lifespan vacuum electron devices.

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.

Optimal temporal decay rates of solutions for combustion of compressible fluids

Optimal temporal decay rates of solutions for combustion of compressible fluids

Global existence and time decay rate of classical solutions to a hybrid Vlasov-Fokker-Planck-MHD equations

In this paper, we prove the existence of global classical solutions to a kinetic-fluid system when initial data is a small perturbation of some given equilibrium state in R3. The system consists of the Vlasov-Fokker-Planck equation coupled with the compressible magnetohydrodynamics (MHD) equations via the nonlinear coupling terms of Lorenz force type. It describes the motion of energetic particles in a fluid with a magnetic field. The proof of global existence mainly relies on the energy method. Due to the complex nonlinear structure of Lorentz force, we need to establish a more refined uniform a prior estimates. Moreover, under additional conditions on initial data, the optimal time decay rate of solutions toward the equilibrium state can be obtained by using the Fourier analysis.

Optimal decay of the Boltzmann equation

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high‐low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity‐weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time‐weighted dissipation estimates via the time‐weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics.

Hermite expansions for spaces of functions with nearly optimal time-frequency decay

We establish Hermite expansion characterizations for several subspaces of the Fréchet space of functions on the real line satisfying|f(x)|≲e−(12−λ)x2,|fˆ(ξ)|≲e−(12−λ)ξ2,∀λ>0. In particular, we extend and improve Fourier characterizations of the so-called proper Pilipović spaces obtained in [21]. The main ingredients in our proofs are the Bargmann transform and some achieved optimal forms of the Phragmén-Lindelöf principle.

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.

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

Convergence to the planar interface for a nonlocal free‐boundary evolution

AbstractWe capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well‐prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one‐dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

The optimal polynomial decay in the extensible Timoshenko system

AbstractIn this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non‐exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds.

Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources

Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources

Global well-posedness for the three dimensional compressible micropolar equations

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to Ḃ2,∞−σ1(R3).

Stability and optimal decay rates for abstract systems with thermal damping of Cattaneo’s type

This paper studies the stability of an abstract thermoelastic system with Cattaneo’s law, which describes finite heat propagation speed in a medium. We introduce a region of parameters containing coupling, thermal dissipation, and possible inertial characteristics. The region is partitioned into distinct subregions based on the spectral properties of the infinitesimal generator of the corresponding semigroup. By a careful estimation of the resolvent operator on the imaginary axis, we obtain distinct polynomial decay rates for systems with parameters located in different subregions. Furthermore, the optimality of these decay rates is proved. Finally, we apply our results to several coupled systems of partial differential equations.

Optimal decay rate for a Rayleigh beam–string coupled system with frictional damping

This paper addresses the stability of a coupled system that consists of a conservative Rayleigh beam and a damped elastic string, the latter being equipped with frictional damping. By estimating the growth of the resolvent along the imaginary axis, we employ the frequency domain method to demonstrate that the coupled Rayleigh beam–string system exhibits polynomial stability, characterized by a decay rate t−1/4. Furthermore, we establish the optimality of the decay rate, t−1/4, through a rigorous spectral analysis. Finally, a numerical example is present to validate the theoretical findings.

Stability and optimal decay for the 3D anisotropic magnetohydrodynamic equations

AbstractThis paper investigates the stability problem and large time behavior of solutions to the three‐dimensional magnetohydrodynamic equations with horizontal velocity dissipation and magnetic diffusion only in the direction. By applying the structure of the system, time‐weighted methods, and the method of bootstrapping argument, we prove that any perturbation near the background magnetic field (1, 0, 0) is globally stable in the Sobolev space . Furthermore, explicit decay rates in are obtained. Motivated by the stability of the three‐dimensional Navier–Stokes equations with horizontal dissipation, this paper aims to understand the stability of perturbations near a magnetic background field and reveal the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.