Optimal Decay Rates Research Articles

Optimal decay rates for the viscous two-phase model without constraints on transition to single-phase flow

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37], so as to derive the decay estimates for the velocity and relax the constraints on the initial data.

Optimal temporal decay rates for the compressible viscoelastic flows

For the Cauchy problem of the three-dimensional compressible viscoelastic flows, we establish the optimal temporal decay rates of the all-order spatial derivatives of the global strong solution in the weaker initial condition. The main novelty of this paper is that the optimal decay estimates of the highest-order derivatives of the solution is obtained by using spectral analysis and energy method, which can be considered as the further investigation to [X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal. 45 (2013) 2815–2833] with only the lower-order derivative estimates.

Optimal energy decay for a viscoelastic Kirchhoff equation with distributed delay acting on nonlinear frictional damping

In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function.

The two-fluid incompressible Navier–Stokes–Maxwell system: Green’s function and optimal decay rate

In this paper, we consider the two-fluid incompressible Navier–Stokes–Maxwell system in three dimensional space. The analysis shows that the effect of the Lorentz force induced by the electromagnetic field leads to some different structures of the spectrum. Moreover, the detailed analysis of the Green’s function to the linearized system is made with applications to derive the optimal time decay rate of the solution converging to the steady state.

Optimal Decay Rates for n-dimensional Generalized Magnetohydrodynamic Equations

Optimal Decay Rates for n-dimensional Generalized Magnetohydrodynamic Equations

Optimal well-posedness for the pressureless Euler–Navier–Stokes system

In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.

Large time behavior of solutions to a diffusion approximation radiation hydrodynamics model

This paper concerns the large time behavior of solutions to a diffusion approximation radiation hydrodynamics model when the initial data is a small perturbation around an equilibrium state. The global-in-time well-posedness of solutions is achieved in Sobolev spaces depending on the Littlewood-Paley decomposition technique together with certain elaborate energy estimates in frequency space. Moreover, the optimal decay rates of solutions are also derived provided the initial data satisfy an additional L1 condition. Meanwhile, the same decay rates of the solutions to the approximation system without thermal conductivity could also be established.

Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in \({\mathbb {R}}^n\): (II) Overdamping Case

This paper is concerned with the large time behavior of the multidimensional compressible Euler equations with time-dependent overdamping of the form in , where , , and . This continues our previous work dealing with the underdamping case for . We show the optimal decay estimates of the solutions such that for and , and , which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of , we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that and for . The overdamping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.

The optimal time decay rates for the compressible Oldroyd-B model in ℝ3

In this paper, we consider the optimal time decay rates of the higher order spatial derivatives of global solution for the three-dimensional compressible Oldroyd-B model, which was derived independently by Barrett-Lu-Süli [Commun. Math. Sci., 15(5): 1265–1323, 2017] via micro–macro analysis of the compressible Navier–Stokes–Fokker–Planck system. We show that under H 3 framework, the third-order spatial derivatives of global small solution of the compressible Oldroyd-B model decay at the rate ( 1 + t ) − 9 4 instead of ( 1 + t ) − 7 4 stated by Wang-Wen [Math. Models Methods Appl. Sci., 30(1): 139–179, 2020]. This decay rate is optimal in the sense that it coincides with that of solutions to heat equation.

A transfer learning approach for automatic mapping of retrogressive thaw slumps (RTSs) in the western Canadian Arctic

ABSTRACT Retrogressive thaw slumps (RTSs) are thermokarst landforms that develop on slopes in permafrost regions when thawing permafrost causes the land surface to collapse. RTSs are an indicator of climate change and pose a threat to human infrastructure and ecosystems in the affected areas. As the availability of ready-to-use high-resolution satellite imagery increases, automatic RTS mapping is being explored with deep learning methods. We employed a pre-trained Mask-RCNN model to automatically map RTSs on Banks Island and Victoria Island in the western Canadian Arctic, where there is extensive RTS activity. We tested the model with different settings, including image band combinations, backbones, and backbone trainable layers, and performed hyper-parameter tuning and determined the optimal learning rate, momentum, and decay rate for each of the model settings. Our final model successfully mapped most of the RTSs in our test sites, with F1 scores ranging from 0.61 to 0.79. Our study demonstrates that transfer learning from a pre-trained Mask-RCNN model is an effective deep learning model that has the potential to be applied for RTS mapping across the Canadian Arctic.

Global existence and large time behaviour for the pressureless Euler–Naver–Stokes system in ℝ3

We investigate the global Cauchy problem for a two–phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations through a drag forcing term. This model was first derived by Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] by taking the hydrodynamic limit of the Vlasov/compressible Navier–Stokes equations. Under the assumption that the initial perturbation is sufficiently small, Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] established the global well–posedness and large time behaviour for the three dimensional periodic domain $\mathbb {T}^3$. However, up to now, the global well–posedness and large time behaviour for the three dimensional Cauchy problem still remain unsolved. In this paper, we resolve this problem by proving the global existence and optimal decay rates of classic solutions for the three dimensional Cauchy problem when the initial data is near its equilibrium. One of key observations here is that to overcome the difficulties arising from the absence of pressure in the Euler equations, we make full use of the drag forcing term and the dissipative structure of the Navier–Stokes equations to closure the energy estimates of the variables for the pressureless Euler equations.

Optimal decay rates for semi-linear non-autonomous evolution equations with vanishing damping

We consider a second order non-autonomous evolution equation in a Hilbert space Hu′′(t)+Au(t)+γ(t)u′(t)+∇F(u(t))=0,where A is a self-adjoint and nonnegative operator on H with a nontrivial kernel, the nonlinear term ∇F is the gradient of a differentiable function F:D(A1/2)→R, and γ(t) is a damping coefficient, vanishing as t goes to infinity. We focus on a limit case γ(t)=α/t (with some α>0), which turns out to be quite tricky to deal with. When α exceeds a critical value, we obtain the optimal decay rate of the solution of the equation, in terms of the exponent associated with F. Moreover, we present two examples to illustrate our theoretical results.

Strong stabilisation and decay estimate for distributed semilinear systems with time-varying delay

In this paper we deal with the problem of strong stabilisation for a class of distributed semilinear systems by means of a nonlinear feedback control. The systems under consideration evolve in a Hilbert space and present periodic time-varying state delay. We first give a proof for the existence and uniqueness of the solution for the considered systems. Then, under a null controllability condition, we establish the stabilisation result and provide an explicit optimal decay rate estimate. The particular case of such systems for which the linear part generates a differentiable semigroup is also investigated. Some illustrating applications to hyperbolic and parabolic equations are displayed. Finally, a conclusion and some perspectives are given.

Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects

Abstract We are concerned with the stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.

Global well-posedness and optimal decay rates for a transmission problem of viscoelastic wave equations with degenerate nonlocal damping

Global well-posedness and optimal decay rates for a transmission problem of viscoelastic wave equations with degenerate nonlocal damping

Optimal decay rates of a nonconservative compressible two‐phase fluid model

AbstractWe are concerned with the time decay rates of strong solutions to a nonconservative compressible viscous two‐phase fluid model in the whole space . Compared to the previous related works, the main novelty of this paper lies in the fact that it provides a general framework that can be used to extract the optimal decay rates of the solution as well as its all‐order spatial derivatives from one‐order to the highest‐order, which are the same as those of the heat equation. Furthermore, for well‐chosen initial data, we also show the lower bounds on the decay rates. Our methods mainly consist of Hodge decomposition, low‐ and high‐frequency decomposition, delicate spectral analysis, and energy method based on finite induction.

On optimal convergence rates of Laguerre polynomial expansions for piecewise functions

By applying Hilb type formula and generalized van der Corput type Lemmas, this paper presents optimal decay rates on the expansion coefficients for given functions expanded in the form of Laguerre polynomial series, from which it leads to the convergence rates on the spectral orthogonal projections, and derives new estimates on the errors for Gauss–Laguerre quadrature. Ample numerical experiments are carried out to validate the optimality and accuracy of these theoretical results.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Mittag-Leffler stability of numerical solutions to linear homogeneous time fractional parabolic equations

<abstract><p>In a bounded domain, the solution of linear homogeneous time fractional parabolic equation is known to exhibit polynomial type decay rate (the so-called Mittag-Leffler stability) over time, which is quite different from the exponential decay of classical parabolic equation. We firstly use the finite element method or finite difference method to discretize the parabolic equation in space to obtain fractional ordinary differential equation, and then use fractional linear multistep method (F-LMM) to discretize in time to obtain a fully discretized schemes. We prove that the strongly $ A $-stable F-LMM method combined with appropriate spatial discretization can accurately maintain the long-term optimal algebraic decay rate of the original continuous equation. Numerical examples are included to confirm the correctness of our theoretical analysis.</p></abstract>

Global Small Solutions to a Special $$2\frac{1}{2}$$-D Compressible Viscous Non-resistive MHD System

This paper solves the global well-posedness and stability problem on a special \(2\frac{1}{2}\)-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in \(L^p\)-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution’s optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.