Optimal Decay Rates Research Articles

Stability analysis of a Timoshenko beam with local degenerate viscoelastic damping

We consider the Timoshenko beam with localized viscoelastic damping of Kelvin-Voigt type. The constitutive law has singularity at the interface of the damped and undamped region. More precisely, the damping coefficients are continuous, but their derivatives have singularities at the interface. We show that the system satisfies a sharper polynomial decay rate, which is consistent with the polynomial decay rate when the constitutive law is discontinuous and exponential stability when the law is of C1-type. This is a big step toward the goal of obtaining eventually the optimal decay rate.

Sharp decay estimates for 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. By using the energy methods, we obtain that this system possesses a global solution when the initial data is small in H3(R3). In addition, the optimal decay rates of this global solution and its first-order derivatives are established by bounding two time-weighted norms and applying a bootstrapping argument.

Optimal time-decay estimates for a diffusive Oldroyd-B model

In this paper, we study the optimal time decay rates for the higher order spatial derivatives of solutions to a diffusive Oldroyd-B model. As pointed out in the Section 1.2 of Huang-Wang-Wen-Zi (J. Differential Equations 306: 456-491, 2022), how to estiblish the optimal decay estimate for the highest-order spatial derivatives of the solution to this model is still an open problem. Motivated by Wang-Wen (Sci. China Math. 65: 1199-1228, 2022), we give a positive answer to this problem via some delicate analyses on the low and high frequency parts of the solution.

Optimal decay of critical derivative for a two-dimensional chemotaxis-shallow water system

In this paper, we focus on the optimal decay rate of critical spatial derivative of global solution to the two-dimensional chemotaxis-shallow water system. The main result improves the N order spatial derivative of the global small solution tends to zero at the rate (1+t)−(N−1+s) stated by Tao and Yao (2018) to the optimal decay rate (1+t)−(N+s).

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

This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent damping of the form \(-\frac{\mu }{(1+t)^\lambda }\rho {\varvec{u}}\) in \({\mathbb {R}}^n\), where \(n\ge 2\), \(\mu >0\), and \(\lambda \in [0,1)\). When \(\lambda >0\) is bigger, the damping effect time-asymptotically gets weaker, which is called under-damping. We show the optimal decay estimates of the solutions such that \(\Vert \partial _x^\alpha (\rho -1)\Vert _{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+\lambda }{2}\left( \frac{n}{2}+|\alpha |\right) }\), and \(\Vert \partial _x^\alpha {\varvec{u}}\Vert _{L^2({\mathbb {R}}^n)}\approx (1+t)^{-\frac{1+\lambda }{2}\left( \frac{n}{2}+|\alpha |\right) -\frac{1-\lambda }{2}}\), and see how the under-damping effect influences the structure of the Euler system. Different from the traditional view that the stronger damping usually makes the solutions decaying faster, here we recognize that the weaker damping with \(0\le \lambda <1\) enhances the faster decay for the solutions. The adopted approach is the technical Fourier analysis and the Green function method. The main difficulties caused by the time-dependent damping lie in twofold: non-commutativity of the Fourier transform of the linearized operator precludes explicit expression of the fundamental solution; time-dependent evolution implies that the Green matrix G(t, s) is not translation invariant, i.e., \(G(t,s)\ne G(t-s,0)\). We formulate the exact decay behavior of the Green matrices G(t, s) with respect to t and s for both linear wave equations and linear hyperbolic system, and finally derive the optimal decay rates for the nonlinear Euler system.

Sharp Exponential Decay Rates for Anisotropically Damped Waves

In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the damping over geodesics, in analogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic dampings which satisfy our hypotheses on the flat torus.

Long time decay of solutions to the 3D incompressible Navier–Stokes equations with nonlinear damping

This paper is interested in the decay of solutions to 3D Navier–Stokes equations with a nonlinear damping term . We discuss the bounds of the global strong solutions in the negative Sobolev space and establish the optimal decay rates in of the global strong solutions. Moreover, the upper bound of the derivative is also obtained. Finally, we investigate the asymptotic stability of the solutions to the system.

Global existence and optimal decay rate of the classical solution to 3-D radiative hydrodynamics model with or without heat conductivity

The classical solution of the 3-D radiative hydrodynamics model is studied in Hk-norm under two different conditions, with and without heat conductivity. We have proved the following results in both cases. First, when the Hk norm of the initial perturbation around a constant state is sufficiently small and the integer k≥2, a unique classical solution to such Cauchy problem is shown to exist. Second, if we further assume that the L1 norm of the initial perturbation is small too, the i-order (0≤i≤k−2) derivatives of the solutions have the decay rate of (1+t)−34−i2 in H2 norm. Third, from the results above we can see that for radiative hydrodynamics, the radiation can do the same job as that the heat conduction does, which means if the thermal conductivity coefficient turns to 0, because of the effect of radiation, the solvability of the system and decay rate of the solutions stay the same.

Global existence and decay estimates of solutions to the MHD–Boussinesq system with stratification effects* *This work was supported by National Natural Science Foundation of China, NSFC (No.: 11926316, 11531010, 12071391).

In this paper, we consider the inviscid Boussinesq system under the presence of magnetic field in porous media. We firstly proved the global well-posedness and large time behaviour of solutions in the whole space . Precisely, when the H 3 norm of initial data is small, but the higher order derivatives can be arbitrary large, the system is globally well-posed by pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the L p − L 2 (1 ⩽ p ⩽ 2) type of the optimal time decay rates are obtained without any smallness assumption on the L p norm of the initial data. At last, we derive a global weak solution with large initial data and give an explicit decay rate of the solution in . Our results mathematically explain the stable phenomenon of the system with stratification effects.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Optimal decay of compressible Navier-Stokes equations with or without potential force

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, we show that the N-th order spatial derivative of global small solution of the CNS equations without potential force tends to zero at the rate (1+t)−(s+N) instead of (1+t)−(s+N−1) stated in [13]. Secondly, we study optimal decay rate for global solution of the CNS equations with potential force. The tricky problem comes from the equilibrium state that will depend on spatial variable rather than constant. Based on energy estimate, spectral analysis, and high-low frequency decomposition, we establish the upper and lower bounds of decay rates for the solution itself and its spatial derivatives. These decay rates are really optimal since the upper bounds of decay rates coincide with the lower ones.

Linear Stability of the 2D Irrotational Circulation Flow around an Elliptical Cylinder

In this article we prove a linear inviscid damping result with optimal decay rates of the 2D irrotational circulation flow around an elliptical cylinder. In our result, all components of the asymptotic velocity field do not vanish and the asymptotic flow lines are not ellipse any more.

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

We investigate time-decay rates of strong solutions to the 3D compressible magnetohydrodynamics equations with large initial data. The main novelty of this paper is two-fold: first, we prove the upper optimal decay rates of the higher-order spatial derivatives of the solution, which are the same as those of the heat equation, and faster than the decay rates in the previous related works. Second, if the initial data satisfy some additional low frequency assumption, we also show the lower optimal decay rates of the solution as well as its all-order spatial derivatives. Therefore, our decay rates are optimal in this sense. Our methods mainly involve the Fourier splitting method, low-frequency and high-frequency decomposition and delicate energy estimates.

Optimal decay for a wave-heat system with Coleman–Gurtin thermal law

We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman–Gurtin thermal law. Our approach is based on the asymptotic theory of C0-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman–Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate t−2 as t→∞. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.

Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force

Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force

Decay properties for a density-dependent liquid crystal system

The purpose of this paper is to establish the optimal temporal decay properties for nonhomogeneous incompressible liquid crystal flows in Rn, n = 2, 3. Under a weaker smallness assumption on the initial datum d0, we construct a uniform (in t) energy estimate for the director field d, which plays a crucial role in giving the optimal L2-decay rates for the liquid crystal flows. If the initial data are more regular, the optimal decay rates for the higher spatial derivatives are also obtained.

Optimal decay rate of solutions to the two‐phase flow model

This article is devoted to the study of the global existence and large time behavior of the three‐dimensional two‐phase flow model derived from the Chapman‐Enskog expansion of the Navier‐Stokes‐Vlasov‐Fokker‐Planck equations around the local Maxwellian. When the initial data are a small perturbation of the equilibrium state in , we prove that the strong solution converges to the equilibrium state at an optimal algebraic rate in ‐norm. It is observed that due to the dispersion effect of the drag force term, the difference of velocities decays at a faster rate in ‐norm.

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.

Optimal decay rate to 3D reduced gravity two-and-a-half-layer model

Optimal decay rate to 3D reduced gravity two-and-a-half-layer model

Sharp stability of a string with local degenerate Kelvin–Voigt damping

AbstractThis paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping where on , and on for . It is known that the optimal decay rate of solution is in the limit case and exponential for . When , the damping coefficient is continuous, but its derivative has a singularity at the interface . In this case, the best known decay rate is , which fails to match the optimal one at . In this paper, we obtain a sharper polynomial decay rate . More significantly, it is consistent with the optimal polynomial decay rate at and uniform boundedness of the resolvent operator on the imaginary axis at (consequently, the exponential decay rate at as ). This is a big step toward the goal of obtaining eventually the optimal decay rate.