Optimal Decay Estimates Research Articles

Global existence and optimal time-decay estimates of solutions to the generalized double dispersion equation on the framework of Besov spaces

We investigate the initial value problem for the generalized double dispersion equation in any dimensions. Inspired by [28] for the hyperbolic system of first order PDEs, we develop Littlewood-Paley pointwise energy estimates for the dissipative wave equation of high-order. Furthermore, with aid of the frequency-localization Duhamel principle, we establish the global existence and optimal decay estimates of solutions in spatially critical Besov spaces. Our results could hold true for any dimensions (n≥1). Indeed, the proofs are different in case of high dimensions and low dimensions owing to interpolation tricks.

Large time behavior for two-dimensional magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion

This paper is devoted to the optimal decay estimates of solution for two dimensional (2D) magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion. Using the properties of heat operator, the optimal L∞-decay estimate for magnetic field b is obtained. Furthermore, using this estimate, we establish the optimal L2-decay estimates for velocity field u and micro-rotational field w, which significantly improve the previous decay results in this direction.

Decay Result for a Delay Viscoelastic Plate Equation

In this work, we consider a fourth-order plate equation that represents the downward displacement of suspension bridge in the presence of viscoelastic and delay damping. We establish an optimal and general decay estimates with partially hinged boundary conditions under suitable assumptions on the relaxation function. This result improves and generalizes earlier results in the literature such as Kirane and Said-Houari (Z Angew Math Phys 62:1065–1082, 2011) and Zhifeng (Z Angew Math Phys 66:727–745, 2015).

Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory

We are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation, which describes the dynamics of many viscoelastic systems and covers the well-known Timoshenko system. We present an optimal energy decay estimate under very weak conditions. More specifically, we prove that the energy of the system decays to zero at least at the rate of t−1 only under the condition that the related kernel functions are non-increasing and integrable. Moreover, we apply our abstract results to three classes of coupled systems of second-order partial differential equations arising in viscoelasticity.

Global existence and optimal decay estimates of the compressible viscoelastic flows in <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> critical spaces

This paper is dedicated to the global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows in the whole space $\mathbb{R}^n$ with any $n\geq2$. We aim at extending those works by Qian \& Zhang and Hu \& Wang to the critical $L^p$ Besov space, which is not related to the usual energy space. With aid of intrinsic properties of viscoelastic fluids as in \cite{QZ1}, we consider a more complicated hyperbolic-parabolic system than usual Navier-Stokes equations. We define "\emph{two effective velocities}", which allows us to cancel out the coupling among the density, the velocity and the deformation tensor. Consequently, the global existence of strong solutions is constructed by using elementary energy approaches only. Besides, the optimal time-decay estimates of strong solutions will be shown in the general $L^p$ critical framework, which improves those decay results due to Hu \& Wu such that initial velocity could be \textit{large highly oscillating}.

Optimal decay estimate of mild solutions to the compressible Navier‐Stokes‐Korteweg system in the critical Besov space

In this paper, we consider the initial value problem for the compressible Navier‐Stokes‐Korteweg system in several space variables. Optimal decay estimate of mild solutions in the critical Besov spaces is established. The proof is based on the decay estimate of solutions operator in the low‐frequency region and the energy estimate in the high‐frequency region.

Optimal decay estimates in the critical $$L^{p}$$ L p framework for flows of compressible viscous and heat-conductive gases

The global existence issue in critical regularity spaces for the full Navier-Stokes equations satisfied by compressible viscous and heat-conductive gases has been first addressed in \cite{D2}, then recently extended to the general $L^p$ framework in \cite{DH}. In the present work, we establish decay estimates for the global solutions constructed in \cite{DH}, under an additional mild integrability assumption that is satisfied if the low frequencies of the initial data are in $L^{p/2}({\mathbb{R}}^d).$ As a by-product we recover in dimension three the classical decay rate $t^{-\frac34}$ for $t\to+\infty$ that has been observed by A. Matsumura and T. Nishida in \cite{MN2} for solutions with high Sobolev regularity. Compared to a recent paper of us \cite{DX} dedicated to the barotropic case, not only we are able to treat the full system, but we also improve the decay rates for the high frequencies of the solution. We believe the correspondant decay exponents to be optimal.

Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

Dissipative structure and global existence in critical space for Timoshenko system of memory type

In this paper, we consider the initial value problem for the Timoshenko system with a memory term in one dimensional whole space. In the first place, we consider the linearized system: applying the energy method in the Fourier space, we derive the pointwise estimate of the solution in the Fourier space, which first gives the optimal decay estimate of the solution. Next, we give a characterization of the dissipative structure of the system by using the spectral analysis, which confirms our pointwise estimate is optimal. In the second place, we consider the nonlinear system: we show that the global-in-time existence and uniqueness result could be proved in the minimal regularity assumption in the critical Sobolev space H2. In the proof we don't need any time-weighted norm as recent works; we use just an energy method, which is improved to overcome the difficulties caused by regularity-loss property of Timoshenko system.

Global well-posedness and polynomial decay for a nonlinear Timoshenko–Cattaneo system under minimal Sobolev regularity

We consider the nonlinear stability of the Timoshenko–Cattaneo system in the one-dimensional whole space. The Timoshenko system consists of two coupled wave equations with non-symmetric relaxation, and describes vibrations of the beam with shear deformation and rotational inertia effect. Generally, if the relaxation is not symmetric, the dissipation is produced through the complicated interaction of the components of the system, and their decay estimates and the energy estimates are of regularity-loss type. In this paper, we introduce the mathematical method to control such a weak dissipativity by investigating the Timoshenko system with Cattaneo’s law, which is the first order approximation of Fourier’s law with its time-delay effect. Racke & Said-Houari (2012), showed the global existence and the decay estimate of solutions by assuming high regularity H8∩L1 on the small initial data to control their weak dissipativity. In contrast, we prove the global existence in H2 by energy methods without any negative weights. Our regularity assumption is the same as that needed to show the local existence. That is, we do not need to assume the extra higher regularity on the initial data. Besides, the optimal decay estimate in H2∩L1 is shown by using the time decay inequality of Lp-Lq-Lr type.

Optimal decay estimates for solutions to damped second order ODE's

Optimal decay estimates for solutions to damped second order ODE's

Optimal decay estimates of a regularity-loss type system with constraint condition

In the paper [14], the authors formulated a new structural condition which includes the Kawashima–Shizuta condition, and analyzed the weak dissipative structure called the regularity-loss type for general systems which contain the Timoshenko system and the Euler–Maxwell system. However, this new structural condition can not cover all of dissipative systems. Indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper. Precisely we first derive the L2 decay estimate of solutions and discuss the type of the corresponding regularity-loss structure. Moreover, in order to show the optimality of the decay estimate, we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity.

A vector field method for relativistic transport equations with applications

We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in $x$ or $v$) for the distribution functions. In the second part of this article, we apply our method to the study of the massive and massless Vlasov-Nordstr\"om systems. In the massive case, we prove global existence and (almost) optimal decay estimates for solutions in dimensions $n \geq 4$ under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions $n \geq 4$ for arbitrarily large data, and in dimension $3$ under some smallness assumptions, exploiting a certain form of the null condition satisfied by the equations. The $3$-dimensional massive case requires an extension of our method and will be treated in future work.

Parabolic models for chemotaxis on weighted networks

In this article we consider the Keller–Segel model for chemotaxis on networks, both in the doubly parabolic case and in the parabolic–elliptic one. Introducing appropriate transition conditions at vertices, we prove the existence of a time global and spatially continuous solution for each of the two systems. The main tool we use in the proof of the existence result are optimal decay estimates for the fundamental solution of the heat equation on a weighted network.

Self‐similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation

In this paper, we obtain optimal decay estimates for the solutions to an evolution equation with critical, structural, dissipation, and absorbing power nonlinearity: urn:x-wiley:mma:media:mma4469:mma4469-math-0001 with μ>0, θ is a positive integer, and p>1+4θ/n, in space dimension n∈(2θ,4θ). We use these estimates to find the self‐similar asymptotic profile of the solutions, when μ≥1.

Energy decay rates for the elasticity system with structural damping in the Fourier space

This paper deals with a Cauchy problem for the elasticity system with a structural damping. The decay rates of the total energy of the system are presented by applying the Fourier transform and multiplier in the Fourier space. Furthermore, almost optimal decay estimates of the total energy are obtained based on the Haraux–Komornik inequality.

A.e. convergence and 2-weight inequalities for Poisson-Laguerre semigroups

We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators L. From these, we solve two problems about the Poisson semigroup \(e^{-t\sqrt{L}}\). First, we find the largest space of initial data f so that \(e^{-t\sqrt{L}}f(x)\rightarrow f(x)\) at \({\,a.e.\,}x\). Secondly, we characterize the largest class of weights w which admit 2-weight inequalities of the form \(\Vert \sup _{0<t\le t_0}|e^{-t\sqrt{L}}f|\,\Vert _{L^p(v)}\lesssim \Vert f\Vert _{L^p(w)}\), for some other weight v.

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

AbstractWe are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L2‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.

Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velazquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordstrom system.

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion‐type source term. Based on a low‐frequency and high‐frequency decomposition, Green's function method and the classical energy method, we not only obtain L2 time‐decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u0(x) satisfies the smallness condition on , but not on . Furthermore, by taking a time‐frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley &amp; Sons, Ltd.