L2 Decay Research Articles

Decay of a model system of radiating gas

This paper concerns the Cauchy problem of a model system to the radiating gas in . Large time behaviors of classical solutions to the Cauchy problem are studied without needing the smallness assumption of initial perturbation in L1-norm. We obtain the optimal HN-norm time-decay rates of the solutions in with 1 ≤ n ≤ 4 by applying the Fourier splitting method introduced by Schonbek (1980) with a slight modification and an energy method. Furthermore, when initial perturbation is bounded in Lp-norm (p ∈ (1,2]), optimal Lp- L2 decay estimates of the derivatives of solutions are shown. Copyright © 2013 John Wiley & Sons, Ltd.

Global existence and asymptotic decay of solutions to the nonlinear Timoshenko system with memory

In this paper we consider the initial-value problem for the nonlinear Timoshenko system with a memory term. Due to the regularity-loss property and weak dissipation, we have to assume stronger nonlinearity than usual. By virtue of the semi-group arguments, we obtain the global existence and optimal decay of solutions to the nonlinear problem under smallness and enough regularity assumptions on the initial data, where we employ a time-weighted L2 energy method combined with the optimal L2 decay of lower-order derivatives of solutions.

Global strong solvability of a quasilinear subdiffusion problem

We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the modeling of dynamic processes in materials with memory. The proof relies heavily on a regularity result on the interior Holder continuity of weak solutions to time fractional diffusion equations, which has been proved recently by the author. We further establish an L2 decay estimate for the special case with vanishing external source term and homogeneous Dirichlet boundary condition.

Global existence and optimal [formula omitted] decay rate for the strong solutions to the compressible fluid models of Korteweg type

In this paper, we consider the compressible Navier–Stokes–Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the global existence of a strong solution to the compressible Navier–Stokes–Korteweg system when the initial perturbation ‖ρ0−ρ¯‖H2+‖u0‖H1 is small. Furthermore, if the L1 norm of the initial perturbation is finite, we can obtain the optimal L2 decay rates.

Decay property of Timoshenko system in thermoelasticity

We investigate the decay property of a Timoshenko system of thermoelasticity in the whole space for both Fourier and Cattaneo laws of heat conduction. We point out that although the paradox of infinite propagation speed inherent in the Fourier law is removed by changing to the Cattaneo law, the latter always leads to a solution with the decay property of the regularity‐loss type. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We derive L2 decay estimates of solutions and observe that for the Fourier law the decay structure of solutions is of the regularity‐loss type if the wave speeds of the first and the second equations in the system are different. For the Cattaneo law, decay property of the regularity‐loss type occurs no matter what the wave speeds are. In addition, by restricting the initial data to with a suitably large s and γ ∈ [0,1], we can derive faster decay estimates with the decay rate improvement by a factor of t−γ/2. Copyright © 2011 John Wiley & Sons, Ltd.

Optimal decay rates for the compressible fluid models of Korteweg type

We consider the compressible Navier–Stokes–Korteweg system that models the motions of the compressible isothermal viscous capillary fluids. We prove the optimal L2 and Lp, p⩾2 decay rates for the classical solutions and their spatial derivatives. In particular, the optimal L2 decay rate of the second-order spatial derivatives is obtained. The proof is based on the detailed study of the linear decay estimates and nonlinear energy estimates.

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper, upper bounds of the L2-decay rate for the Boussinesq equations are considered. Using the L2 decay rate of solutions for the heat equation, and assuming that the solutions of the Boussinesq equations are smooth, we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data. The decay results may then be obtained by passing to the limit of approximating sequences of solutions. The main tool is the Fourier splitting method.

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L2 decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.

L2 decay of the incompressible Navier-Stokes equations with damping

In this article, we show large time behavior of weak solutions to the Cauchy problem of the Navier-Stokes equations with damping α| u| β-1 u (α > 0).

Central Limit Theorem for Zero-Range Processes

We consider additive functionals INTt-0 V(ns )ds of symmetric zero-range processes, where V is a mean zero local function. In dimensions 1 and 2 we obtain a central limit theorem for a-1(t) INTt-0 V(ns)ds with a(t) = SQRROOT (tlogt) in d =2 and a(t) = t 3/4 in d = 1 and an explicit form for the asymptotic variance SIGMA2. The transient case d greater than or equal to 3 can be handled by standard arguments [KV, SX,S]. We also obtain corresponding invariance principles. This generalizes results obtained by Port (see [CG]) for noninteracting random walks and Kipnis [K] for the symmetric simple exclusion process. Our main tools are the martingale method together with L2 decay estimates [JLQY] for the process semigroup.

L2 decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space

L2 decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space

Scattering Theory for Conformally Compact Metrics with Variable Curvature at Infinity

We develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum.

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi-discrete piecewise optimal control problem and the desired states in L2 and H1 norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.

Algebraic L2 decay for the solution to a class system of non-Newtonian fluid in Rn

In this paper, we investigate the L2 decay rates for the solution to a class system of non-Newtonian fluid in Rn(n⩾2). The decay rates are optimal in the sense that they coincide with the decay rates of the solution to the heat system.

L2Decay for the Compressible Navier-Stokes Equations in Unbounded Domains

(1993). L2 Decay for the Compressible Navier-Stokes Equations in Unbounded Domains. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1445-1476.

Algebraic L2 decay for Navier-Stokes flows in exterior domains

Algebraic L2 decay for Navier-Stokes flows in exterior domains

L2 decay for weak solutions of the Navier-Stokes equations

L2 decay for weak solutions of the Navier-Stokes equations

Lepton-Hadron interactions from supersymmetry radiative corrections

The new particles and interactions implied by supersymmetric theories may induce, via radiative corrections, new effective low energy interactions in the “standard” sector of the theory. We calculate the effects of lepton-hadron “contact” interactions thus induced as they appear in the π ℓ2 decay and in the energy levels of hydrogenic atoms (induced Lamb shift).

New tests for and bounds on neutrino masses and lepton mixing

We propose a new class of correlated tests for neutrino masses and lepton mixing. Two particular tests based on (π, K)ℓ2 decay and nuclear β decay are discussed and applied to present data to derive bounds on these quantities.

SU(3) symmetry breaking in semileptonic decays

The first-order SU(3) symmetry breaking in semi-leptonic decays is studied by use of the techniques of equal time commutators. The correction for the baryon axial-vector decay constants is related to that for the strong derivative coupling constants of pseudoscalar meson-baryon interaction. It is further found that the K ℓ2 and π ℓ2 decay constants remain unrenormalized so that ƒ K = ƒ π .