Sharp Decay Research Articles

Energy decay rate for a quasi-linear wave equation with localized strong dissipation

The purpose of this paper is to derive a sharp energy decay estimate for a quasi-linear wave equation with localized strong dissipation of the type − ∇ ⋅ ( a ( x ) ∇ u t ) in a domain Ω of R N , where a ( x ) is a nonnegative function supported only on a part of the boundary ∂ Ω . We note that the index of algebraic decay depends on dimension N and no geometrical condition is imposed on the boundary ∂ Ω .

Formula omitted]-decay rates for perturbations of degenerate scalar viscous shock waves

This contribution shows how to obtain L p -decay rates, with 2 ⩽ p ⩽ + ∞ , for zero-mass perturbations of degenerate scalar viscous shock waves using energy methods. The proof is based upon previous work by Matsumura and Nishihara (Comm. Math. Phys. 165 (1) (1994) 83–96), by extending their weighted energy estimates to L p -norms, and by obtaining sharp decay rates for the antiderivative of the perturbation with the aid of basic interpolation inequalities. The analysis applies to shocks of all orders of degeneracy and it is elementary.

On sharp decay estimates of solutions for mildly degenerate dissipative wave equations of Kirchhoff type

We consider the initial data boundary value problem for the degenerate dissipative wave equations of Kirchhoff type ρu′′ + ∥A1/2u∥2γAu+ u′ = 0. When either the coefficient ρ or the initial data are appropriately small at least, we show the global existence theorem by using suitable identities together with the energy. Moreover, under the same assumption for ρ and the initial data, we derive the sharp decay estimates of the solutions and their second derivatives. Copyright © 2011 John Wiley & Sons, Ltd.

Diffusion phenomenon in Hilbert spaces and applications

We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.

Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system

We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite-dimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (Alabau-Boussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell's principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems.

Global analysis of cloud field coverage and radiative properties, using morphological methods and MODIS observations

Abstract. The recently recognized continuous transition zone between detectable clouds and cloud-free atmosphere ("the twilight zone") is affected by undetectable clouds and humidified aerosol. In this study, we suggest to distinguish cloud fields (including the detectable clouds and the surrounding twilight zone) from cloud-free areas, which are not affected by clouds. For this classification, a robust and simple-to-implement cloud field masking algorithm which uses only the spatial distribution of clouds, is presented in detail. A global analysis, estimating Earth's cloud field coverage (50° S–50° N) for 28 July 2008, using the Moderate Resolution Imaging Spectroradiometer (MODIS) data, finds that while the declared cloud fraction is 51%, the global cloud field coverage reaches 88%. The results reveal the low likelihood for finding a cloud-free pixel and suggest that this likelihood may decrease as the pixel size becomes larger. A global latitudinal analysis of cloud fields finds that unlike oceans, which are more uniformly covered by cloud fields, land areas located under the subsidence zones of the Hadley cell (the desert belts), contain proper areas for investigating cloud-free atmosphere as there is 40–80% probability to detect clear sky over them. Usually these golden-pixels, with higher likelihood to be free of clouds, are over deserts. Independent global statistical analysis, using MODIS aerosol and cloud products, reveals a sharp exponential decay of the global mean aerosol optical depth (AOD) as a function of the distance from the nearest detectable cloud, both above ocean and land. Similar statistical analysis finds an exponential growth of mean aerosol fine-mode fraction (FMF) over oceans when the distance from the nearest cloud increases. A 30 km scale break clearly appears in several analyses here, suggesting this is a typical natural scale of cloud fields. This work shows different microphysical and optical properties of cloud fields, urging to separately investigate cloud fields and cloud-free atmosphere in future climate research.

Rate of Convergence for Vanishing Viscosity Approximations to Hyperbolic Balance Laws

The rate of convergence for vanishing viscosity approximations to hyperbolic balance laws is established. The result applies to systems that are strictly hyperbolic and genuinely nonlinear with a source term satisfying a special mechanism that induces dissipation. The proof relies on error estimates that measure the interaction of waves. Shock waves are treated by monitoring the evolution of suitable Lyapunov functionals, whereas interactions involving rarefaction waves are accommodated by employing a sharp decay estimate.

Energy decay estimates for the dissipative wave equation with space-time dependent potential

We study the asymptotic behavior of solutions of dissipative wave equations with space–time-dependent potential. When the potential is only time-dependent, Fourier analysis is a useful tool to derive sharp decay estimates for solutions. When the potential is only space-dependent, a powerful technique has been developed by Todorova and Yordanov to capture the exact decay of solutions. The presence of a space–time-dependent potential, as in our case, requires modifications of this technique. We find the energy decay and decay of the L2 norm of solutions in the case of space–time-dependent potential. Copyright © 2010 John Wiley & Sons, Ltd.

Dispersive stability of infinite-dimensional Hamiltonian systems on lattices

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein–Gordon chain. If the nonlinearity is of degree higher than four, then small localized initial data decay exactly as in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different ℓ p spaces.

A stable boundary layer perspective on global temperature trends

One of the most significant signals in the thermometer-observed temperature record since 1900 is the decrease in the diurnal temperature range over land, largely due to warming of the minimum temperatures. While some data sets have indicated this asymmetrical warming has been reduced since 1979, regional analyses (e.g. East Africa) indicate that the nocturnal warming continues at a pace greater than daytime temperatures. The cause for this night time warming in the observed temperatures has been attributed to a variety of causes. Climate models have in general not replicated the change in diurnal temperature range well. Here we would like to try to distinguish between warming in the nocturnal boundary layer due to a redistribution of heat and warming due to the accumulation of heat. The temperature at night at shelter height is a result of competition between thermal stability and mechanical shear. If stability wins then turbulence is suppressed and the cooling surface becomes cut-off from the warmer air aloft, which leads to sharp decay in surface air temperature. If shear wins, then turbulence is maintained and warmer air from aloft is continually mixed to the surface, which leads to significantly lower cooling rates and warmer temperatures. This warming occurs due to a redistribution of heat. As will be shown by techniques of nonlinear analysis the winner of the stability and shear contest can be very sensitive to changes in greenhouse gas forcing, surface roughness, cloudiness, and surface heat capacity (including soil moisture). Further, the minimum temperatures measured in the nocturnal boundary layer represent only a very shallow layer of the atmosphere which is usually only a few hundred meters thick. It is likely that the observed warming in minimum temperature, whether caused by additional greenhouse forcing or land use changes or other land surface dynamics, is reflecting a redistribution of heat by turbulence-not an accumulation of heat. Because minimum temperatures in the stable boundary layer are not very robust measures of the heat content in the deep atmosphere and climate models do not predict minimum temperatures well, minimum temperatures should not be used as a surrogate for measures of deep atmosphere global warming.

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

In this paper we establish pointwise decay estimates of solutions to some linear elliptic equations by using the Nash–Moser iteration arguments and the ODE method. As applications we obtain sharp Gaussian decay estimates for solutions to nonlinear elliptic equations that are related with self-similar solutions to nonlinear heat equations and standing wave solutions to nonlinear Schrodinger equations with harmonic potential.

ESR response of powder samples of clear fused quartz material to high doses of 10 MeV electrons: possible applications for industrial radiation processing

Abstract Powder samples of clear fused quartz were irradiated by 10 MeV electrons at doses between 6–15 kGy and 25–37 kGy and analyzed with electron spin resonance. The reproducibility of the material dose response curve was studied, in which the irradiated samples were annealed and reused for further investigations. Results indicate a good linearity between the absorbed dose and the ESR intensity of the signals. They also showed that in the reuse processes, the electron spin resonance signal intensity, in both dose ranges was decreased. The thermal fading represented a relative sharp decay at room temperature through 15 days, after which the decay-rate was negligible for a measurement period of 60 days.

Influence of low pressure on electrets behaviour at various air gaps

Polypropylene samples, 20 μm thick, charged in a negative corona were studied. Each sample was placed on a metal pad. The samples were charged to different initial surface potentials, by means of a conventional corona triode system. The negative corona voltage was 5 kV and the grid voltages of the same polarity were 350 V, 500 V, 650 V, 800 V and 950 V. After measuring the surface potential, the samples, together with their metal pads, were placed for 30 min between two short circuited plate electrodes in a vacuum chamber under various low pressures from 10 Pa to 1 × 10 5 Pa. The air gap thicknesses between the charged surface of electrets and the upper electrode were 0.28 mm, 0.84 mm, 1.69 mm and 3.00 mm, respectively. The results obtained show that there is a pressure range within which the surface potential sharply decays. The sharp surface potential decay depends on the pressure-to-initial surface potential ratio and it is different for different air gap thicknesses and for different initial surface potentials. A spark breakdown in the air gap according to the Paschen’s law and processes of desorption from the electret surface are assumed to occur.

Nonlinear exact controllability and nonlinear stabilization of hyperbolic or hyperbolic-like evolution equations

The present manuscript provides a comprehensive and detailed version of the content of the lectures delivered by the author at the Winter School on Nonlinear Analysis, Instituto de Matematica/UFRJ, August 3-7, 2009, Rio de Janeiro, Brazil. They encompassed two distinct yet related topics: (1) nonlinear global exact controllability, in the space of finite energy, for a second-order hyperbolic equation with Neumannboundary control; and(2) sharp uniform decay rates, at the L2(Ω)-level, of the Schrodinger equation with nonlinear boundary dissipation. ∗Research partially supported by the National Science Foundation under grant NSF-DMS01043-5 and by the Army Research Office under grant ARO-DAAD19-02-1-0179. Mathematics Subject Classification. Primary: 35L05, 35L20. Secondary: 93.

Unusual 1,6-diphenyl-1,3,5-hexatriene (DPH) spectrophotometric behavior in water/ethanol and water/DMSO mixtures

The absorption spectra of DPH at fixed concentration do not change with water content in organic solvents. It exhibits monomer bands, such as those obtained in ethanol. The absorption did not change for solutions up to 54 and 46% of water in ethanol and DMSO, respectively, for [DPH] = 5.0 × 10-6 mol L-1 at 30 °C. However, at the same experimental conditions, a gradual sharp decay of the DPH fluorescence is observed. It is proposed that water molecules below these water concentration limits act as quenchers of the excited states of DPH. Stern-Volmer quenching constants by intensities measurements are 7.4 × 10-2 (water/ethanol) and 2.6 × 10-2 L mol-1 (water/DMSO). DPH lifetime measurements in the absence and presence of water resulted in 7.1 × 10-2 L mol-1 in water/ethanol, which pointed out that the process is a dynamic quenching by water molecules. For experiments using DPH as probe, this process can affect data, leading to misunderstanding interpretation.

Study of the thermoluminescence dosimetric properties of window glass

This paper presents the main thermoluminescence (TL) dosimetric characteristics of commercial Turkish transparent window glass. The structure of the glow curves, including the number of peaks, was found to be dose-dependent. A low-temperature glow peak that at 160 °C shifts to higher temperatures was also observed with increasing storage time at room temperature. This result suggests that this TL glow peak is actually made up of two or more overlapping peaks. These we have attributed to the glow peaks at lower temperatures, which decay faster than the ones at higher temperatures with storage time. The thermal fading of the window glass sample at room temperature showed a relatively sharp decay of about 60% occurring over a period of 28 days, after which the decay rate is small for a measured period of 250 days. In order to the improve the post-irradiation stability of the glow curve, the glass samples were heated after irradiation. To remove the unstable TL peaks responsible for the initial rapid fading, post-irradiation heating at 160 °C for 10 min was found to be the most suitable procedure. The dosimetric characteristics of the post-irradiation heated window glass examined in this study include fading, gamma photon dose-response, reproducibility, batch sensitivity, humidity influence, a dose-rate effect and photon energy response. Dose-response was found to be appropriate for dosimetry in the range 5 Gy to 10 kGy. The post-irradiation heating procedure did not affect the main dosimetric characteristics of the window glass samples. The results in this work suggest that the materials could, by using the TL technique, be a suitable candidate for alternative dose measurements in radiation processing, provided that a judicious choice of the post-irradiation heat temperature is made to minimize fading.

A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials

The aim of this paper is to establish a sharp decay estimate forradially symmetric solutions of the following type of nonlinearSchrödinger equations:$-\Delta u + V(|x|)u =Q(|x|)|u|^{p-2}u, x\in R^N, $ $u(x)\rightarrow 0$ as $|x|\rightarrow+\infty,$ where $N\geq 3$, $p\in(2,+\infty)$, $V(x)$ and $Q(x)$ arecontinuous functions which vanishes at infinity and may changesign. As a special case, our result shows that the solutionsobtained by Su-Wang-Willem in [11, Theorem 3] must decayprecisely like $|x|^{-(N-2)}$ as $|x|\rightarrow+\infty$ if$V(|x|)$ decays faster than $|x|^{-2}$ at infinity.

Decay estimate for a new measure of rarefaction waves in general systems of conservation laws

The sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations, cf. Bressan and Yang (2004) [10]. Such an estimate is well established for genuinely nonlinear system of conservation laws, cf. Bressan and Yang (2004) [9]. But similar result is not available for non-genuinely nonlinear system. In this paper, we give a new measure about the rarefaction waves. In addition, a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws.

Properties of positive harmonic functions on the half-space with a nonlinear boundary condition

In this paper we study existence and properties of solutions of the problem Δ w = 0 on the half-space R + N with nonlinear boundary condition ∂ w / ∂ η + w = | w | p − 2 w where 2 < p < 2 ( N − 1 ) / ( N − 2 ) and N ⩾ 3 . We obtain a ground state solution w = w ( x 1 , … , x N − 1 , t ) which is radial and has exponential decay in the first N − 1 variables. Moreover, w has sharp polynomial decay in the variable t.

Estimates for oscillatory strongly singular integral operators

In this paper we consider oscillatory integral operators with strong singularities on R . We obtain sharp decay estimates for L 2 operator norm. We also obtain H p estimates for difference of oscillatory strongly singular integral operators and strongly singular integral operators, which gives L p mapping properties of oscillatory strongly singular integral operators.