Virial Estimates Research Articles

Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

In this paper, we consider the dynamics of even solutions of the one-dimensional nonlinear Klein–Gordon equation \partial_t^2 \phi - \partial_x^2 \phi + \phi - |\phi|^{2\alpha} \phi =0 for \alpha>1 , in the vicinity of the unstable soliton \smash{Q} . Our main result is that stability in the energy space H^1(\mathbb{R})\times L^2(\mathbb{R}) implies asymptotics stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition \alpha>1 corresponds to cases where the linearized operator around Q has no resonance and no internal mode. Recall that the case \alpha>2 is treated by Krieger, Nakanishi and Schlag [Math. Z. 272 (2012)] using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed by Chang, Gustafson, Nakanishi and Tsai [SIAM J. Math. Anal. 39 (2007/08)].

Blow-up criteria for coupled nonlinear Schödinger equations

This paper deals with the blw-up problem of a system of two coupled nonlinear focusing Schrödinger equations. By using localized virial estimates, we establish a blow-up criteria for non-radial solutions without the hypothesis of finite variance.

Blow-up criteria and instability of standing waves for the inhomogeneous fractional Schrodinger equation

In this article, we study the blow-up and instability of standing waves for theinhomogeneous fractional Schrodinger equation $$ i\partial_tu-(-\Delta)^su+ |x|^{-b}|u|^{p}u=0, $$ where \(s\in (\frac{1}{2},1)\), \(0<b<\min \{2s,N\}\) and \(0<p< \frac{4s-2b}{N-2s}\). In the \(L^2\)-critical and \(L^2\)-supercritical cases, i.e.,\(\frac{4s-2b}{N}\leq p< \frac{4s-2b}{N-2s}\), we establish general blow-up criteriafor non-radial solutions by using localized virial estimates. Based on theseblow-up criteria, we prove the strong instability of standing waves.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/39/abstr.html

Star Formation Efficiency and Dispersal of Giant Molecular Clouds with UV Radiation Feedback: Dependence on Gravitational Boundedness and Magnetic Fields

Abstract Molecular clouds are supported by turbulence and magnetic fields, but quantifying their influence on cloud life cycle and star formation efficiency (SFE) remains an open question. We perform radiation magnetohydrodynamic simulations of star-forming giant molecular clouds (GMCs) with UV radiation feedback, in which the propagation of UV radiation via ray tracing is coupled to hydrogen photochemistry. We consider 10 GMC models that vary in either initial virial parameter (1 ≤ α vir,0 ≤ 5) or dimensionless mass-to-magnetic flux ratio (0.5 ≤ μ Φ,0 ≤ 8 and ∞ ); the initial mass 105 M ⊙ and radius 20 pc are fixed. Each model is run with five different initial turbulence realizations. In most models, the duration of star formation and the timescale for molecular gas removal (primarily by photoevaporation) are 4–8 Myr. Both the final SFE (ε *) and time-averaged SFE per freefall time (ε ff) are reduced by strong turbulence and magnetic fields. The median ε * ranges between 2.1% and 9.5%. The median ε ff ranges between 1.0% and 8.0%, and anticorrelates with α vir,0, in qualitative agreement with previous analytic theory and simulations. However, the time-dependent α vir(t) and ε ff,obs(t) based on instantaneous gas properties and cluster luminosity are positively correlated due to rapid evolution, making observational validation of star formation theory difficult. Our median ε ff,obs(t) ≈ 2% is similar to observed values. We show that the traditional virial parameter estimates the true gravitational boundedness within a factor of 2 on average, but neglect of magnetic support and velocity anisotropy can sometimes produce large departures from traditional virial parameter estimates. Magnetically subcritical GMCs are unlikely to represent sites of massive star formation given their unrealistic columnar outflows, prolonged lifetime, and low escape fraction of radiation.

Global dynamics below the ground states for NLS under partial harmonic confinement

We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schrodinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.

Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

Abstract In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i ∂ t ψ − ( − Δ ) s ψ + ( I α ∗ | ψ | p ) | ψ | p − 2 ψ = 0. $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L 2-critical and L 2-supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in Hs . Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.

No Significant Evolution of Relations between Black Hole Mass and Galaxy Total Stellar Mass Up to z ∼ 2.5

Abstract We investigate the cosmic evolution of the ratio between black hole (BH) mass (M BH) and host galaxy total stellar mass (M stellar) out to z ∼ 2.5 for a sample of 100 X-ray-selected moderate-luminosity, broad-line active galactic nuclei (AGNs) in the Chandra-COSMOS Legacy Survey. By taking advantage of the deep multiwavelength photometry and spectroscopy in the COSMOS field, we measure in a uniform way the galaxy total stellar mass using an spectral energy distribution decomposition technique and the BH mass based on broad emission line measurements and single-epoch virial estimates. Our sample of AGN host galaxies has total stellar masses of 1010−12 M ⊙, and BH masses of 107.0–9.5 M ⊙. Combining our sample with the relatively bright AGN samples from the literature, we find no significant evolution of the M BH–M stellar relation with the BH-to-host total stellar mass ratio of M BH/M stellar ∼ 0.3% at all redshifts probed. We conclude that the average BH-to-host stellar mass ratio appears to be consistent with the local value within the uncertainties, suggesting a lack of evolution of the M BH–M stellar relation up to z ∼ 2.5.

Decay in the one dimensional generalized Improved Boussinesq equation

We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $$H^1\times H^2$$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho–Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm $$L^2\times H^1$$, for any $$p>1$$, in two almost complementary regimes: (1) outside the light cone for all small, bounded in time $$H^1\times H^2$$ solutions, and (2) decay on compact sets of arbitrarily large bounded in time $$H^1\times H^2$$ solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.

Quasar emission lines as virial luminosity estimators

Quasars accreting matter at very high rates (known as extreme Population A [xA]) may provide a new class of distance indicators covering cosmic epochs from present day up to less than 1 Gyr from the Big Bang. We report on the developments of a method that is based on "virial luminosity" estimates from measurements of emission line widths of xA quasars. The approach is conceptually equivalent to the virial estimates based on early and late type galaxies. The main issues related to the cosmological application of luminosity estimates from xA quasar line widths are the identification of proper emission lines whose broadening is predominantly virial over a wide range of luminosity, and the assessment of the effect of the emitting region orientation with respect to the line of sight. We report on recent developments concerning the use of the AlIII 1860 intermediate ionisation line and of the Hydrogen Balmer line H$\beta$ as "virial broadening estimators."

Anomalously Narrow Line Widths of Compact Massive Star-forming Galaxies at z ∼ 2.3: A Possible Inclination Bias in the Size–Mass Plane

Abstract Compact, massive star-forming galaxies at z ∼ 2.5 are thought to be building the central regions of giant elliptical galaxies today. However, a significant fraction of these objects were previously shown to have much smaller Hα line widths than expected. A possible interpretation is that Hα emission from their central regions, where the highest velocities are expected, is typically obscured by dust. Here we present Atacama Large Millimeter/submillimeter Array observations of the CO(3–2) emission line of three compact, massive galaxies with Hα line widths of FWHM(Hα) ∼ 125–260 km s−1 to test this hypothesis. Surprisingly, in all three galaxies, the CO line width is similar to the Hα line width: we find FWHM(CO) ∼ 165 km s−1 for all three galaxies whereas FWHM(CO) ∼ 450–700 km s−1 was expected from a simple virial estimator. These results show that the narrow Hα line widths of many compact massive star-forming galaxies are not due to preferential obscuration of the highest velocity gas. An alternative explanation for the narrow line widths is that the galaxies are disks that are viewed nearly face-on. We suggest that there may be an inclination bias in the size–mass plane, such that the apparent rest-frame optical sizes of face-on galaxies are smaller than those of edge-on galaxies. Although not conclusive, this hypothesis is supported by an observed anti-correlation between size and axis ratio of massive galaxies.

Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

Consider the Hamiltonian a b c d abcd system in one dimension, with data posed in the energy space H 1 × H 1 H^1\times H^1 . This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where a , c > 0 a,c>0 and b = d > 0 b=d>0 . Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this 2 × 2 2\times 2 system was given in [J. Math. Pure Appl. (9) 127 (2019), 121–159] in a strongly dispersive regime, i.e., under essentially the conditions \[ b = d > 2 9 , a , c > − 1 18 . b=d > \frac 29, \quad a,c>-\frac 1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone ( − | t | , | t | ) (-|t|,|t|) . In this paper, we improve [J. Math. Pure Appl. (9) 127 (2019), 121–159] in three directions. First, we enlarge the set of parameters ( a , b , c , d ) (a,b,c,d) for which decay to zero is the only available option, considering now the so-called weakly dispersive regime a , c ∼ 0 a,c\sim 0 : we prove decay if now \[ b = d > 3 16 , a , c > − 1 48 . b=d > \frac 3{16}, \quad a,c>-\frac 1{48}. \] This result is sharp in the case where a = c a=c , since for a , c a,c bigger, some a b c d abcd linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form | x | ∼ | v | t |x|\sim |v|t for any | v | > 1 |v|>1 . This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small a b c d abcd solutions in exterior regions | x | ≫ | t | |x|\gg |t| , also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators.

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH4-xDx/CH4 and CH5-xDx +/CH5 +, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

X-ray properties of reverberation-mapped AGNs with super-Eddington accreting massive black holes

AbstractActive Galactic Nuclei (AGN) exhibit multi-wavelength properties that are representative of the underlying physical processes taking place in the vicinity of the accreting supermassive black hole. The black hole mass and the accretion rate are fundamental for understanding the growth of black holes, their evolution, and the impact on the host galaxies. Recent results on reverberation-mapped AGNs show that the highest accretion rate objects have systematic shorter time-lags. These super-Eddington accreting massive black holes (SEAMBHs) show BLR size 3-8 times smaller than predicted by the Radius-Luminosity (R-L) relationship. Hence, the single-epoch virial black hole mass estimates of highly accreting AGNs have an overestimation of a factor of 3-8 times. SEAMBHs likely have a slim accretion disk rather than a thin disk that is diagnostic in X-ray. I will present the extreme X-ray properties of a sample of dozen of SEAMBHs. They indeed have a steep hard X-ray photon index, Γ, and demonstrate a steeper power-law slope, ασx.

Are Narrow-line Seyfert 1 Galaxies Powered by Low-mass Black Holes?

Abstract Narrow-line Seyfert 1 galaxies (NLS1s) are believed to be powered by the accretion of matter onto low-mass black holes (BHs) in spiral host galaxies with BH masses M BH ∼ 106–108 M ⊙. However, the broadband spectral energy distribution of the γ-ray-emitting NLS1s are found to be similar to flat-spectrum radio quasars. This challenges our current notion of NLS1s having low M BH. To resolve this tension of low M BH values in NLS1s, we fitted the observed optical spectrum of a sample of radio-loud NLS1s (RL-NLS1s), radio-quiet NLS1s (RQ-NLS1s), and radio-quiet broad-line Seyfert 1 galaxies (RQ-BLS1s) of ∼500 each with the standard Shakura–Sunyaev accretion disk (AD) model. For RL-NLS1s we found a mean log( ) of 7.98 ± 0.54. For RQ-NLS1s and RQ-BLS1s we found mean log( ) of 8.00 ± 0.43 and 7.90 ± 0.57, respectively. While the derived values of RQ-BLS1s are similar to their virial masses, for NLS1s the derived values are about an order of magnitude larger than their virial estimates. Our analysis thus indicates that NLS1s have M BH similar to RQ-BLS1s and their available virial M BH values are underestimated, influenced by their observed relatively small emission line widths. Considering Eddington ratio as an estimation of the accretion rate and using , we found the mean accretion rate of our RQ-NLS1s, RL-NLS1s, and RQ-BLS1s as , and , respectively. Our results therefore suggest that NLS1s have BH masses and accretion rates that are similar to BLS1s.

Atomistic mechanisms of crack nucleation and propagation in amorphous silica

This paper presents an atomistic understanding of the nanoscale processes that govern crack nucleation and propagation in amorphous silica using a combination of theoretical calculations and molecular dynamics simulations with the ReaxFF potential. We show that both crack nucleation and propagation are governed by chainlike nanoscale virial stress-fibers formed by an intricate mixture of Si and O atoms. The stress-fibers are of a few nanometers in length, aligned parallel to the loading direction, and spatially localized at the evolving crack front at the nucleating site or a propagating crack tip. They form and break continuously during the crack nucleation and propagation process and are responsible for localizing stress at the crack nucleation site or propagating crack front. As soon as the stress-fibers reach a critical density the material starts nucleating cracks or leads to the propagation of initial crack. Additionally, the virial stress fields in the domain is highly heterogeneous and species-dependent---and the O and Si atoms play fundamentally distinct roles throughout the deformation process. With increased loading, heterogeneity in virial stress for the Si atoms goes up, whereas for the O atoms it goes down. Furthermore, from the virial and Hardy estimates of atomic stress, it is found that the stress field emanating from the crack tip decays as $1/r$. Also, presence of holes or pores within the interaction distance of a crack tip intensifies the stress state of the stress-fibers near the crack front leading to an improved effective toughness compared to the situation where the pore is far from the crack tip. Nucleation and propagation of cracks are strictly mediated by a complex admixture of localized bond rupture processes across a set of interacting stress-fibers. The details of the atomistic process regulating the underlying mechanisms are undetectable from the macroscopic stress-strain data.

Asymptotic dynamic for Dipolar Quantum Gases below the ground state energy threshold

We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters.

On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

We consider the focusing $L^2$-supercritical fractional nonlinear Schr\"odinger equation \[ i\partial_t u - (-\Delta)^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d\geq 2, \frac{d}{2d-1} \leq s <1$ and $\frac{4s}{d}<\alpha<\frac{4s}{d-2s}$. By means of the localized virial estimate, we prove that the ground state standing wave is strongly unstable by blow-up. This result is a complement to a recent result of Peng-Shi [J. Math. Phys. 59 (2018), 011508] where the stability and instability of standing waves were studied in the $L^2$-subcritical and $L^2$-critical cases.

Blow-up criteria for fractional nonlinear Schrödinger equations

We consider the focusing fractional nonlinear Schrödinger equation i∂tu−(−Δ)su=−|u|αu,(t,x)∈R+×Rd,where s∈(1∕2,1) and α>0. By using localized virial estimates, we establish general blow-up criteria for non-radial solutions to the equation. As consequences, we obtain blow-up criteria in both L2-critical and L2-supercritical cases which extend the results of Boulenger–Himmelsbach–Lenzmann (Boulenger et al., 2016) for non-radial initial data.

Highly accreting quasars: The SDSS low-redshift catalog

Context. The most highly accreting quasars are of special interest in studies of the physics of active galactic nuclei (AGNs) and host galaxy evolution. Quasars accreting at high rates (L/LEdd ∼ 1) hold promise for use as “standard candles”: distance indicators detectable at very high redshift. However, their observational properties are still largely unknown.Aims. We seek to identify a significant number of extreme accretors. A large sample can clarify the main properties of quasars radiating nearL/LEdd ∼ 1 (in this paper they are designated as extreme Population A quasars or simply as extreme accretors) in theHβspectral range for redshift ≲0.8.Methods. We use selection criteria derived from four-dimensional Eigenvector 1 (4DE1) studies to identify and analyze spectra for a sample of 334 candidate sources identified from the SDSS DR7 database. The source spectra were chosen to show a ratioRFeIIbetween the FeII emission blend atλ4570 andHβ,RFeII&gt; 1. Composite spectra were analyzed for systematic trends as a function of Fe IIstrength, line width, and[OIII]strength. We introduced tighter constraints on the signal-to-noise ratio (S/N) andRFeIIvalues that allowed us to isolate sources most likely to be extreme accretors.Results. We provide a database of detailed measurements. Analysis of the data allows us to confirm thatHβshows a Lorentzian function with a full width at half maximum (FWHM) ofHβ≤ 4000 km s−1. We find no evidence for a discontinuity at 2000 km s−1in the 4DE1, which could mean that the sources below this FWHM value do not belong to a different AGN class. Systematic[OIII]blue shifts, as well as a blueshifted component inHβare revealed. We interpret the blueshifts as related to the signature of outflowing gas from the quasar central engine. The FWHM ofHβis still affected by the blueshifted emission; however, the effect is non-negligible if the FWHMHβis used as a “virial broadening estimator” (VBE). We emphasize a strong effect of the viewing angle onHβbroadening, deriving a correction for those sources that shows major disagreement between virial and concordance cosmology luminosity values.Conclusions. The relatively large scatter between concordance cosmology and virial luminosity estimates can be reduced (by an order of magnitude) if a correction for orientation effects is included in the FWHMHβvalue; outflow and sample definition yield relatively minor effects.

Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schrödinger equation (see work of Zakharov, Glassey, and Ogawa–Tsutsumi) to show that certain classes of negative energy solutions must blowup in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.