Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and diffusive effects

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.Comment: 12 pages, 20 figure

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to appl

Fluorescence-Controlled Er:YAG Laser For Caries Removal In Permanent Mandibular Molars

Poster presentation of research proposal addressing the question: Do fluorescence-controlled Er:YAG lasers more effectively remove dentinal carious hard tissue on permanent mandibular molars than conventional rotary burs?https://dune.une.edu/cdm_studpost/1000/thumbnail.jp

Dust Formation in Very Massive Primordial Supernovae

At redshift z>5 Type II supernovae (SNII) are the only known dust sources with evolutionary timescales shorter than the Hubble time. We extend the model of dust formation in the ejecta of SNII by Todini & Ferrara (2001) to investigate the same process in pair-instability supernovae (PISN), which are though to arise from the explosion of the first, metal free, very massive (140-260 Msun) cosmic stars. We find that 15%-30% of the PISN progenitor mass is converted into dust, a value >10 times higher than for SNII; PISN dust depletion factors (fraction of produced metals locked into dust grains) range between 0.3 and 0.7. These conclusions depend very weakly on the mass of the PISN stellar progenitor, which instead affects considerably the composition and size distribution. For the assumed temperature evolution, grain condensation starts 150-200 days after the explosion; the dominant compounds for all progenitor masses are SiO2 and Mg2SiO4 while the contribution of amorphous carbon and magnetite grains grows with progenitor mass; typical grain sizes range between 0.001 and a few 0.1 micron and are always smaller than 1 micron. We give a brief discussion of the implications of dust formation for the IMF evolution of the first stars, cosmic reionization and the intergalactic medium.Comment: 10 pages, 8 figures, accepted for publication in MNRA

Rapidly Accreting Supergiant Protostars: Embryos of Supermassive Black Holes?

Direct collapse of supermassive stars (SMSs) is a possible pathway for generating supermassive black holes in the early universe. It is expected that an SMS could form via very rapid mass accretion with Mdot ~ 0.1 - 1 Msun/yr during the gravitational collapse of an atomic-cooling primordial gas cloud. In this paper we study how stars would evolve under such extreme rapid mass accretion, focusing on the early evolution until the stellar mass reaches 1000 Msun. To this end we numerically calculate the detailed interior structure of accreting stars with primordial element abundances. Our results show that for accretion rates higher than 0.01 Msun/yr, stellar evolution is qualitatively different from that expected at lower rates. While accreting at these high rates the star always has a radius exceeding 100 Rsun, which increases monotonically with the stellar mass. The mass-radius relation for stellar masses exceeding ~ 100 Msun follows the same track with R_* \propto M_*^0.5 in all cases with accretion rates > 0.01 Msun/yr; at a stellar mass of 1000 Msun the radius is about 7000 Rsun (~= 30 AU). With higher accretion rates the onset of hydrogen burning is shifted towards higher stellar masses. In particular, for accretion rates exceeding Mdot > 0.1 Msun/yr, there is no significant hydrogen burning even after 1000 Msun have accreted onto the protostar. Such "supergiant" protostars have effective temperatures as low as Teff ~= 5000 K throughout their evolution and because they hardly emit ionizing photons, they do not create an HII region or significantly heat their immediate surroundings. Thus, radiative feedback is unable to hinder the growth of rapidly accreting stars to masses in excess of 1000 Msun, as long as material is accreted at rates Mdot > 0.01 Msun/yr.Comment: 11 pages, 10 figure

Lyalpha heating and its impact on early structure formation

In this paper we have calculated the effect of Lyalpha photons emitted by the first stars on the evolution of the IGM temperature. We have considered both a standard Salpeter IMF and a delta-function IMF for very massive stars with mass 300 M_sun. We find that the Lyalpha photons produced by the stellar populations considered here are able to heat the IGM at z<25, although never above ~100 K. Stars with a Salpeter IMF are more effective as, due to the contribution from small-mass long-living stars, they produce a higher Lyalpha background. Lyalpha heating can affect the subsequent formation of small mass objects by producing an entropy floor that may limit the amount of gas able to collapse and reduce the gas clumping.We find that the gas fraction in halos of mass below ~ 5 x 10^6 M_sun is less than 50% (for the smallest masses this fraction drops to 1% or less) compared to a case without Lyalpha heating. Finally, Lyalpha photons heat the IGM temperature above the CMB temperature and render the 21cm line from neutral hydrogen visible in emission at z<15.Comment: 7 pages, 5 figures, to be printed in MNRA

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we include in an appendix the corresponding treatment of the Swift-Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.Comment: 78 pages, 11 figure

Probing intergalactic radiation fields during cosmic reionization through gamma-ray absorption

We discuss expectations for the absorption of high-energy gamma-rays by gamma-gamma pair production with intergalactic radiation fields (IRFs) at very high redshifts (z~5-20), and the prospects thereof for probing the cosmic reionization era. For the evolving IRF, a semi-analytical model incorporating both Population II and Population III stars is employed, which is consistent with a wide variety of existing high-z observations including QSO spectral measurements, WMAP Thomson depth constraints, near-IR source count limits, etc. We find that the UV IRF below the Lyman edge energy with intensities in the range of a few times 10^{-19} erg cm^{-2} s^{-1} Hz^{-1} sr^{-1} can cause appreciable attenuation above ~12 GeV at z~5, down to ~6-8 GeV at z>~8-10. This may be observable in the spectra of blazars or gamma-ray bursts by the Fermi Gamma-ray Space Telescope or next generation facilities such as the Cherenkov Telescope Array, Advanced Gamma-ray Imaging System or 5@5, providing invaluable insight into early star formation and cosmic reionization.Comment: MNRAS in press with minor revisions, 5 pages, 5 figures. Numerical data of the model results will be available at http://www-tap.scphys.kyoto-u.ac.jp/~inoue/hizabs