Domain invariance for local solutions of semilinear evolution equations in Hilbert spaces

A closed set K of a Hilbert space H is said to be invariant under the evolution equation X'(t) = AX(t) + f(t,X(t)) (t > 0), whenever all solutions starting from a point of K, at any time t0 0, remain in K as long as they exist. For a self-adjoint strictly dissipative operator A, perturbed by a (possibly unbounded) nonlinear term f, we give necessary and sufficient conditions for the invariance of K, formulated in terms of A, f, and the distance function from K. Then, we also give sufficient conditions for the viability of K for the control system X'(t) = AX(t) + f(t,X(t), u(t)) (t > 0, u(t) ∈ U). Finally, we apply the above theory to a bilinear control problem for the heat equation in a bounded domain of RN, where one is interested in keeping solutions in one fixed level set of a smooth integral functional

Efficiency of wastewater disposal in mountain areas

Bibliography: pages 88-92.Supported in part by Colorado State University Experiment Station Project 264, WR-118, and Eisenhower Consortium, U. S. Forest Service, Cooperative Agreement 16-660-GR

Star Spot Induced Radial Velocity Variability in LkCa 19

We describe a new radial velocity survey of T Tauri stars and present the first results. Our search is motivated by an interest in detecting massive young planets, as well as investigating the origin of the brown dwarf desert. As part of this survey, we discovered large-amplitude, periodic, radial velocity variations in the spectrum of the weak line T Tauri star LkCa 19. Using line bisector analysis and a new simulation of the effect of star spots on the photometric and radial velocity variability of T Tauri stars, we show that our measured radial velocities for LkCa19 are fully consistent with variations caused by the presence of large star spots on this rapidly rotating young star. These results illustrate the level of activity-induced radial velocity noise associated with at least some very young stars. This activity-induced noise will set lower limits on the mass of a companion detectable around LkCa 19, and similarly active young stars.Comment: ApJ accepted, 27 pages, 12 figures, aaste

Rimozione di sedimenti per fluitazione dal serbatoio di Sernio (SO)

Il presente lavoro descrive la rimozione di un ingente quantitativo di sedimenti (circa 100'000 tonnellate) dal serbatoio di Sernio, in provincia di Sondrio, effettuata tra maggio e luglio del 2009. Il sedimento \ue8 stato evacuato per fluitazione (flushing), nel sostanziale rispetto dei vincoli preventivamente stabiliti sulla concentrazione di solidi sospesi (CSS) delle acque scaricate. Tali limitazioni hanno lo scopo di contenere l\u2019impatto delle operazioni sugli ecosistemi acquatici coinvolti. La CSS \ue8 stata controllata regolando il livello nel serbatoio, la portata in uscita e, in un secondo tempo, mediante l\u2019utilizzo di escavatori meccanici. La gestione delle operazioni si \ue8 basata sul costante monitoraggio della CSS poco a valle dell\u2019area di intervento. La campagna di misura \ue8 stata ulteriormente estesa a valle, per un tratto di circa 40 km lungo l\u2019asta dell\u2019Adda, al fine di quantificare la riduzione della CSS per effetto combinato di diluizione e deposizione

Statistical properties of stochastic 2D Navier-Stokes equations from linear models

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$. The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.Comment: 23 pages; improved versio

Electron-Electron Bremsstrahlung Emission and the Inference of Electron Flux Spectra in Solar Flares

Although both electron-ion and electron-electron bremsstrahlung contribute to the hard X-ray emission from solar flares, the latter is normally ignored. Such an omission is not justified at electron (and photon) energies above $\sim 300$ keV, and inclusion of the additional electron-electron bremsstrahlung in general makes the electron spectrum required to produce a given hard X-ray spectrum steeper at high energies. Unlike electron-ion bremsstrahlung, electron-electron bremsstrahlung cannot produce photons of all energies up to the maximum electron energy involved. The maximum possible photon energy depends on the angle between the direction of the emitting electron and the emitted photon, and this suggests a diagnostic for an upper cutoff energy and/or for the degree of beaming of the accelerated electrons. We analyze the large event of January 17, 2005 observed by RHESSI and show that the upward break around 400 keV in the observed hard X-ray spectrum is naturally accounted for by the inclusion of electron-electron bremsstrahlung. Indeed, the mean source electron spectrum recovered through a regularized inversion of the hard X-ray spectrum, using a cross-section that includes both electron-ion and electron-electron terms, has a relatively constant spectral index $\delta$ over the range from electron kinetic energy $E = 200$ keV to $E = 1$ MeV. However, the level of detail discernible in the recovered electron spectrum is not sufficient to determine whether or not any upper cutoff energy exists.Comment: 7 pages, 5 figures, submitted to Astrophysical Journa

On the Navier-Stokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier-Stokes modeling viscous fluid flow past a moving or rotating obstacle in $\mathbb{R}^d$ subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use $L^p$-techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in $\mathbb{R}^d$ is solved by an evolution system on $L^p_{\sigma}(\mathbb{R}^d)$ for $1<p<\infty$. For this we use results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove, for $p\geq d$ and initial data $u_0\in L^p_{\sigma}(\mathbb{R}^d)$, the existence of a unique mild solution to the full Navier-Stokes system.Comment: 18 pages, to appear in J. Math. Fluid Mech. (published online first

Approximating the coefficients in semilinear stochastic partial differential equations

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt + G(t,X(t))dW_H(t), X(0)=X_0, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^{tA}X_0 in the norms L^p(\Omega;C^\lambda([0,T];E)) assuming that the approximating operators A_n are uniformly sectorial and converge to A in the strong resolvent sense, and that the approximating nonlinearities F_n and G_n are uniformly Lipschitz continuous in suitable norms and converge to F and G pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite-dimensional multiplicative noise.Comment: Referee's comments have been incorporate

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.Comment: 26 pages, double columns; 15 figure