We study a two-layer one-dimensional energy balance model, which allows for vertical energy exchanges between a surface layer and the atmosphere, as well as meridional energy transport across latitudes via a diffusion law. The evolution equations of the surface temperature and the atmospheric temperature are coupled by exchange of infrared radiation as well as other non-radiative energy exchanges. The energy enters the system as solar radiation, which is partially absorbed and partially reflected by the two layers. The system is then composed of two degenerate parabolic equations coupled by nonlinear terms, the growth of these terms being crucial for the choice of the functional setting. An essential parameter is the absorptivity of the atmosphere, denoted ε _a, whose value depends critically on greenhouse gases. We prove that blow up in finite time occurs if ε _a >2, while global existence of solutions and the existence of a global attractor hold when ε _a ∈ (0,2). Proofs are based on comparison principles that derive from the cooperative structure of the problem, and that provide invariant rectangles for smooth initial conditions, and on regularity properties.

In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the L^p framework, with p∈ [1,∞). We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak and strong solutions for p∈ [1,∞), under suitable assumptions on the damping function. Then we study the asymptotic behaviour of the associated energy when p ∈ (1,∞), and we provide decay estimates that appear to be almost optimal as compared to a similar problem with boundary damping. Our study is motivated by earlier works, in particular, . Our proofs combine arguments from (wave equation in the L^p framework with a linear damping) with a technique of weighted energy estimates () and new integral inequalities when p>2, and with convex analysis tools when p∈ (1,2).

Solid Composite Propellants (SCPs) are extensively used in the field of propulsion for their chemical and mechanical stability in long-term storage, and for having simple production and operation processes. Computational modeling enables cost reduction, increased efficiency, and greater coverage of the configuration space in the SCP design process. However, accurate and efficient SCP modeling presents a number of numerical challenges. A primary obstacle in modeling these systems is capturing the complex evolving interface. Recently, it has been shown that the phase-field method has a strong ability to model the combustion behavior of SCPs, implicitly capturing the topological evolution at a relatively low computational cost. Initial phase-field methods show promise in their ability to do predictive regression modeling but require a number of approximations and heuristic modeling methods. This work presents a new formulation for the phase field regression model that combines a thermal solver with an Arrhenius rate law to model interface regression using a comprehensive and physics-based approach. This improves the capability of previous methods by increasing the number of kinematic forces that are accounted for, and allowing the study of thermal diffusivity in the system. It also enables the integration of a fully coupled solid-fluid interface. To demonstrate the efficacy of the model, it is applied and calibrated to a homogeneous monopropellant (ammonium perchlorate). The model is validated for a range of temperatures, with a reasonable quantitative match to experimental data. It was also demonstrated that the model recovered the relationship between ignition time and heat flux, with no fitting required. Overall, the method showed great capability of reproducing experimental data by matching temperature, burn rate, and thermal diffusivity profiles. Statement of Significance: A new diffuse interface method is developed for calculating ignition and regression rates in SCPs. While this builds on previous work in diffuse interface modeling of SCPs, it is novel in its coupling to thermal transport, as well as its development of (and coupling to) a approximation for gas phase heat flux. The model is shown to reproduce experimental regression rate data with reasonable accuracy. Moreover, it quantitatively captures ignition behavior with no additional modifications to the model. The model is implemented in a high performance computational framework that is able to resolve large, three-dimensional mesostuctures. (c) 2023 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

We consider the linear degenerate wave equation, on the interval (0,1) w(tt)-(x(alpha)wx)(x)=p(t)mu(x)w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the "ground state." Under some classical and generic assumptions on mu, we prove that there exists a threshold value for time, T-alpha=4 / 2-alpha, such that the reachable set is a neighborhood of the ground state. If T<T-alpha it is contained in a C-1-submanifold of infinite codimension. Finally, it T = T-alpha and alpha is an element of [0,1) it is a C-1-submanifold of codimension 1 and if alpha is an element of(1,2) the reachable set is a neighborhood of the ground state. The case alpha=1 remains open. This extends to the degenerate case the work [K. Beauchard, J. Differential Equations, 250 (2011), pp. 2064-2098] and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, SIAM J. Control Optim., 55 (2017), pp. 2052-2087]. Our proofs are based on a careful analysis of the spectral problem and on Ingham type results, which are extensions of Kadec's 1/4 theorem.

We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted εa. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2. (Note that the physical range of values for εa is (0,1].) Next, we explain the long time dynamics for εa∈(0,2), and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove, in particular, that the surface temperature increases monotonically with respect to εa. This is the key mathematical manifestation of the greenhouse effect.

We consider a reaction-diffusion model of biological invasion in which the evolution of the population is governed by several parameters among them the intrinsic growth rate \begin{document}$ \mu(x) $\end{document}. The knowledge of this growth rate is essential to predict the evolution of the population, but it is a priori unknown for exotic invasive species. We prove uniqueness and unconditional Lipschitz stability for the corresponding inverse problem, taking advantage of the positivity of the solution inside the spatial domain and studying its behaviour near the boundary with maximum principles. Our results complement previous works by Cristofol and Roques [11,13].

This work is motivated by the study of null controllability for the typical degenerate parabolic equation with interior degeneracy and one-sided control:u(t) - (vertical bar x vertical bar(alpha)u(x))(x) = h(x, t)chi((a,b)), x is an element of(-1, 1),with 0 < a < b < 1. It was proved in [7] that this equation is null controllable (in any positive time T) if and only if alpha < 1, and that the cost of null controllability blows up as alpha -> 1(-). This is related to the following property of the eigenvalues: the gap between an eigenvalue of odd order and the consecutive one goes to 0 as alpha -> 1(-) (see [7]).The goal of the present work is to provide optimal upper and lower estimates of the null controllability cost, with respect to the degeneracy parameter (when alpha -> 1(-)) and in short time (when T -> 0(+)). We prove that the null controllability cost behaves as 1/1-alpha as a alpha -> 1(-) and as e(1/T) as T -> 0(+). Our analysis is based on the construction of a suitable family biorthogonal to the sequence (e(lambda nt)) n in L-2(0, T), under some general gap conditions on the sequence (lambda(n))(n), conditions that are suggested by a motivating example.

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell, based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behaviour: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and so to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results of Fattorini-Russell concerning biorthogonal families, using complex analysis techniques developed by Seidman, G\"uichal, Tenenbaum-Tucsnak, and Lissy.

In this paper, we study two Energy Balance Models with Memory arising in climatology, which consist in a 1D degenerate nonlinear parabolic equation involving a memory term, and possibly a set-valued reaction term (of Sellers type and of Budyko type, in the usual terminology). We provide existence and regularity results, and obtain uniqueness and stability estimates that are useful for the determination of the insolation function in Sellers' model with memory.

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

The goal of this paper is to analyze the cost of boundary null controllability for the 1 - D linear heat equation with the so-called inverse square potential: u(t) - u(xx) - mu/x(2) u = 0; x is an element of(0, 1); t is an element of(0, T); where mu is a real parameter such that mu <= 1/4. Since the works by Baras and Goldstein [4, 5], it is known that such problems are well-posed for any mu <= 1/4 (the constant appearing in the Hardy inequality) whereas instantaneous blowup may occur when mu > 1/4. For any mu <= 1/4, it has been proved in [52] (via Carleman estimates) that the equation can be controlled (in any time T > 0) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at x = 1. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters T > 0 and mu is an element of(-infinity, 1/4]. Our proofs are based on the moment method and very recent results on biorthogonal sequences.

For $\alpha\in (0,2)$ we study the null controllability of the parabolic operator $Pu= u_t - (\vert x \vert ^\alpha u_x)_x \ (-1

We consider the one-dimensional degenerate parabolic equationu(t) - (x(alpha)u(x))(x) = 0 x is an element of (0, 1), t is an element of (0, T),controlled by a boundary force acting at the degeneracy point x = 0.We study the reachable targets at some given time T using H-1 controls, studying the influence of the degeneracy parameter alpha is an element of [0, 1). First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as alpha -> 1(-) and exponentially fast when T -> 0(+).Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states.Our approach is based on the moment method developed by Fattorini and Russell [19, 20]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [48], Guichal [26], Tenenbaum-Tucsnak [49] and Lissy [35, 36].

In this paper, we introduce a new age-structured population model with diffusion and gestation processes and make a complete study of the qualitative properties of its solutions. The model is in the spirit of a model introduced in [13, 15] and studied in [10]. We aim here to correct some weakness of the model that was pointed out in [10].

Uranium ore concentrates from two different sources were examined using scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS). The ore powders are referred to as Namibia (id. no. 90036, LIMS id. no. 18775) and Canada Key Lake (id. no. 90019, LIMS id. no. 18774). Earlier work identified the ores as the U₃O₈ phase of uranium oxide using x-ray diffraction. Both sets of powders were in the form of dark brown to black powder fines. However, the Canada Key Lake concentrates contained larger chunks of material on the millimeter scale that were easily visible to the unaided eye. The powders were mounted for SEM examination by hand dispersing a small amount onto conductive sticky tape. Two types of applicators were used and compared: a fine-tipped spatula and a foam-tipped applicator. The sticky tape was on a standard SEM “tee” mount, which was tapped to remove loose contamination before being inserted into the SEM.

Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics.This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models.Global Carleman estimates are a priori estimates in weighted Sobolev norms for solutions of linear partial differential equations subject to boundary conditions. Such estimates proved to be extremely useful for several kinds of uniformly parabolic equations and systems. This is the first work where such estimates are derived for degenerate parabolic operators in dimension higher than one. Applications to null controllability with locally distributed controls and inverse source problems are also developed in full detail.Compared to nondegenerate parabolic problems, the current context requires major technical adaptations and a frequent use of Hardy type inequalities. On the other hand, the treatment is essentially self-contained, and only calls upon standard results in Lebesgue measure theory, functional analysis and ordinary differential equations.

In this paper, we are interested in some inverse problem that consists in recovering the so-called insolation function in the 2-D Sellers model on a Riemannian manifold that materializes the Earth's surface. For this nonlinear problem, we obtain a Lipschitz stability result in the spirit of the result by Imanuvilov-Yamamoto in the case of the determination of the source term in the linear heat equation. The paper complements an analogous study by Tort-Vancostenoble in the case of the 1-D Sellers model.

A simple, robust analytical chemistry method has been developed to dissolve plutonium containing particles in a complex matrix. The aerosol particles collected on Marple cascade impactor substrates were shown to be dissolved completely with an acid mixture of 12 M HNO3 and 0.1 M HF. A pressurized closed vessel acid digestion technique was utilized to heat the samples at 130 °C for 16 h to facilitate the digestion. The dissolution efficiency for plutonium particles was 99 %. The resulting particle digestate solution was suitable for trace elemental analysis and isotope composition determination, as well as radiochemistry measurements.

Solid debris was recovered from the previously-emptied nitrate salt waste drum S855793. The bulk sample was nondestructively assayed for radionuclides in its as-received condition. Three monoliths were selected for further characterization. Two of the monoliths, designated Specimen 1 and 3, consisted primarily of sodium nitrate and lead nitrate, with smaller amounts of lead nitrate oxalate and lead oxide by powder x-ray diffraction. The third monolith, Specimen 2, had a complex composition; lead carbonate was identified as the predominant component, and smaller amounts of nitrate, nitrite and carbonate salts of lead, magnesium and sodium were also identified. Microfocused x-ray fluorescence (MXRF) mapping showed that lead was ubiquitous throughout the cross-sections of Specimens 1 and 2, while heteroelements such as potassium, calcium, chromium, iron, and nickel were found in localized deposits. MXRF examination and destructive analysis of fragments of Specimen 3 showed elevated concentrations of iron, which were broadly distributed through the sample. With the exception of its high iron content and low carbon content, the chemical composition of Specimen 3 was within the ranges of values previously observed in four other nitrate salt samples recovered from emptied waste drums.

This study presents a method for destructive analysis of irradiated uranium (U) targets, with a focus on collection and measurement of long-lived ( t 1/2 > ~10 years) and stable fission product isotopes of ruthenium and cesium. Long-lived and stable isotopes of these elements can provide information on reactor conditions (e.g. flux, irradiation time, cooling time) in old samples (>5–10 years) whose short-lived fission products have decayed away. The separation and analytical procedures were tested on archived U reactor targets at Los Alamos National Laboratory as part of an effort to evaluate reactor models at low-burnup.