We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of
N
≥
2
components occupying a bounded region
Ω
⊂
R
d
,
d
≤
3
. Our system is ...subject to a constant mobility matrix of coefficients, a free energy functional given in terms of
singular entropy
generated potentials and
localized
capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if
d
≤
2
, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if
d
=
3
, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If
d
=
3
, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.
The natural resistance of plants to disease is based not only on preformed mechanisms, but also on induced mechanisms. The defense mechanisms present in resistant plants may also be found in ...susceptible ones. This study attempted to analyze the metabolic alterations in plants of the potato Solanum tuberosum L. cv. Agata that were inoculated with the incompatible plant-pathogenic bacteria X. axonopodis and R. solanacearum, and the compatible bacterium E. carotovora. Levels of total phenolic compounds, including the flavonoid group, and the activities of polyphenol oxidase (PPO) and peroxidase (POX) were evaluated. Bacteria compatibility was evaluated by means of infiltration of tubers. The defense response was evaluated in the leaves of the potato plants. Leaves were inoculated depending on their number and location on the stem. Multiple-leaf inoculation was carried out on basal, intermediate, and apical leaves, and single inoculations on intermediate leaves. Leaves inoculated with X. axonopodis and with R. solanacearum showed hypersensitive responses within 24 hours post-inoculation, whereas leaves inoculated with E. carotovora showed disease symptoms. Therefore, the R. solanacearum isolate used in the experiments did not exhibit virulence to this potato cultivar. Regardless of the bacterial treatments, the basal leaves showed higher PPO and POX activities and lower levels of total phenolic compounds and flavonoids, compared to the apical leaves. However, basal and intermediate leaves inoculated with R. solanacearum and X. axonopodis showed increases in total phenolic compounds and flavonoid levels. In general, multiple-leaf inoculation showed the highest levels of total phenolics and flavonoids, whereas the single inoculations resulted in the highest increase in PPO activity. The POX activity showed no significant difference between single- and multiple-leaf inoculations. Plants inoculated with E. carotovora showed no significant increase in defense mechanisms such as enzyme activity and phenolic compounds. Therefore, resistance or susceptibility in S. tuberosum cv. Agata might depend on leaf age, type of inoculation performed (single or multiple), and the interaction between plant and pathogen.
This paper investigates how changes in expectations regarding the ability of the European Monetary Union to address the debt crisis have asymmetrically impacted the cost of sovereign borrowing in ...central and peripheral European countries. It shows that most of the variations in sovereign spreads can be explained by fundamentals in a model that allows for structural breaks. We test for both the presence and the time of structural breaks, deriving their asymptotic distribution and confidence intervals. The two estimated breakpoints are: the second quarter of 2010, a period when financial markets lacked confidence in a resolution for the crisis; and the third quarter of 2010, when financial markets regained confidence following Mario Draghi's 'whatever it takes' announcement. Market fears, measured by the degree of international risk aversion, became more important to price sovereign debt only for peripheral economies during the crisis.
In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The ...control enters the governing
state system
in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the
cost functional
to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with
ad hoc
Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.
Abstract
This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and ...different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points
$\pm 1$
, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.
We consider a multi-component version of the conserved Allen–Cahn equation proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for phase separation. In our case, the free ...energy is characterized by a mixing entropy density which belongs to a large class of physically relevant entropies like, for example, the Boltzmann–Gibbs entropy. We establish the well-posedness of the Cauchy–Neumann problem with respect to a natural notion of (finite) energy solution which is more regular under appropriate assumptions and is strictly separated from pure phases if the initial datum is. We then prove that the energy solution becomes more regular and strictly separated instantaneously. Also, we show that any finite energy solution converges to a unique equilibrium. The validity of a dissipative inequality (identity for strong solutions) allows us to analyze the problem within the theory of infinite-dimensional dissipative dynamical systems. On account of the obtained results, we can associate to our problem a dissipative dynamical system and we can prove that it has a global attractor as well as an exponential attractor.
The link between compressible models of tissue growth and the Hele–Shaw free boundary problem of fluid mechanics has recently attracted a lot of attention. In most of these models, only repulsive ...forces and advection terms are taken into account. In order to take into account long range interactions, we include a surface tension effect by adding a nonlocal term which leads to the degenerate nonlocal Cahn–Hilliard equation, and study the incompressible limit of the system. The degeneracy and the source term are the main difficulties. Our approach relies on a new
L
∞
estimate obtained by De Giorgi iterations and on a uniform control of the energy despite the source term. We also prove the long-term convergence to a single constant stationary state of any weak solution using entropy methods, even when a source term is present. Our result shows that the surface tension in the nonlocal (and even local) Cahn–Hilliard equation will not prevent the tumor from completely invading the domain.
The 1994 United Nations International Conference on Population and Development (ICPD) identified the centrality of defending reproductive rights and the importance of ensuring voluntary reproductive ...choices (UNFPA, 2019). This paper presents a review of the main theoretical models of demand for children and, by assessing the empirical counterpart of these models in an international database, points out the main socioeconomic variables determining fertility. This study shows that female schooling is the most important socioeconomic variable to explain the number of children per woman, as increased female education can expand access to information on contraceptive methods, improve employment opportunities and “empower” women in defending their sexual and reproductive choices.
A Conferência Internacional sobre População e Desenvolvimento das Nações Unidas (CIPD) de 1994 identificou a centralidade da defesa dos direitos reprodutivos e a importância de se garantir a realização de escolhas reprodutivas voluntárias (UNFPA, 2019). O presente artigo apresenta uma revisão sobre os principais modelos teóricos de demanda por filhos e, avaliando a contrapartida empírica desses modelos em uma base de dados internacional, aponta as principais variáveis socioeconômicas determinantes da fecundidade. Esse estudo mostra que a escolaridade feminina é a variável socioeconômica mais importante para explicar o número de filhos por mulher, visto que o aumento da educação feminina pode expandir o acesso à informação sobre os métodos contraceptivos, melhorar as oportunidades de emprego e “empoderar” as mulheres na defesa de suas escolhas sexuais e reprodutivas.
La Conferencia Internacional de las Naciones Unidas sobre Población y Desarrollo (CIPD) de 1994 identificó la centralidad de defender los derechos reproductivos y la importancia de garantizar opciones reproductivas voluntarias (UNFPA 2019). Este artículo presenta una revisión de los principales modelos teóricos de demanda de niños y, al evaluar la contraparte empírica de estos modelos en una base de datos internacional, señala las principales variables socioeconómicas que determinan la fertilidad. Este estudio muestra que la escolarización femenina es la variable socioeconómica más importante para explicar el número de hijos por mujer, ya que una mayor educación femenina puede ampliar el acceso a la información sobre métodos anticonceptivos, mejorar las oportunidades de empleo y "empoderar" a las mujeres en defensa de sus elecciones sexuales y reproductivas.
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