From kinetic to fluid models of liquid crystals by the moment method

<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>

Brittle fracture in linearly elastic plates

In this work we derive by $\Gamma$ -convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$ . The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

The Model of Multi Limits of the Quantity of Electrons in Central Coulomb force Field

Based on the theory of periodic bifurcation of iterative equation, a conjectural model of periodic bifurcation of number of electrons in a central Coulomb force field is proposed. After that with the help of the methods Zeng’s The Course of Quantum Mechanics and Wu’s Methods of Mathematical Physics, [1], [2] the wave function of the electrons under the approximate state is solved in the central Coulomb force field. By using the method of separating variables for solving partial differential equations and some transformation and construction techniques, the strict mathematical solution of the Schrödinger equation for the electron in the field of central Coulomb force is obtained, and the iterative formula of the level of electron number is given theoretically. And using MATLAB, the multi-limit model of electron number is simulated under different initial value problems, to explore the change of the limit with the initial value and the factors affecting the limit number to a certain extent. Some potential research value of this model is also proposed.

The three dimensional Maxwell equations with strongly anisotropic electric permittivity

The subject matter of this work concerns the propagation of the electro-magnetic fields through strongly anisotropic media, in the three dimensional setting. We concentrate on the asymptotic behavior for the solutions of the Maxwell equations when the electric permittivity tensor is strongly anisotropic. We derive limit models and prove their well-posedness. We appeal to the variational framework and study the propagation speed of the solutions. We prove that almost all the electro-magnetic energy concentrates inside the propagation cone of the limit model.

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system

In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].

Constraining the Eurasian biogenic boreal carbon-cycle with satellite-SIF

&lt;p&gt;The Eurasian boreal ecosystem acts as a major terrestrial carbon sink in the northern hemisphere. Under changing climatic conditions, it is crucial to monitor biogenic carbon fluxes in this area. The Siberian in-situ CO&lt;sub&gt;2&lt;/sub&gt; data are, however, sparse in spatial coverage and limit model-validation there. Satellite observations of CO&lt;sub&gt;2&lt;/sub&gt; and Sun-Induced Fluorescence (SIF) can provide essential information to constrain the Eurasian boreal biogenic carbon-cycle and further, to improve carbon cycle inverse models.&lt;/p&gt;&lt;p&gt;In this study, we investigate the Eurasian boreal carbon cycle with satellite observations of the Orbiting Carbon Observatory 2 (OCO-2) and the Greenhouse gase Observing SATellite (GOSAT). We compare the observed carbon cycle dynamics to model data such as provided by CarbonTracker (CT2019, CT-NRT.v2020-1) and find differences in the ppm range. Various sensitivity studies with respect to region selection, sampling biases and model choices are used to consolidate the robustness of the detected pattern. Using SIF and FLUXCOM GPP data, we will show first attempts to attribute the model-measurement differences to uncertainties in biogenic carbon fluxes.&lt;/p&gt;