On the extremal solutions of superlinear Helmholtz problems

In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

Sobolev regularity solutions for a class of singular quasilinear ODEs

This paper considers an initial-boundary value problem for a class of singular quasilinear second-order ordinary differential equations with the constraint condition stemming from fluid mechanics. We prove that the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs by means of a suitable Nash-Moser iteration scheme Meanwhile, asymptotic expansion of those positive solutions is shown.

On regular movements of a material rod

We study the equations of movement of a rod and other simplest rectilinear bodies. We have found some of their regular solutions, which can be used to construct periodic solutions, close to these regular ones.

Yang-Baxter and the Boost: splitting the difference

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times1616×16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2)𝔰𝔲(2)⊕𝔰𝔲(2) symmetry, which include the one-dimensional Hubbard model and the SS-matrix of the {AdS}_5 \times {S}^5AdS5×S5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

Existence and Uniqueness of Isothermal, Slightly Compressible Stratified Flow

We show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.

On General Properties of Degenerate Systems of Second Order Partial Differential Equations of Hypergeometric Type

For the first time, the general properties of degenerate related hypergeometric systems such as Horn, Whittaker, Bessel and Laguerre are investigated together. The joint research allowed to reveal their various common properties and to establish a number of new degenerate related systems. They are all private cases of the common system offered by the authors for consideration. For the full study, it is important to classify its regular and irregular special curves and to identify the types of corresponding solutions. In this paper, they are implemented using simple rules. Special attention is paid to the construction of normal and regular solutions, because the solutions of all related degenerate systems such as Horn, Whittaker, Bessel and Laguerre near the irregular singularity on infinity relate to this species. Peculiarities of building normal-regular solutions by the Frobenius-Latysheva method are shown. All constructed normal-regular solutions are expressed through the function of Humbert  variables, which is the solution of degenerate hypergeometric system of Horn type. As an example, the cases  where, along with the application of the Frobenius-Latysheva method, the possibility of outputting new degenerate related systems is demonstrated.

Τhe role of surface energy in the apparent solubility of two different calcite crystal habits

The interplay between polymorphism and facet-specific surface energy on the dissolution of crystals is examined in this work. It is shown that, using cationic additives, it is possible to produce star-shaped calcite crystals at very high supersaturations. In crystallization processes following the Ostwald rule of stages these star-shaped crystals appear to have higher solubility than both their rhombohedral counterparts and needle-shaped aragonite crystals. The vapour pressures of vaterite, aragonite, star-shaped calcite and rhombohedral calcite crystals are measured using thermogravimetric analysis and the corresponding enthalpies of melting are obtained. Using inverse gas chromatography, the surface energy of the aforementioned crystals is measured as well and the surface energy of the main crystal facets is calculated. Combining the effect of facet-specific surface energies and the enthalpies of melting on a modified version of the classical solubility equation for regular solutions, it is proved that the star-shaped calcite crystals can indeed have higher apparent solubility than aragonitecrystals.