Authors:Ivan Boronin; Andrey Shevlyakov Pages: 1 - 8 Abstract: Buckley–Leverett equations describe non viscous, immiscible, two-phase filtration, which is often of interest in modelling of oil production. For many parameters and initial conditions, the solutions of these equations exhibit non-smooth behaviour, namely discontinuities in form of shock waves. In this paper we obtain a novel method for the solution of Buckley–Leverett equations, which is based on geometry of differential equations. This method is fast, accurate, stable, and describes non-smooth phenomena. The main idea of the method is that classic discontinuous solutions correspond to the continuous surfaces in the space of jets - the so-called multi-valued solutions (Bocharov et al., Symmetries and conservation laws for differential equations of mathematical physics. American Mathematical Society, Providence, 1998). A mapping of multi-valued solutions from the jet space onto the plane of the independent variables is constructed. This mapping is not one-to-one, and its singular points form a curve on the plane of the independent variables, which is called the caustic. The real shock occurs at the points close to the caustic and is determined by the Rankine–Hugoniot conditions. PubDate: 2018-03-01 DOI: 10.1007/s13324-016-0157-1 Issue No:Vol. 8, No. 1 (2018)

Authors:Ranis N. Ibragimov Pages: 11 - 24 Abstract: The nonlinear Euler equations are used to model two-dimensional atmosphere dynamics in a thin rotating spherical shell. The energy balance is deduced on the basis of two classes of functorially independent invariant solutions associated with the model. It it shown that the energy balance is exactly the conservation law for one class of the solutions whereas the second class of invariant solutions provides and asymptotic convergence of the energy balance to the conservation law. PubDate: 2018-03-01 DOI: 10.1007/s13324-016-0158-0 Issue No:Vol. 8, No. 1 (2018)

Authors:Anatoly Golberg; Evgeny Sevost’yanov Pages: 25 - 35 Abstract: We prove that open discrete mappings of Sobolev classes \(W_\mathrm{loc}^{1, p},\) \(p>n-1,\) with locally integrable inner dilatations admit \(ACP_p^{\,-1}\) -property, which means that these mappings are absolutely continuous on almost all preimage paths with respect to p-module. In particular, our results extend the well-known Poletskiĭ lemma for quasiregular mappings. We also establish the upper bounds for p-module of such mappings in terms of integrals depending on the inner dilatations and arbitrary admissible functions. PubDate: 2018-03-01 DOI: 10.1007/s13324-016-0159-z Issue No:Vol. 8, No. 1 (2018)

Authors:P. Bibikov Pages: 37 - 42 Abstract: In this work the group classification of Buckley–Leverett system is studied. The symmetry algebras of Buckley–Leverett systems are calculated and the fields of differential invariants for the actions of these algebras are found. PubDate: 2018-03-01 DOI: 10.1007/s13324-016-0161-5 Issue No:Vol. 8, No. 1 (2018)

Authors:Xing Wang; Xinqiang Qin; Gang Hu Pages: 43 - 55 Abstract: In this paper we consider the existence of weak positive solutions for an elliptic problems with the nonlinearity containing both singular and supercritical terms. By means of a priori estimate and sub-and supersolutions method, a positive weak solution is obtained. PubDate: 2018-03-01 DOI: 10.1007/s13324-016-0162-4 Issue No:Vol. 8, No. 1 (2018)

Authors:Mats K. Brun; Henrik Kalisch Pages: 57 - 75 Abstract: The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface boundary condition. The condition for breaking can be conveniently formulated as a convective breaking criterion based on the local Froude number at the wave crest. This breaking criterion can also be applied to time-dependent situations, and one case of interest is the development of an undular bore created by an influx at a lateral boundary. It is shown that this boundary forcing leads to wave breaking in the leading wave behind the bore if a certain threshold is surpassed. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0163-y Issue No:Vol. 8, No. 1 (2018)

Authors:Hüseyin Bor Pages: 77 - 83 Abstract: In this paper, we proved a known theorem dealing with \( \bar{N},p_{n} _{k}\) summability factors of infinite series under weaker conditions. Also we applied this theorem to the Fourier series. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0164-x Issue No:Vol. 8, No. 1 (2018)

Authors:Sergey Tychkov Pages: 85 - 91 Abstract: Rapoport–Leas model of motion of a two-phase flow on a plane is considered. Travelling-wave solutions for these equations are found. Wavefronts of these solutions may be circles, lines and parabolae. Ordinary differential equations for pressure and saturation on the wavefronts are established. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0165-9 Issue No:Vol. 8, No. 1 (2018)

Authors:Yuri Kondratiev; Yuri Kozitsky Pages: 93 - 121 Abstract: The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in \({\mathbbm {R}}^d\) in which the constituents appear (immigrate) with rate b(x) and disappear, also due to competition. For this model, we prove the existence of the evolution of states \(\mu _0 \mapsto \mu _t\) such that the moments \(\mu _t(N_\Lambda ^n)\) , \(n\in {\mathbbm {N}}\) , of the number of entities in compact \(\Lambda \subset {\mathbbm {R}}^d\) remain bounded for all \(t>0\) . Under an additional condition, we prove that the density of entities and the second correlation function remain point-wise bounded globally in time. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0166-8 Issue No:Vol. 8, No. 1 (2018)

Authors:Guangyue Huang; Zhi Li Pages: 123 - 134 Abstract: In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\begin{aligned} \Delta _V u+au\log u=0, \end{aligned}$$ where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of \( \nabla u \) with respect to \(\sup u\) and the lower bound of Bakry-Emery Ricci curvature. In particular, for complete noncompact manifolds with \(a<0\) , we prove that any positive solution must be \(u\equiv 1\) under a suitable condition for a with respect to the lower bound of Bakry-Emery Ricci curvature. It generalizes a classical result of Yau. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0168-6 Issue No:Vol. 8, No. 1 (2018)

Authors:Anna Duyunova; Valentin Lychagin; Sergey Tychkov Pages: 135 - 154 Abstract: Algebras of symmetries and the corresponding algebras of differential invariants for plane flows of inviscid fluids are given. Their dependence on thermodynamical states of media are studied and a classification of thermodynamical states is given. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0169-5 Issue No:Vol. 8, No. 1 (2018)

Authors:Natalia P. Bondarenko Pages: 155 - 168 Abstract: The Sturm–Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved. PubDate: 2018-03-01 DOI: 10.1007/s13324-017-0172-x Issue No:Vol. 8, No. 1 (2018)

Authors:Matteo Gallone; Alessandro Michelangeli Abstract: We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreĭn-Višik-Birman extension scheme, or also on Grubb’s universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension. PubDate: 2018-02-24 DOI: 10.1007/s13324-018-0219-7

Authors:Steven R. Bell Abstract: The adjoint of the classic composition operator on the Hardy space of the unit disc determined by a holomorphic self map of the unit disc is well known to send the Szegő kernel function associated to a point in the unit disc to the Szegő kernel associated to the image of that point under the self map. The purpose of this paper is to show that a constructive proof that holomorphic functions that extend past the boundary can be well approximated by complex linear combinations of the Szegő kernel function gives an explicit formula for the adjoint of a composition operator that yields a new way of looking at these objects and provides inspiration for new ways of thinking about operators that act on linear spans of the Szegő kernel. Composition operators associated to multivalued self mappings will arise naturally, and out of necessity. A parallel set of ideas will be applied to composition operators on the Bergman space. PubDate: 2018-02-22 DOI: 10.1007/s13324-018-0215-y

Authors:Zhaohong Sun; Youfa Lei Abstract: In this paper we study the existence of multiple sign-changing solutions for the following nonlocal Kirchhoff-type boundary value problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\Omega } \nabla u ^2{ dx}\right) \triangle {u}=\lambda u ^{p-1}u,&{}\quad \text{ in }\quad \Omega ,\\ u=0,&{} \quad \text{ on }\quad \partial \Omega . \\ \end{array}\right. \end{aligned}$$ Using a new method, we prove that this problem has infinitely many sign-changing solutions and has a least energy sign-changing solution for \(p\in (3,5)\) . Few existence results of multiple sign-changing solutions are available in the literature. This new method is that, by choosing some suitable subsets which separate the action functional and on which the functional is bounded, so that we can use genus and the method of invariant sets of descending flow to construct the minimax values of the functional. Our work generalize some results in literature. PubDate: 2018-02-21 DOI: 10.1007/s13324-018-0218-8

Authors:Vladimir Jaćimović Abstract: We demonstrate that the Douady–Earle extension can be computed by solving the specific set of ODE’s. This system of ODE’s has several interpretations in Mathematical Physics, such as Kuramoto model of coupled oscillators or Josephson junction arrays. This method emphasizes the key role of conformal barycenter in some theories of Mathematical Physics. On the other hand, it indicates that variations of Kuramoto model might be used in different problems of computational quasiconformal geometry. The idea can be extended to compute the D–E extension in higher dimensions as well. PubDate: 2018-02-19 DOI: 10.1007/s13324-018-0214-z

Authors:Guowei Jiang; Yu Liu Abstract: We consider the Schrödinger operator \(L = -\Delta _{G}+V\) on the stratified Lie group G, where \(\Delta _{G}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \( B_{q_{1}}\) for \(q_1\ge \frac{Q}{2}\) , where Q is the homogeneous dimension of G. Let \(q_2 = 1\) when \(q_1\ge Q\) and \(\frac{1}{q_2}=1-\frac{1}{q_1}+\frac{1}{Q}\) when \(\frac{Q}{2}<q_1<Q\) . The commutator \([b,\mathcal {R}]\) is generated by a function \(b\in \varLambda ^{\theta }_{\nu }(G)\) for \(\theta >0,0<\nu <1\) , where \(\varLambda ^{\theta }_{\nu }(G)\) is a new function space on the stratified Lie group which is larger than the classical Companato space, and the Riesz transform \(\mathcal {R}=\nabla _{G}(-\Delta _{G}+V)^{-\frac{1}{2}}\) . We prove that the commutator \([b,\mathcal {R}]\) is bounded from \(L^{p}(G)\) into \(L^{q}(G)\) for \(1<p<q^{'}_{2}\) , where \(\frac{1}{q}=\frac{1}{p}-\frac{\nu }{Q}\) . PubDate: 2018-02-19 DOI: 10.1007/s13324-018-0216-x

Authors:Stephen J. Gardiner; Hermann Render Abstract: The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions. PubDate: 2018-02-10 DOI: 10.1007/s13324-018-0213-0

Authors:Timothy Ferguson; William T. Ross Abstract: We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees. PubDate: 2018-02-05 DOI: 10.1007/s13324-018-0212-1