Authors:Ovidiu Calin; Der-Chen Chang; Jishan Hu Pages: 9 - 18 Abstract: Abstract This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere \(\mathbb {S}^3\) . We prove that a certain sub-Riemannian system \(Xf =a\) , \(Yf =b\) on \(\mathbb {S}^3\) has a solution if and only if the following integrability conditions hold: \(X^2 b + 4b = (XY + [X, Y]) a \) , \(Y^2 a + 4a = (YX-[X, Y]) b\) . We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0126-8 Issue No:Vol. 7, No. 1 (2017)

Authors:Valentin Lychagin; Valeriy Yumaguzhin Pages: 19 - 29 Abstract: Abstract We construct explicit solutions of the Einstein–Maxwell equations in the case when distributions defined by the Faraday tensor are completely integrable and totally geodesic. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0127-7 Issue No:Vol. 7, No. 1 (2017)

Authors:Yu Liu; Guobin Tang Pages: 31 - 45 Abstract: Abstract Let \({\mathbb {H}^n}\) be the Heisenberg group and \(Q=2n+2\) be its homogeneous dimension. The Schrödinger operator is denoted by \( - {\Delta _{{\mathbb {H}^n}}} + V\) , where \({\Delta _{{\mathbb {H}^n}}}\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \({B_{{q_1}}}\) for \({q_1} \ge \frac{Q}{2}\) . Let \(H^p_L(\mathbb {H}^n)\) be the Hardy space associated with the Schrödinger operator for \(\frac{Q}{Q+\delta _0}<p\le 1\) , where \(\delta _0=\min \{1,2-\frac{Q}{q_1}\}\) . In this note we show that the operators \({T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}\) and \({T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}\) are bounded from \(H_L^p({\mathbb {H}^n})\) into \({L^p}({\mathbb {H}^n})\) . Our results are also valid on the stratified Lie group. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0128-6 Issue No:Vol. 7, No. 1 (2017)

Authors:M. Mabdaoui; H. Moussa; M. Rhoudaf Pages: 47 - 76 Abstract: Abstract We shall give the proof of existence results for the entropy solutions of the following nonlinear parabolic problem where A is a Leray–Lions operator having a growth not necessarily of polynomial type. The lower order term \(\Phi \) : \(\Omega \times (0,T)\times \mathbb {R}\rightarrow \mathbb {R}^N\) is a Carathéodory function, for a.e. \((x,t)\in Q_T\) and for all \(s\in \mathbb {R}\) , satisfying only a growth condition and the right hand side f belongs to \(L^1(Q_T)\) . PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0129-5 Issue No:Vol. 7, No. 1 (2017)

Authors:Natalia Bondarenko Pages: 77 - 92 Abstract: Abstract The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0131-y Issue No:Vol. 7, No. 1 (2017)

Authors:S. Rahman; T. Hayat; B. Ahmad Pages: 93 - 105 Abstract: Abstract The purpose of present paper is to establish the regularity criteria for nonlinear problem of unsteady flow of third grade fluid in a rotating frame. The fluid is between two plates and the lower plate is porous. The main result of this paper is to establish the global regularity of classical solutions when \(\left\ F\right\ _{BMO}^{2}\) , \(\left\ g\right\ _{BMO}^{2}\) , \(\left\ \frac{\partial g}{\partial y}\right\ _{BMO}^{2}\) and \(\left\ \frac{\partial ^{2} g}{\partial y^{2}}\right\ _{BMO}^{2}\) are sufficiently small. In addition uniqueness of weak solution is also verified. Here BMO denotes the homogeneous space of bounded mean oscillations, F is the velocity and \(g=\nabla \times F=\frac{\partial F}{\partial z}\) is the vorticity of the rotating fluid. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0132-x Issue No:Vol. 7, No. 1 (2017)

Authors:Natalia P. Bondarenko Abstract: 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: 2017-04-24 DOI: 10.1007/s13324-017-0172-x

Authors:Yehonatan Salman Abstract: Abstract The aim of the article is to recover a certain type of finite parametric distributions and functions using their spherical mean transform which is given on a certain family of spheres whose centers belong to a finite set \(\Gamma \) . For this, we show how the problem of reconstruction can be converted to a Prony’s type system of equations whose regularity is guaranteed by the assumption that the points in the set \(\Gamma \) are in general position. By solving the corresponding Prony’s system we can extract the set of parameters which define the corresponding function or distribution. PubDate: 2017-04-18 DOI: 10.1007/s13324-017-0171-y

Authors:Anna Duyunova; Valentin Lychagin; Sergey Tychkov Abstract: 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: 2017-03-04 DOI: 10.1007/s13324-017-0169-5

Authors:Guangyue Huang; Zhi Li Abstract: 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: 2017-03-01 DOI: 10.1007/s13324-017-0168-6

Authors:Yuri Kondratiev; Yuri Kozitsky Abstract: 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: 2017-03-01 DOI: 10.1007/s13324-017-0166-8

Authors:Sergey Tychkov Abstract: 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: 2017-02-28 DOI: 10.1007/s13324-017-0165-9

Authors:Hüseyin Bor Abstract: 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: 2017-02-23 DOI: 10.1007/s13324-017-0164-x

Authors:Mats K. Brun; Henrik Kalisch Abstract: 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: 2017-02-02 DOI: 10.1007/s13324-017-0163-y

Authors:Xing Wang; Xinqiang Qin; Gang Hu Abstract: 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: 2017-01-17 DOI: 10.1007/s13324-016-0162-4

Authors:P. Bibikov Abstract: 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: 2017-01-04 DOI: 10.1007/s13324-016-0161-5

Authors:Ranis N. Ibragimov Abstract: 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: 2016-12-28 DOI: 10.1007/s13324-016-0158-0

Authors:Anatoly Golberg; Evgeny Sevost’yanov Abstract: 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: 2016-12-28 DOI: 10.1007/s13324-016-0159-z