Authors:Ovidiu Calin; Der-Chen Chang; Jishan Hu Pages: 9 - 18 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: 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: 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: 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: 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: 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:Anna Duyunova; Valentin Lychagin; Sergey Tychkov 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: 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: 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: 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: 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: 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: 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: 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:Leandro M. Del Pezzo; Julio D. Rossi Pages: 365 - 391 Abstract: We study the first eigenvalue of the \(p-\) Laplacian (with \(1<p<\infty \) ) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases \(p\rightarrow \infty \) and \(p\rightarrow 1\) . PubDate: 2016-12-01 DOI: 10.1007/s13324-016-0123-y Issue No:Vol. 6, No. 4 (2016)

Authors:Ranis N. Ibragimov 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: 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

Authors:Boronin Ivan; Shevlyakov Andrey 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: 2016-12-16 DOI: 10.1007/s13324-016-0157-1