ALBERT

Description: This paper deals with two stability aspects of linear systems of the form \({I\ddot{x}+B\dot{x}+Cx=0}\) given by the triple ( I , B , C ). A general transformation scheme is given for a structure and Jordan form preserving transformation of the triple. We investigate how a system can be transformed by suitable choices of the transformation parameters into a new system ( I , B 1 , C 1 ) with a symmetrizable matrix C 1 . This procedure facilitates stability investigations. We also consider systems with a Hamiltonian spectrum which discloses marginal stability after a Jordan form preserving transformation.

Description: In this paper, we study the existence of critical points for the following functional constrained on \({S_c=\{u\in H^1(\mathbb{R}^N)| |u|_2=c\}}\) : $$I(u)=\frac{a}{2}\int_{\mathbb{R}^N}|\nabla{u}|^{2}+\frac{b}{4}\left(\int_{\mathbb{R}^N}|\nabla{u}|^{2}\right)^{2}-\frac{N}{2N+8}\int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}},$$ where N  = 1, 2, 3 and a , b  〉 0 are constants. The constraint problem is L 2 -critical. We showed that I ( u ) has a constraint critical point with a mountain pass geometry on S c if \({c 〉 c^*:=(2^{-1}b|Q|_2^{\frac{8}{N}})^{\frac{N}{8-2N}}}\) , where Q is the unique positive radial solution of \({-2\Delta Q+(\frac{4}{N}-1)Q=|Q|^{\frac{8}{N}} Q}\) in \({\mathbb{R}^N}\) . For 0 〈  c  〈  c * , I ( u ) has no critical point on S c , and we proved the existence of minimizers for a new perturbation functional on S c : $$E_{a,b}(u)=\frac{a}{2} \int_{\mathbb{R}^N}|\nabla u|^2+\frac{b}{4} \left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^2-\frac{1}{4} \int_{\mathbb{R}^N}V(x)|u|^{4}-\frac{N}{2N+8} \int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}}.$$

Description: In this paper, we analyze the following abstract system $$\left\{\begin{array}{ll} u_{tt} +Au+ Bu_t =0,\\ u(0) =u_0,\,\,u_t(0) = u_1,\end{array}\right.$$ where A is a self-adjoint, positive definite operator on a Hilbert space H , B (the dissipation operator) is another positive operator satisfying \({cA^{\alpha}u \leq Bu \leq CA^{\alpha}u}\) for some constants 0 〈  c 〈  C . The case of \({0 \leq \alpha \leq 1}\) has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when \({\alpha 〈 0}\) . Moreover, we obtain the optimal order of polynomial stability.

Description: In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi’s theory. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation.

Description: In this article, we study the rigidity of planar central configurations in the non-collinear n -body problem relative to the change of masses. More precisely, we study central configurations for which it is possible to change the values of k masses keeping fixed all the positions and the values of the masses of the other n − k bodies and still have central configurations. Here, we consider the cases k  = 1 and k  = 2. The central configurations that have such properties are closely related to the so-called stacked central configurations and super central configurations.

Description: Moving boundary problems of generalised Stefan type are considered for the Harry Dym equation via a Painlevé II symmetry reduction. Exact solutions of such nonlinear boundary value problems are obtained in terms of Yablonski–Vorob’ev polynomials corresponding to an infinite sequence of values of the Painlevé II parameter. The action of two kinds of reciprocal transformation on the moving boundary problems is described.

Description: We construct a family of periodic piezoelectric waveguides Π ɛ , depending on a small geometrical parameter, with the following property: as ɛ → +0, the number of gaps in the essential spectrum of the piezoelectricity problem on Π ɛ grows unboundedly.

Description: The elastostatic bending of an arbitrarily loaded bimaterial plate with a circular interface is analysed. It is shown that the deflections in the composite solid are directly related to the deflection in the corresponding homogeneous material by integral and differential operators. It is further shown that, by a simple transformation of elastic constants, the Airy stress function induced in the composite by a stretching singularity can be deduced from the deflection induced by a bending singularity. This result is significant for reduction of mathematical labour and for systematic construction of solutions for more complex structures with circular geometry.

Description: In this paper, we derive the sampling theorem associated with a Sturm–Liouville problem which has two points of discontinuity and contains an eigenparameter in a boundary condition and also two transmission conditions. We establish briefly spectral properties of the problem, and then, we prove the sampling theorem associated with the problem.

Description: In this work, we consider a semi-infinite expanse of a rarefied gas bounded by its plane condensed phase on which evaporation takes place. The analysis is based on the BGK model derived from the Boltzmann equation. In particular, the strong evaporation problem is considered, where nonlinear aspects have to be taken into account. We present the complete development of a closed form solution for evaluating density, velocity and temperature perturbations. Numerical results are presented and discussed.

Description: It is shown that the analysis of radiative heat transport across a nanoscale gap cannot ignore the correlation between radiations from the different sides of the gap. This correlation can be neglected in two cases: when the gap is considerably wider than the dominant wavelength of radiation and when the temperatures on different sides of the gap are equal.

Description: In large-area organic light-emitting diodes (OLEDs), spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors, we present a stationary thermistor model based on the heat equation for the temperature coupled to a p -Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p -Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions, we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder’s fixed-point theorem.

Description: In this paper, we consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient  μ is large enough.

Description: The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. It is a nonlinear evolution equation. In this paper, the well-posedness of bounded solutions for the homogeneous initial boundary value problem and the Cauchy problem associated with this equation are studied.

Description: In this paper, we study a system of Schrödinger–Poisson equation $$\left\{\begin{array}{ll}-\Delta u+(\lambda a(x)+a_0(x))u+K(x)\phi u=|u|^{p-2}u, & \quad x \in\mathbb{R}^3, \\- \Delta \phi=K(x)u^2,& \quad x \in \mathbb{R}^3 \end{array} \right.$$ where \({p \in (4,6)}\) and \({\lambda}\) is a parameter. We require that \({a(x) \geq 0}\) and has a bounded potential well \({\Omega = a^{-1}(0)}\) . Combining this with other suitable assumptions on Ω, a 0 and K , we obtain the existence of multi-bump-type solution \({u_\lambda}\) when \({\lambda}\) is large via variational methods.

Description: For \({m=1,2,3,}\) we consider differential systems of the form $$x'\,=\,F_{0}(t,x)\,+\,\sum_{i=1}^{m}\,\varepsilon^{i} F_{i}(t,x)+\varepsilon^{m+1} R(t,x,\varepsilon),$$ where \({F_i:\mathbb{R} \times \mathcal{D}\,\rightarrow\,\mathbb{R}^{n}}\) and \({R:\mathbb{R}\,\times\,\mathcal{D}\,\times\,(-\varepsilon_{0},\varepsilon_{0})\,\rightarrow\,\mathbb{R}^{n}}\) are \({\mathcal{C}^{m+1}}\) functions, and \({T}\) -periodic in the first variable, being \({\mathcal{D}}\) an open subset of \({\mathbb{R}^{n}}\) , and \({\varepsilon}\) a small parameter. For such system, we assume that the unperturbed system x ′  =   F 0 ( t , x ) has a k -dimensional manifold of periodic solutions with k  ≤  n . We weaken the sufficient assumptions for studying the periodic solutions of the perturbed system when \({|\varepsilon|\, 〉 \,0}\) is sufficiently small.

Description: We study the behaviour as \({t \to \infty}\) of solutions ( c j ( t )) to the Redner–Ben-Avraham–Kahng coagulation system with positive and compactly supported initial data, rigorously proving and slightly extending results originally established in Redner et al. (J Phys A Math Gen 20:1231–1238, 1987 ) by means of formal arguments.

Description: We prove the global-in-time and uniform-in- \({(\epsilon_1,\epsilon_2)}\) of strong solutions to the isentropic Navier–Stokes–Maxwell system in a bounded domain, when \({\epsilon_1}\) is the Mach number, and \({\epsilon_2}\) is the dielectric constant. Consequently, we obtain the convergences of compressible Navier–Stokes–Maxwell system to the incompressible Navier–Stokes–Maxwell system ( \({\epsilon_1\rightarrow 0}\) and \({\epsilon_2}\) fixed), the compressible magnetohydrodynamic equations ( \({\epsilon_1}\) fixed and \({\epsilon_2\rightarrow 0}\) ) or the incompressible magnetohydrodynamic equations ( \({\epsilon_1\rightarrow 0}\) and \({\epsilon_2\rightarrow 0}\) ) for well-prepared data.

Description: This paper concerns about the Cauchy problem for the three-dimensional Navier–Stokes equations and provides a regularity criterion in terms of the gradient of one velocity component. This improves previous results.

Description: This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t 〉 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t 〉 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\) , where the diffusion function D ( u ) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u 〉 0 \end{array}$$ with some c D  〉 0 and m  〉 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014 ) under the condition that \({m 〉 2-\frac{2}{n}}\) can be relaxed to \({m 〉 2-\frac{6}{n+4}}\) .

Description: In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.

Description: The aim of the work described in this paper is to determine, via an asymptotic analysis, the limiting form of the free energy governing in the first case 3D ferromagnetic nanowires of infinite length in the limit and in the second case 3D thin films which become infinite when their thickness is vanished. A 1D limit problem on the nanowires and a 2D limit problem on the thin films are obtained.

Description: We analytically investigate the contribution of surface piezoelectricity to the interaction between a piezoelectric screw dislocation and a finite crack in a hexagonal piezoelectric solid. The piezoelectric screw dislocation suffers jumps in the displacement and in the electric potential across the slip plane, and meanwhile it is subjected to a line force and a line charge at its core. The original boundary value problem is reduced to two sets of coupled first-order Cauchy singular integro-differential equations by considering a distribution of line dislocations, electric-potential-dislocations, line forces and line charges on the crack. By using a diagonalization method, the two sets of equations are decoupled into four independent singular integro-differential equations, each of which can be numerically solved by means of the collocation method. Our analysis reveals that in general the stresses, strains, electric displacements and electric fields exhibit both the weak logarithmic and the strong square root singularities at the two crack tips. The image force acting on the piezoelectric screw dislocation due to its interaction with the finite crack is calculated.

Description: We propose a thermodynamically consistent general-purpose model describing diffusion of a solute or a fluid in a solid undergoing possible phase transformations and damage, beside possible visco-inelastic processes. Also heat generation/consumption/transfer is considered. Damage is modelled as rate-independent. The applications include metal-hydrogen systems with metal/hydride phase transformation, poroelastic rocks, structural and ferro/para-magnetic phase transformation, water and heat transport in concrete, and if diffusion is neglected, plasticity with damage and viscoelasticity, etc. For the ensuing system of partial differential equations and inclusions, we prove existence of solutions by a carefully devised semi-implicit approximation scheme of the fractional-step type.

Description: We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\Delta u+V(x)u+\lambda \phi(x)u =f(u), \ &\quad x \in \mathbb{R}^3,\\ -\Delta \phi=u^2, \ &\quad x \in \mathbb{R}^3,\end{array}\right.$$ where V ( x ) is a smooth function and λ is a positive parameter. Because the so-called nonlocal term \({\lambda \phi_u(x)u}\) is involving in the equation, the variational functional of the equation has totally different properties from the case of \({\lambda=0}\) . Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution \({u_\lambda}\) . Moreover, we show that any sign-changing solution of the problem has an energy exceeding twice the least energy, and for any sequence \({\{\lambda_n\} \rightarrow 0^+(n \rightarrow \infty)}\) , there is a subsequence \(\{\lambda_{n_k}\}\) , such that \({u_{\lambda_{n_k}}}\) converges in \({H^1(\mathbb{R}^3)}\) to \({u_0}\) as \({k\rightarrow \infty}\) , where \({u_0}\) is a sign-changing solution of the following equation $$-\Delta u+V(x)u=f(u),\quad \ x \in \mathbb{R}^3$$ .

Description: Agrawal’s (Q J Mech Appl Math, 10:42–44, 1957 ) stagnation-point flow problem is extended to flow impingement normal to a uniformly rotating disk. This is the analog of the extension of Homann’s (Z Angew Math Mech (ZAMM), 16:153–164, 1936 ) stagnation flow when impinging on a rotating disk as reported by Hannah (Rep Mem Aerosp Res Coun Lond 2772, 1947 ). While both oncoming stagnation flows are axisymmetric, in the far field Homann’s stagnation flow is irrotational while Agrawal’s is rotational. A similarity reduction of the Navier–Stokes equations yields a pair of coupled ordinary differential equations governed by a dimensionless rotation rate σ . Integrations were carried out up to σ = 30 beyond which the equations become stiff and solution independence of integration length cannot be ensured. Results for the radial and azimuthal shear stresses are presented along with the strength of the flow induced into the boundary layer and the thickness of the azimuthal flow boundary layer. Analytic results found at σ = 0 are shown to be in excellent agreement with the numerical calculations. Sample velocity profiles for the radial and azimuthal flows are presented.

Description: We consider two-fluid Euler–Maxwell equations for magnetized plasmas composed of electrons and ions. By using the method of asymptotic expansions, we analyze the combined non-relativistic and quasi-neutral limit for periodic problems with well-prepared initial data. It is shown that the small parameter problems have a unique solution existing in a finite time interval where the corresponding limit problems (compressible Euler equations) have smooth solutions. The proof is based on energy estimates for symmetrizable hyperbolic equations and on the exploration of the coupling between the Euler equations and the Maxwell equations.

Description: Resorting to the characteristic polynomial of Lax matrix for the Mikhailov–Shabat–Sokolov hierarchy associated with a \({3 \times 3}\) matrix spectral problem, we introduce a trigonal curve, from which we deduce the associated Baker–Akhiezer function, meromorphic functions and Dubrovin-type equations. The straightening out of the Mikhailov–Shabat–Sokolov flows is exactly given through the Abel map. On the basis of these results and the theory of trigonal curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire Mikhailov–Shabat–Sokolov hierarchy.

Description: This paper is devoted to the following chemotaxis system $$\left\{ \begin{array}{llll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v),\quad &x\in \Omega,\quad t〉0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t〉0,\end{array} \right.$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) ( \({n\geq2}\) ), not necessarily being convex. There are some constants \({c_D 〉 0}\) , \({c_S 〉 0}\) , \({m\in\mathbb{R}}\) and \({q\in\mathbb{R}}\) such that $$D(u) \geq c_D(u+1)^{m-1} \quad\text{and} \quad S(u)\leq c_S(u+1)^{q-1}\quad for all \,\,\,u\geq0.$$ If \({q 〈 m+\frac{n+2}{2n}}\) , it is shown that the model possesses a unique global classical solution which is uniformly bounded; if \({q 〈 \frac{m}{2}+\frac{n+2}{2n}}\) , the global existence of solution is established.

Description: In order to design the microstructure of metamaterials showing high toughness in extension (property to be shared with muscles), it has been recently proposed (Dell’Isola et al. in Z Angew Math Phys 66(6):3473–3498, 2015 ) to consider pantographic structures. It is possible to model such structures at a suitably small length scale (resolving in detail the interconnecting pivots/cylinders) using a standard Cauchy first gradient theory. However, the computational costs for such modelling choice are not allowing for the study of more complex mechanical systems including for instance many pantographic substructures. The microscopic model considered here is a quadratic isotropic Saint-Venant first gradient continuum including geometric nonlinearities and characterized by two Lamé parameters. The introduced macroscopic two-dimensional model for pantographic sheets is characterized by a deformation energy quadratic both in the first and second gradient of placement. However, as underlined in Dell’Isola et al. (Proc R Soc Lond A 472(2185):20150790, 2016 ), it is needed that the second gradient stiffness depends on the first gradient of placement if large deformations and large displacements configurations must be described. The numerical identification procedure presented in this paper consists in fitting the macro-constitutive parameters using several numerical simulations performed with the micro-model. The parameters obtained by the best fit identification in few deformation problems fit very well also in many others, showing that the reduced proposed model is suitable to get an effective model at relevantly lower computational effort. The presented numerical evidences suggest that a rigorous mathematical homogenization result most likely holds.

Description: The dynamics of mechanical systems with a finite number of degrees of freedom (discrete mechanical systems) is governed by the Lagrange equation which is a second-order differential equation on a Riemannian manifold (the configuration manifold). The handling of perfect (frictionless) unilateral constraints in this framework (that of Lagrange’s analytical dynamics) was undertaken by Schatzman and Moreau at the beginning of the 1980s. A mathematically sound and consistent evolution problem was obtained, paving the road for many subsequent theoretical investigations. In this general evolution problem, the only reaction force which is involved is a generalized reaction force, consistently with the virtual power philosophy of Lagrange. Surprisingly, such a general formulation was never derived in the case of frictional unilateral multibody dynamics. Instead, the paradigm of the Coulomb law applying to reaction forces in the real world is generally invoked. So far, this paradigm has only enabled to obtain a consistent evolution problem in only some very few specific examples and to suggest numerical algorithms to produce computational examples (numerical modeling). In particular, it is not clear what is the evolution problem underlying the computational examples. Moreover, some of the few specific cases in which this paradigm enables to write down a precise evolution problem are known to show paradoxes: the Painlevé paradox (indeterminacy) and the Kane paradox (increase in kinetic energy due to friction). In this paper, we follow Lagrange’s philosophy and formulate the frictional unilateral multibody dynamics in terms of the generalized reaction force and not in terms of the real-world reaction force. A general evolution problem that governs the dynamics is obtained for the first time. We prove that all the solutions are dissipative; that is, this new formulation is free of Kane paradox. We also prove that some indeterminacy of the Painlevé paradox is fixed in this formulation.

Description: Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible and illustrate that while epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of “epigenetic load” and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a “proof of concept” of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.

Description: This paper is dedicated to studying the following Schrödinger–Poisson system $${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$ where V , K are positive continuous potentials, f is a continuous function and \({\lambda}\) is a positive parameter. We develop a direct approach to establish the existence of one ground state sign-changing solution \({u_\lambda}\) with precisely two nodal domains, by introducing a weaker condition that there exists \({\theta_0\in (0,1)}\) such that $$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t 〉 0, \tau\ne 0$$ than the usual increasing condition on \({f(t)/|t|^3}\) . Under the above condition, we also prove that the energy of any sign-changing solution is strictly larger than two times the least energy, and give a convergence property of \({u_\lambda}\) as \({\lambda\searrow 0}\) .

Description: We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.

Description: In this paper, we study the large time behavior of the isentropic compressible Navier–Stokes–Maxwell system introduced by Jiang and Li (Nonlinearity 25(6):1735–1752, 2012 ) in the whole space \({{\mathbb{R}}^3}\) when the initial data are a small perturbation of some given constant state. We obtain the desired result through taking the refined analysis on the time decay property and Green’s function of the linearized system. Moreover, we also obtain the optimal time rate of the solution.

Description: Motivated by the theory of kinetic models in gas dynamics, we obtain an integral representation of lower semicontinuous functions on \({{\mathbb{R}}^d,}\) \({d\geq1}\) . We use the representation to study the problem of compactness of a family of the solutions of the discrete time BGK model for the compressible Euler equations. We determine sufficient conditions for strong compactness of moments of kinetic densities, in terms of the measures from their integral representations.

Description: Viscous flow past an ensemble of polydisperse spherical drops is investigated under thermocapillary effects. We assume that the collection of spherical drops behaves as a porous media and estimates the hydrodynamic interactions analytically via the so- called cell model that is defined around a specific representative particle. In this method, the hydrodynamic interactions are assumed to be accounted by suitable boundary conditions on a fictitious fluid envelope surrounding the representative particle. The force calculated on this representative particle will then be extended to a bed of spherical drops visualized as a Darcy porous bed. Thus, the “effective bed permeability” of such a porous bed will be computed as a function of various parameters and then will be compared with Carman–Kozeny relation. We use cell model approach to a packed bed of spherical drops of uniform size (monodisperse spherical drops) and then extend the work for a packed bed of polydisperse spherical drops, for a specific parameters. Our results show a good agreement with the Carman–Kozeny relation for the case of monodisperse spherical drops. The prediction of overall bed permeability using our present model agrees well with the Carman–Kozeny relation when the packing size distribution is narrow, whereas a small deviation can be noted when the size distribution becomes broader.

Description: In this paper, we obtain global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Our works are consistent with the corresponding works by Elgindi–Widmayer (SIAM J Math Anal 47:4672–4684, 2015 ) in the special case \({A=\kappa=1}\) . In addition, our result concerning the SQG equation can be regarded as the borderline case of the work by Cannone et al. (Proc Lond Math Soc 106:650–674, 2013 ).

Description: In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier–Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number \({\epsilon\in(0,\overline{\epsilon}]}\) and time \({t\in[0,\infty)}\) are established by deriving a differential inequality with decay property, where \({\overline{\epsilon} \in(0,1]}\) is a constant. Based on these uniform estimates, the global solution of full compressible Navier–Stokes equations with “well-prepared” initial conditions converges to the one of isentropic incompressible Navier–Stokes equations as the Mach number goes to zero.

Description: Umeda et al. (Jpn J Appl Math 1:435–457, 1984 ) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices \({L}\) and \({B^{jk}(j,k=1,\ldots,n)}\) are both assumed to be real symmetric. So far there are no general results in case that \({L}\) and \({B^{jk}}\) are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of \({L^{1}({\mathbb{R}}^3)}\) - \({L^2({\mathbb{R}}^3)}\) type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of \({L^p({\mathbb{R}}^n)}\) - \({L^{q}({\mathbb{R}}^n)}\) - \({L^{r}({\mathbb{R}}^n)}\) estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.

Description: The present work aims to investigate the mechanical oscillatory behavior of ions, and in particular \({{\rm Li}^{+}, {\rm Na}^{+}, {\rm Rb}^{+}}\) and \({{\rm Cl}^{-}}\) ions, inside a cyclo[(– d -Ala– l -Ala) 4 –] peptide nanotube using the continuum approximation along with the 6–12 Lennard–Jones (LJ) potential function. Assuming that each peptide unit is comprised of an inner and an outer tube, the van der Waals (vdW) potential energy and interaction force between an ion and a cyclic peptide nanotube (CPN) are determined analytically. With respect to the present formulations, a detailed parametric study is conducted on the vdW potential energy and interaction force distributions by varying the number of peptide units. Employing the conservation of mechanical energy principle, a novel expression for precise evaluation of oscillation frequency is introduced. To verify the accuracy of the proposed frequency expression, the results obtained from energy equation are compared with the ones predicted through solving the equation of motion numerically. The effects of number of peptide units and initial conditions including initial separation distance and velocity on the oscillatory behavior of various ions inside CPNs are explored. Among the considered ions, \({{\rm Cl}^{-}}\) ion is found to generate the highest frequency. According to the potential energy profile, one oscillatory zone for one peptide unit and different oscillatory zones for more than one peptide unit are observed. Numerical results indicate that optimal frequency decreases with increasing the number of peptide units and almost remains unchanged when the number of peptide units exceeds four.

Description: This paper is concerned with the long-time behaviour of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows introduced in Lin and Liu (Commun Pure Appl Math, 48:501–537, 1995 ) with a non-autonomous forcing bulk term and order parameter field boundary conditions. In this paper, we prove the existence of pullback attractors and estimate the upper bound of its fractal dimension under some suitable assumptions.

Description: The essential ideas and equations of classic plasticity and hyperplasticity are successively recalled and compared, in order to highlight their differences and complementarities. The former is based on the mathematical framework proposed by Hill (The mathematical theory of plasticity. Oxford University Press, Oxford, 1950 ), whereas the latter is founded on the orthogonality hypothesis of Ziegler (An introduction to thermomechanics. Elsevier, North-Holland, 1983 ). The main drawback of classic plasticity is the possibility of violating the second principle of thermodynamics, while the relative ease to conjecture the yield function in order to approach experimental results is its main advantage. By opposition, the a priori satisfaction of thermodynamic principles constitutes the chief advantage of hyperplasticity theory. Noteworthy is also the fact that this latter approach allows a finer energy partition; in particular, the existence of frozen energy emerges as a natural consequence from its theoretical formulation. On the other hand, the relative difficulty to conjecture an efficient dissipation function to produce accurate predictions is its main drawback. The two theories are thus better viewed as two complementary approaches. Following this comparative study, a methodology to extend the hyperplasticity approach initially developed for dry or saturated materials to the case of partially saturated materials, accounting for interface energies and suction effects, is developed. A particular example based on the yield function of modified Cam–Clay model is then presented. It is shown that the approach developed leads to a model consistent with other existing works.

Description: In this paper, we study the existence and decay of a transmission problem for the plate equation with a memory condition on part of the boundary. First, we prove the global existence of weak solution by using Faedo–Galerkin’s method and compactness arguments. Then, without imposing \({u_0 = \frac{\partial u_0}{\partial\nu} = 0}\) on \({\Gamma_2}\) , two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data and thus generalize some previous results existing in the literature.

Description: We study the stationary Keller–Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing’s instability as the chemotaxis rate \({\chi}\) surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover, we show that solutions stay strictly positive in the continuum of each branch. The stabilities of these steady-state solutions are well studied when the creation and degradation rate of the chemical is assumed to be a linear function. Finally, we investigate the asymptotic behaviors of the monotone steady states. We construct solutions with interesting patterns such as a boundary spike when the chemotaxis rate is large enough and/or the cell motility is small.

Description: The problem of ring molecules and macromolecules arises in a number of contexts in physical chemistry. Perhaps the simplest example of a seven-membered loop is cycloheptane \({\rm C_{7}\rm H_{14}}\) , which is a molecule where the carbon–carbon bonds form a regular seven-membered loop. However, it is possible to envisage much more complicated arrangements of proteins in chains comprising straight rigid sections linked in ways that enforce the same angle at all of the joins. In this paper, we present a coordinate system that reduces the problem to four free variables and three constraints. We then survey the solutions numerically and find that there are families of solutions for all join angles \({\theta}\) between \({\pi/7}\) and \({5\pi/7}\) with fixed planar solutions existing for \({\theta = \pi/7}\) , \({3\pi/7}\) and \({5\pi/7}\) . The available families of solutions undergo a major reorganisation at the join angle \({\theta = \pi/3}\) , where 28 intersecting solutions form a single connected network of configurations.

Description: Reaction–diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N -dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.

Description: In this paper, a novel evolution equation for capillaries growth is proposed. An essential ingredient is given by the consideration of nutrient supply, for which a novel constitutive equation is also proposed. The biological and mechanical stimuli are assumed to depend in non-local way on relevant kinematical descriptors. The integro-differential equations governing the system evolution are extremely sensitive to parameter variations. However, it was possible to perform some meaningful numerical simulations in which osteophyte onset has been observed. While the choice of these parameters was judiciously driven by biomechanical “a priori” knowledge, the mathematical problems concerning well-posedness, stability and continuous dependence of solutions seem to be very challenging and will be object of future investigations. This effort seems motivated by the fact that proposed equations are, to our knowledge, the first ones allowing for the prediction of osteophytes onset.

Description: An initial boundary value problem for the 3D Kawahara equation posed on a channel-type domain was considered. The existence and uniqueness results for global regular solutions as well as exponential decay of small solutions in the H 2 -norm were established.

Description: In this paper, we study the asymptotic behavior of viscosity solutions to boundary blow-up elliptic problem \({\Delta_{\infty}u=b(x)f(u),\, x\in\Omega,\,u|_{\partial\Omega}=+\infty,}\) where \({\Omega}\) is a bounded domain with C 2 -boundary in \({\mathbb{R}^{N}}\) , \({b\in \rm C(\bar{\Omega})}\) is positive in \({\Omega}\) , which may be vanishing on the boundary, \({f\in C^{1}([0, \infty))}\) is regularly varying or is rapidly varying at infinity.

Description: This paper is devoted to studying the simplified nonlinear chromatography equations by introducing the change of state variables. The Riemann solutions containing delta shock waves are presented. In order to study wave interactions of delta shock waves with elementary waves, the global structure of solutions is constructed completely when the initial data are taken as three pieces of constants and the delta shock waves are included. In particular, the strength of delta shock wave is expressed explicitly and the delta contact discontinuity is discovered during the process of wave interactions. Moreover, by analyzing the limits of the solutions as the middle region vanishes, we observe that the Riemann solutions are stable for such a local small perturbation of the Riemann initial data.

Description: In this paper, we consider the quasilinear chemotaxis–haptotaxis system ⋆ $$\begin{aligned}\left\{\begin{array}{ll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S_1(u)\nabla v)-\nabla\cdot(S_2(u)\nabla w)+uf(u,w),\quad x\in\Omega,~t 〉 0,v_t=\Delta v-v+u,\quad x\in\Omega,~t 〉 0,w_t=-vw,\quad x\in\Omega,~t 〉 0\end{array} \right.\end{aligned}$$ in a bounded smooth domain \({\Omega\subset\mathbb{R}^n~(n\geq1)}\) under zero-flux boundary conditions, where the nonlinearities \({D,~S_1}\) and \({S_2}\) are assumed to generalize the prototypes $$D(u)=C_{D}(u+1)^{m-1},~S_1(u)=C_{S_1}u(u+1)^{q_1-1} \quad {\mathrm{and}} \quad S_2(u)=C_{S_2}u(u+1)^{q_2-1}$$ with \({C_D,C_{S_1},C_{S_2} 〉 0,~m,q_1,q_2\in\mathbb{R}}\) and \({f(u,w)\in C^1([0,+\infty)\times[0,+\infty))}\) fulfills $$f(u,w)\leq r-bu\quad {\mathrm{for all}}~~u\geq 0\quad {\mathrm{and}} \quad w\geq 0,$$ where \({r 〉 0,~b 〉 0.}\) Assuming nonnegative initial data \({u_0(x)\in W^{1,\infty}(\Omega),v_0(x)\in W^{1,\infty}(\Omega)}\) and \({w_0(x)\in C^{2,\alpha}(\bar\Omega)}\) for some \({\alpha\in(0,1),}\) we prove that (i) for \({n\leq2,}\) if \({\max\{q_1,q_2\} 〈 m+\frac{2}{n}-1,}\) then \({(\star)}\) has a unique nonnegative classical solution which is globally bounded, (ii) for \({n 〉 2,}\) if \({\max\{q_1,q_2\} 〈 m+\frac{2}{n}-1}\) and \({m 〉 2-\frac{2}{n}}\) or \({\max\{q_1,q_2\} 〈 m+\frac{2}{n}-1}\) and \({m\leq 1,}\) then \({(\star)}\) has a unique nonnegative classical solution which is globally bounded.

Description: In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015 ).

Description: The aim of this paper is to establish the global well-posedness and large-time asymptotic behavior of the strong solution to the Cauchy problem of the two-dimensional compressible Navier–Stokes equations with vacuum. It is proved that if the shear viscosity \({\mu}\) is a positive constant and the bulk viscosity \({\lambda}\) is the power function of the density, that is, \({\lambda=\rho^{\beta}}\) with \({\beta \in [0,1],}\) then the Cauchy problem of the two-dimensional compressible Navier–Stokes equations admits a unique global strong solution provided that the initial data are of small total energy. This result can be regarded as the extension of the well-posedness theory of classical compressible Navier–Stokes equations [such as Huang et al. (Commun Pure Appl Math 65:549–585, 2012 ) and Li and Xin ( http://arxiv.org/abs/1310.1673 ) respectively]. Furthermore, the large-time behavior of the strong solution to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations had been also obtained.

Description: We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves (Hamilton et al. in J Acoust Soc Am 97:891–897, 1995 ), surface waves on current-vortex sheets in incompressible MHD (Alì and Hunter in Q Appl Math 61(3):451–474, 2003 ; Alì et al. in Stud Appl Math 108(3):305–321, 2002 ) and on the incompressible plasma–vacuum interface (Secchi in Q Appl Math 73(4):711–737, 2015 ). The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in Hunter (J Hyperbolic Differ Equ 3(2):247–267, 2006 ), Secchi (Q Appl Math 73(4):711–737, 2015 ). In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

Description: The problem of free vibrations of the Timoshenko beam model has been addressed in the first part of this paper. A careful analysis of the governing equations has shown that the vibration spectrum consists of two parts, separated by a transition frequency , which, depending on the applied boundary conditions, might be itself part of the spectrum. Here, as an extension, the case of a doubly clamped beam is considered. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided: this allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. This case is a meaningful example of more general ones, where the wave-numbers equation cannot be written in a factorized form and hence must be solved by general root-finding methods for nonlinear transcendental equations. These theoretical results can be used as further benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. finite element models.

Description: The distribution of collagen fibers across articular cartilage layers is statistical in nature. Based on the concepts proposed in previous models, we developed a methodology to include the statistically distributed fibers across the cartilage thickness in the commercial FE software COMSOL which avoids extensive routine programming. The model includes many properties that are observed in real cartilage: finite hyperelastic deformation, depth-dependent collagen fiber concentration, depth- and deformation-dependent permeability, and statistically distributed collagen fiber orientation distribution across the cartilage thickness. Numerical tests were performed using confined and unconfined compressions. The model predictions on the depth-dependent strain distributions across the cartilage layer are consistent with the experimental data in the literature.

Description: In this work, the three-dimensional numerical resolution of a complex mathematical model for the blood coagulation process is presented. The model was illustrated in Fasano et al. (Clin Hemorheol Microcirc 51:1–14, 2012 ), Pavlova et al. (Theor Biol 380:367–379, 2015 ). It incorporates the action of the biochemical and cellular components of blood as well as the effects of the flow. The model is characterized by a reduction in the biochemical network and considers the impact of the blood slip at the vessel wall. Numerical results showing the capacity of the model to predict different perturbations in the hemostatic system are discussed.

Description: 〈h3〉Abstract〈/h3〉 〈p〉Based on a symbolic computation approach and the Hirota’s bilinear method, the multiple rogue wave solutions of the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation, which can be regarded as a generalization of the generalized rational solutions of Boussinesq equation proposed by Clarkson and Dowie, are constructed. The first-order, second-order, third-order and fourth-order rogue waves are systematically discussed. Moreover, the maximal amplitude and the minimum amplitude of the first-order rogue wave solutions are given. By choosing some specific parameters of these rogue wave solutions, their dynamic behaviors are analyzed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The article constructs the dual variational form of the model based on the presented differential form of a mathematical model describing the steady-state process of thermal conductivity in a plane or circular cylindrical layer of a solid dielectric at an alternating voltage. This form includes the main and alternative functionals, whose stationary points analysis makes it possible to establish a combination of the determining parameters corresponding to the occurrence of thermal breakdown of the dielectric layer. An example of the analysis of stationary points for two variants of test functions that are admissible for these functionals and approximating the steady temperature distribution in the dielectric layer is given. The estimation of the approximation error, which makes it possible to choose the function closest to the limiting temperature distribution in the layer preceding the thermal breakdown of the dielectric, is presented.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we investigate a nonlocal SIS epidemic model with double free boundaries. The existence, uniqueness and some estimates of the global solution are discussed first. Then, with the help of studying the long-time behavior of the solution to a Cauchy problem for a nonhomogeneous heat equation, the long-time behavior of the solution to the SIS free boundary problem is obtained for the disease vanishing case. At last, some sufficient conditions for the disease vanishing are established.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This article addresses several problems that are related to the elastic stability of thin shells and that are due to inconsistencies between the experimental data and predictions based on the equations of the shallow-shell theory. The above contradictions are solved within the three-dimensional nonlinear theory of elasticity. In particular, the dynamic approach enables a prediction of the moment of bifurcation for very thin shells under momentless stress through the analysis of asymptotically built two-dimensional equations. Appearance and development of the dents and patterns in initially ideal shells that precede the buckling are also explained in this article. The dents represent solitonic waves that are detectible by acoustic devices. Therefore, it is possible to reduce the risk of failure of thin shells by using acoustic devices to monitor their conditions. The article covers two types of experiments with thin shells when the loads are close to buckling. These experiments enable to assess the safety buckling factor under technical operation of shells.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper deals with the 3D incompressible MHD equations with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions is established in the vacuum cases, provided the assumption that 〈span〉 〈span〉\({\bar{\rho }}+\Vert H_0\Vert _{L^3}\)〈/span〉 〈/span〉 is suitably small with large velocity, which extends the recent work (Song in Z Angew Math Phys 69(2):23, 〈span〉2018〈/span〉) to the global one. Furthermore, the exponential decay of the solution is also obtained. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In the present work, we search for the propagation of nonlinear shear horizontal waves (SH) in a finite thickness plate which consists of heterogeneous, isotropic, and generalized neo-Hookean materials. In the analysis, we apply the method of multiple scales and strike a balance between the nonlinearity and the dispersion. Then, the self-modulation of nonlinear SH waves can be given by a nonlinear Schrödinger equation which has the well-known dark solitary solution. Consequently, we show that the dark solitary SH waves can propagate in this plate. Moreover, we take the effects of heterogeneity and the nonlinearity into account for these waves.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study a reaction–diffusion system involving coupled sources and nonlinear nonlocal boundary flux. By using the comparison principle, we give the criteria on blow-up and global solutions. There exist no nontrivial global solutions under some conditions on the coefficients. After classifying simultaneous and non-simultaneous blow-up of the components of solutions, we obtain blow-up rates and sets. Lower and upper bounds of blow-up time are determined quantificationally for all dimensions of the space domains.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we determine in second order in the fine structure constant the energy levels of Weber’s Hamiltonian that admit a quantized torus. Our formula coincides with the formula obtained by Wesley using the Schrödinger equation for Weber’s Hamiltonian. We follow the historical approach of Sommerfeld. This shows that Sommerfeld could have discussed the fine structure of the hydrogen atom using Weber’s electrodynamics if he had been aware of the at-his-time-already-forgotten theory of Wilhelm Weber (1804–1891). 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this note, we make two revisions of the paper [〈span〉2〈/span〉]. The first one is the asymptotic behavior of the energy functional as 〈span〉 〈span〉\(t\rightarrow T\)〈/span〉 〈/span〉 (see [〈span〉2〈/span〉, Theorem 1.6]), where 〈em〉T〈/em〉 is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [〈span〉2〈/span〉, Theorem 1.8]).〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Consider the Cauchy problem of incompressible Navier–Stokes equations in three dimension with axisymmetric initial data. If the swirl component satisfies some certain regularity criteria in weighted spaces, the existence of the time-global solution has been proved in our previous work. However, it is more or less evident that the solution is anisotropic in vertical direction (i.e., the direction parallel to the axis of symmetry) and the horizontal directions (in the plane perpendicular to the axis of symmetry). In this paper, we are interested in constructing regularity criteria and time-global solution in anisotropic Lebesgue spaces.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study the well-posedness of a simple model of boundary layer for rotating fluids between two concentric spheres near the equator. We show that this model can be seen as a degenerate elliptic equation, for which we prove an existence result thanks to a Lax–Milgram-type lemma. We also prove uniqueness under an additional integrability assumption and present a transparent boundary condition for such layers. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We are concerned with global well-posedness of the three-dimensional Vlasov–Poisson system with radiation damping. First, we show that global 〈span〉 〈span〉\(C^1\)〈/span〉 〈/span〉 solutions verifying specified decay conditions are stable under small perturbations. As a consequence, we obtain that a small perturbation of a monopolar and spherically symmetric plasma launches a global 〈span〉 〈span〉\(C^1\)〈/span〉 〈/span〉 solution that preserves quasi-spherical symmetry at the macroscopic level. Second, we show that an initially quasi-neutral datum with 〈span〉 〈span〉\(C^1\)〈/span〉 〈/span〉 regularity launches a global classical solution that propagates quasi-neutrality at the macroscopic level. Finally, we obtain better decay estimates for the radiation damping in both cases.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper deals with a drift-diffusion system being subjected to Robin boundary conditions. Under appropriate hypotheses on the data, a local existence result in time is obtained by using a fixed-point argument combined with some a priori estimates. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Singular limits of the rotating stratified Boussinesq equations with ill-prepared data are investigated when the Froude number and Rossby number tend to zero at different rates. The reduced systems are derived, respectively, for the rotation-dominant and stratification-dominant cases through the developed three-scale fast averaging method.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study a diffusive prey–predator model with Beddington–DeAngelis functional response when the intraspecific crowding effect for the prey population disappears in some subdomain 〈span〉 〈span〉\(\Omega _0\)〈/span〉 〈/span〉 of their whole habitat. We are concerned about the global dynamics of the system and discuss it based on two coefficients: the growth rate of prey 〈span〉 〈span〉\(\lambda \)〈/span〉 〈/span〉 and that of predator 〈span〉 〈span〉\(\mu \)〈/span〉 〈/span〉. In particular, the results show that, with the degeneracy of prey population’s intraspecific crowding effect in the subdomain 〈span〉 〈span〉\(\Omega _0\)〈/span〉 〈/span〉, the density of prey may tend to infinity in 〈span〉 〈span〉\(\Omega _0\)〈/span〉 〈/span〉. On the other hand, the unboundedness of prey means that the solutions of the system lose compactness which may bring difficulty to investigate the long-time behavior of the solutions. Finally, some numerical simulations are presented to support and strengthen our theoretical analysis.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The main objective of this paper is to study the temporal approximation scheme of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics. We first prove the existence of a global attractor in 〈em〉V〈/em〉 by using the Sobolev compactness embedding theorem, and then we prove its global attractor and stationary statistical properties of the temporal approximation scheme convergence to those of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics as the time step goes to zero.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉A class of generalised Schrödinger elliptic problems involving concave–convex and other types of nonlinearities is studied. A reasonable overview about the set of solutions is provided when the parameters involved in the equation assume different real values. 〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Five two-dimensional steady-state general solutions for isotropic hygrothermoelastic media are established in this article. Firstly, four general solutions which are expressed in one function can be obtained based on the differential operator theory. Then, because of the Almansi’s theorem and appropriate transformation, the four general solutions can be converted to another four general solutions expressed in two harmonic functions. Finally, the more complete general solution expressed in four harmonic functions is obtained by superposition principle. As a checking example, based on the received complete general solution, the Green’s function for a line heat source combined with moisture source in the interior of infinite hygrothermoelastic plane is obtained. In addition, the Green’s function for a line heat source combined with moisture source on the surface of semi-infinite plane is obtained.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we obtain the global well-posedness and analyticity of the 3D fractional magnetohydrodynamics equations in the critical variable Fourier–Besov spaces, which can be seen as a meaningful complement to the corresponding results of the magnetohydrodynamics equations in usual Fourier–Besov spaces. Moreover, our results are also new for the MHD equations (i.e., in the case of the classical dissipation 〈span〉 〈span〉\(\alpha = 1\)〈/span〉 〈/span〉).〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl’s analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace–Beltrami operator.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent 〈span〉 〈span〉\(\gamma \)〈/span〉 〈/span〉 is close to 1, the global stability is established. In this paper, the initial perturbation can be large.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The scattering of quasiperiodic waves for a two-dimensional Helmholtz equation with a constant refractive index perturbed by a function which is periodic in one direction and of finite support in the other is considered. The scattering problem is uniquely solvable for almost all frequencies and formulas of Breit–Wigner and Fano type for the reflection and transmission coefficients are obtained in a neighborhood of the resonance (a pole of the reflection coefficient). We indicate also the values of the parameters involved which provide total transmission and reflection. For some exceptional frequencies and perturbations (when the imaginary part of the resonance vanishes) the scattering problem is not uniquely solvable and in the latter case there exist embedded Rayleigh–Bloch modes whose frequencies are explicitly calculated in terms of infinite convergent series in powers of the small parameter characterizing the magnitude of the perturbation.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we obtain the existence of two non-trivial solutions for a class of non-periodic Schrödinger lattice systems with perturbed terms when the nonlinearities are super-linear at infinity. In addition, several examples are given to illustrate our results. Our theorems appear to be the first such result.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉 In this paper, we consider a nonlinear divergence form parabolic equation with time-dependent coefficient and inhomogeneous Neumann boundary condition. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution 〈span〉 〈span〉\(u(\pmb {x},t )\)〈/span〉 〈/span〉 exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space and constructing the unified functionals, we obtain upper and lower bounds of the blow-up time 〈span〉 〈span〉\(t^*\)〈/span〉 〈/span〉.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we consider the following Kirchhoff problem 〈span〉 〈span〉$$\begin{aligned} -\left( a+b\int \limits _{\mathbb R^3} |\nabla u|^2\mathrm{d}x\right) \Delta u+V(|x|)u=Q(|x|)u^{p},\quad x\in \mathbb {R}^3, \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(a,b〉0\)〈/span〉 〈/span〉, 〈span〉 〈span〉\(1〈p〈5\)〈/span〉 〈/span〉, 〈em〉V〈/em〉(〈em〉r〈/em〉) and 〈em〉Q〈/em〉(〈em〉r〈/em〉) are bounded and positive functions. Assume that 〈em〉V〈/em〉(〈em〉r〈/em〉) and 〈em〉Q〈/em〉(〈em〉r〈/em〉) have the following expansions 〈span〉 〈span〉$$\begin{aligned} V(r)=1+\frac{d_1}{r^m}+O\left( \frac{1}{r^{m+\theta }}\right) ,\quad Q(r)=1+\frac{d_2}{r^n}+O\left( \frac{1}{r^{n+\kappa }}\right) , \end{aligned}$$〈/span〉 〈/span〉as 〈span〉 〈span〉\(r\rightarrow \infty \)〈/span〉 〈/span〉, for some 〈span〉 〈span〉\(d_1〉0,d_2\in \mathbb {R}\)〈/span〉 〈/span〉, 〈span〉 〈span〉\(m,n〉2\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\theta ,\kappa 〉0\)〈/span〉 〈/span〉. Infinitely many nonradial positive solutions are constructed either 〈span〉 〈span〉\(d_2〉0\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(m〈n\)〈/span〉 〈/span〉 or 〈span〉 〈span〉\(d_2〈0\)〈/span〉 〈/span〉.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of “shearing”: the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in opposite regimes. In particular, Hardy-type inequalities are derived in the regime of repulsive shearing.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper considers the chemotaxis–Navier–Stokes system with nonlinear diffusion and logistic-type degradation term 〈span〉 〈span〉$$\begin{aligned} {\left\{ \begin{array}{ll} n_t + u\cdot \nabla n = \nabla \cdot (D(n)\nabla n) - \nabla \cdot (n \chi (c) \nabla c) + \kappa n - \mu n^\alpha , &{}\quad x\in \Omega ,\ t〉0, \\ c_t + u\cdot \nabla c = \Delta c - nf(c), &{}\quad x \in \Omega ,\ t〉0, \\ u_t + (u\cdot \nabla )u = \Delta u + \nabla P + n\nabla \Phi + g, \quad \nabla \cdot u = 0, &{}\quad x \in \Omega ,\ t〉0, \end{array}\right. } \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(\Omega \subset \mathbb {R}^3\)〈/span〉 〈/span〉 is a bounded smooth domain; 〈span〉 〈span〉\(D \ge 0\)〈/span〉 〈/span〉 is a given smooth function such that 〈span〉 〈span〉\(D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}\)〈/span〉 〈/span〉 for all 〈span〉 〈span〉\(s\ge 0\)〈/span〉 〈/span〉 with some 〈span〉 〈span〉\(D_2 \ge D_1 〉 0\)〈/span〉 〈/span〉 and some 〈span〉 〈span〉\(m 〉 0\)〈/span〉 〈/span〉; 〈span〉 〈span〉\(\chi ,f\)〈/span〉 〈/span〉 are given smooth functions satisfying 〈span〉 〈span〉$$\begin{aligned} { \left( \frac{f}{\chi } \right) ' 〉0, \quad \left( \frac{f}{\chi } \right) '' \le 0, \quad (\chi f)' \ge 0 \quad \text{ on } \ [0,\infty ); } \end{aligned}$$〈/span〉 〈/span〉〈span〉 〈span〉\(\kappa \in \mathbb {R},\mu \ge 0,\alpha 〉1\)〈/span〉 〈/span〉 are constants. This paper shows existence of global weak solutions to the above system under the condition that 〈span〉 〈span〉$$\begin{aligned} m〉\frac{2}{3},\quad \mu \ge 0 \quad \text{ and }\quad \alpha 〉1 \end{aligned}$$〈/span〉 〈/span〉hold or that 〈span〉 〈span〉$$\begin{aligned} m〉 0, \quad \mu〉0 \quad \text{ and } \quad \alpha 〉 \frac{4}{3} \end{aligned}$$〈/span〉 〈/span〉hold. This result asserts that “strong” diffusion effect or “strong” logistic damping derives existence of global weak solutions even though the other effect is “weak” and can include previous works (Kurima and Mizukami in Nonlinear Anal Real World Appl, 〈span〉2018〈/span〉. 〈a href="http://arxiv.org/abs/1802.08807"〉arXiv:1802.08807〈/a〉 [math.AP]; Lankeit in Math Models Methods Appl Sci 26:2071–2109, 〈span〉2016〈/span〉; Winkler in Ann Inst H Poincaré Anal Non Linéaire 33:1329–1352, 〈span〉2016〈/span〉; Zhang and Li in J Differ Equ 259:3730–3754, 〈span〉2015〈/span〉).〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We analyze the Bresse system with partial boundary dissipation. Our main result is to prove that these dissipative mechanisms are enough to stabilize exponentially the whole system provided the wave propagation speeds are equal.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we consider an advection–diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions, we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive, but it becomes negative in some interval between them. Also the vanishing viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉An equilibrium problem for a 2D elastic body with thin inclusions and defects is analyzed. The presence of defects means that the problem is formulated in a non-smooth domain. The defects are characterized by a positive damage parameter. Nonlinear boundary conditions at the defect faces are imposed to prevent a mutual penetration between the faces. An existence of solutions is proved, and different formulations of the problem are proposed. We study an asymptotics of solutions with respect to the damage parameter and analyze the limit models. Moreover, we study the dependence of the solution on the rigidity parameter of the inclusions. In particular, passages to infinity and to zero of the rigidity parameter are investigated.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We are concerned with the following Schrödinger–Poisson systems 〈span〉 〈span〉$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+u+K(x)\phi (x)u=q(x)f(u), &{}\quad x\in {\mathbb {R}}^3, \\ -\Delta \phi =K(x)u^2, &{} \quad x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$〈/span〉 〈/span〉where 〈em〉f〈/em〉 is asymptotically cubic, 〈span〉 〈span〉\(\lim _{|x|\rightarrow \infty }K(x)=0\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\lim _{|x|\rightarrow \infty }q(x)=q_{\infty }〉0\)〈/span〉 〈/span〉. We establish the existence of bound state solutions to this problem by using the method developed in Szulkin and Weth [〈span〉20〈/span〉, 〈span〉21〈/span〉].〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The Lie symmetry analysis method and Bäcklund transformation method are proposed for finding similarity reduction and exact solutions to Euler equation and Navier–Stokes equation, respectively. By using symmetry reduction method, we reduce nonlinear partial differential equation to nonlinear ordinary differential equation. The infinitesimal generators and the soliton solutions to the Euler equation are obtained by Lie symmetry analysis method. Furthermore, the Bäcklund transformation of the Navier–Stokes equation is proposed to obtain the exact solution. We obtain the exact solutions to Navier–Stokes equation on background flow.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional calculus approach is a popular approach to analyze the anomalous relaxation processes and has been intensively studied in various dielectric materials. However, Capelas de Oliveira et al. proved that the memory kernels of Caputo type fractional derivatives must satisfy the initial value condition (IVC). But both kernels of the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative do not satisfy this condition. In this paper, we prove that the Caputo type derivative with a Prabhakar-like kernel satisfies the IVC and this derivative is in the framework of general Caputo fractional derivative (GC derivative). Corresponding anomalous relaxation model and its solution are discussed. Analysis result shows our model, as a direct extension of the Cole–Cole model, contains the Debye model and the fractional relaxation model as particular cases.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉This paper considers the dynamical behavior of solutions of constitutive systems for 1D compressible viscous and heat-conducting micropolar fluids. With proper constraints on initial data, we prove the existence of global attractors in generalized Sobolev spaces 〈span〉 〈span〉\(H_{\delta }^{(1)}\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(H_{\delta }^{(2)}\)〈/span〉 〈/span〉. These attractors are unique in corresponding phase spaces.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then, we use this result to prove the unique weak solvability of a new class of Moreau’s sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples, we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this work, we study the existence of nontrivial solutions for the following class of partial differential inclusion problem: 〈span〉 〈span〉$$\begin{aligned} -\Delta u+\left( 1+\lambda V(x)\right) u\in \partial _t F(x,u) \quad \text {in} \quad \mathbb {R}^N, \qquad \quad (P_\lambda ) \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(N\ge 1\)〈/span〉 〈/span〉, 〈span〉 〈span〉\(\lambda 〉0\)〈/span〉 〈/span〉, 〈em〉V〈/em〉 is a continuous function verifying some conditions and 〈span〉 〈span〉\(\partial _t F(x,u)\)〈/span〉 〈/span〉 is the generalized gradient of 〈em〉F〈/em〉(〈em〉x〈/em〉, 〈em〉t〈/em〉) with respect to 〈em〉t〈/em〉. Assuming that 〈em〉F〈/em〉(〈em〉x〈/em〉, 〈em〉t〈/em〉) is a mensurable function for each 〈span〉 〈span〉\(t\in \mathbb {R}\)〈/span〉 〈/span〉 and locally Lipschitzian for each 〈span〉 〈span〉\(x\in \mathbb {R}^N\)〈/span〉 〈/span〉, we have applied variational methods for locally Lipschitz functionals to get a solution for 〈span〉 〈span〉\((P_\lambda )\)〈/span〉 〈/span〉 when 〈span〉 〈span〉\(\lambda \)〈/span〉 〈/span〉 is large enough.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The feasibility of a realization method for the three-dimensional surface of an aircraft moving with hypersonic velocity is numerically investigated. An approximate mathematical method, which has the same accuracy at much lower computational cost, is used for calculating convective heat transfer over complex geometric shape of a vehicle. Unstructured grids are applied for numerical tests. The hypersonic aircraft computational aerodynamic predictions and comparisons with experimental data are performed.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We consider the focusing 〈span〉 〈span〉\(L^2\)〈/span〉 〈/span〉-supercritical fractional nonlinear Schrödinger equation 〈span〉 〈span〉$$\begin{aligned} i\partial _t u - (-\varDelta )^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb {R}^+ \times \mathbb {R}^d, \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(d\ge 2, \frac{d}{2d-1} \le s 〈1\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(\frac{4s}{d}〈\alpha 〈\frac{4s}{d-2s}\)〈/span〉 〈/span〉. By means of the localized virial estimate, we prove that the ground-state standing wave is strongly unstable by blowup. This result is a complement to a recent result of Peng–Shi (J Math Phys 59:011508, 〈span〉2018〈/span〉) where the stability and instability of standing waves were studied in the 〈span〉 〈span〉\(L^2\)〈/span〉 〈/span〉-subcritical and 〈span〉 〈span〉\(L^2\)〈/span〉 〈/span〉-critical cases.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Distributed-order fractional model of viscoelastic body is used to describe wave propagation in an infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem is obtained explicitly. The wave propagation speed is found to be related to the material properties at initial time. The fundamental solutions corresponding to four thermodynamically acceptable classes of linear fractional constitutive models and power-type distributed-order models are also obtained.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We discuss drift–diffusion models for charge carrier transport in organic semiconductor devices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The former leads to statistical relations with Gauss–Fermi integrals, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlinear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift–diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauder’s fixed-point theorem.〈/p〉