The Two-Dimensional Euler Equations on Singular Domains

We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L ^p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).     

Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.     

Stability Results for Second-Order Evolution Equations with Switching Time-Delay

We consider second-order evolution equations in an abstract setting with intermittently delayed/not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalizing our previous results from Nicaise and Pignotti (Adv Differ Equ 17:879–902, 2012). In particular, under suitable conditions, we can consider unbounded damping operators. Some concrete examples are finally presented.     

Experimental Measurement of Local Burning Velocity Within a Rotating Flow

The work presented in this paper details the implementation of a new technique for the measurement of local burning velocity using asynchronous particle image velocimetry. This technique uses the local flow velocity ahead of the flame front to measure the movement of the flame by the surrounding fluid. This information is then used to quantify the local burning velocity by taking into account the translation of the flame via convection. In this paper the developed technique is used to study the interaction between a flame front and a single toroidal vortex for the case of premixed stoichiometric methane and air combustion. This data is then used to assess the impact of vortex structure on flame propagation rates. The burning velocity data demonstrates that there is a significant enhancement to the rate of flame propagation where the flame directly interacts with the rotating vortex. The increases found were significantly higher than expected but are supported by burning velocities (Filatyev et al, Combust Flame 141:1?C21, 2005; Kobayashi et al, Proc Combust Inst 29:1793?C1800, 2002; Shepherd et al. 1998) found in turbulent flames of the same mixture composition. Away from this interaction with the main vortex core, the flame exhibits propagation rates around the value recorded in literature for unperturbed laminar combustion (Tahtouh et al, Combust Flame 159:1735?C1743, 2009; Hassan et al, Combust Flame 115:539?C550, 1998); Halter et al, Proc Combust Inst 30:201?C208, 2005; Coppens et al, Exp Therm Fluid Sci 31:437?C444, 2007).     

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.     

On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Öttinger in Phys. Rev. E 56: 6620–6632, 1997; Öttinger and Grmela in Phys. Rev. E 56: 6633–6655, 1997; Öttinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980–998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hütter and Tervoort in J. Non-Newton. Fluid Mech. 152: 45–52, 2008; 53–65, 2008; Adv. Appl. Mech. 42: 254–317, 2009) as well as to damage mechanics (Hütter and Tervoort in Acta Mech. 201: 297–312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9–12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts.     

On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ ^γ , γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.     

Zero singularity of codimension two or three in a four-neuron BAM neural network model with multiple delays

In this paper, a four-neuron BAM neural network model with multiple delays is considered. The existence conditions under which that the origin of the system is Bogdanov–Takens (B–T) or triple zero singularity are given. By choosing the connected weights as bifurcation parameters and using the center manifold reduction and the normal form theory and the formula developed by Xu and Huang (J Differ Equ 244:582–598 2008) and Qiao et al. (Chinese Ann Math Ser A 31:59–70 2010), the versal unfoldings and the normal forms for this singularity were given to analyze the behaviors of the system. This paper is a further study of paper Cao and Xiao (IEEE Trans Neural Netw 18:416–430 2007).     

Comparison of dynamical cores for NWP models: comparison of COSMO and Dune

We present a range of numerical tests comparing the dynamical cores of the operationally used numerical weather prediction (NWP) model COSMO and the university code Dune, focusing on their efficiency and accuracy for solving benchmark test cases for NWP. The dynamical core of COSMO is based on a finite difference method whereas the Dune core is based on a Discontinuous Galerkin method. Both dynamical cores are briefly introduced stating possible advantages and pitfalls of the different approaches. Their efficiency and effectiveness is investigated, based on three numerical test cases, which require solving the compressible viscous and non-viscous Euler equations. The test cases include the density current (Straka et al. in Int J Numer Methods Fluids 17:1–22, 1993), the inertia gravity (Skamarock and Klemp in Mon Weather Rev 122:2623–2630, 1994), and the linear hydrostatic mountain waves of (Bonaventura in J Comput Phys 158:186–213, 2000).     

Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells

In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Kármán equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Kármán type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Kármán shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969).     

Experimental and Numerical Study of the Scalar Turbulent/Non-Turbulent Interface Layer in a Jet Flow

Based on two large-eddy simulations (LES) of a non-reacting turbulent round jet with a nozzle based Reynolds number of 8,610 with the same configuration as the one that has recently been investigated experimentally (Gampert et al., 2012; J Fluid Mech, 2012; J Fluid Mech 724:337, 2013), we examine the scalar turbulent/non-turbulent (T/NT) interface layer in the mixture fraction field of the jet flow between ten and thirty nozzle diameters downstream. To this end, the LES—one with a coarse grid and one with a fine grid—are in a first step validated against the experimental data using the axial decay of the mean velocity and the mean mixture fraction as well as based on radial self-similar profiles of mean and root mean square values of these two quantities. Then, probability density functions (pdf) of the mixture fraction at various axial and radial positions are compared and the quality of the LES is discussed. In general, the LES results are consistent with the experimental data. However, in the flow region where the imprint of the T/NT interface layer is dominant in the mixture fraction pdf, discrepancies are observed. In a next step, statistics of the T/NT interface layer are studied, where a satisfactory agreement for the pdf of the location of the interface layer from the higher resolved LES with the experimental data is observed, while the one with the coarse grid exhibits considerable deviations. Finally, the mixture fraction profile across the interface is investigated where the same trend as for the pdf of the location is present. In particular, it is found that the sharp interface that is present in experimental studies (Gampert et al., J Fluid Mech, 2013; Westerweel et al., J Fluid Mech 631:199, 2009) is less distinct in the LES results and rather diffused in radial direction outside of the T/NT interface layer.     

Existence and Stability of Solutions for Steady Flows of Fibre Suspension Flows

We establish existence, uniqueness, convergence and stability of solutions to the equations of steady flows of fibre suspension flows. The existence of a unique steady solution is proven by using an iterative scheme. One of the restrictions imposed on the data confirms a well known fact proven in Galdi and Reddy (J Non-Newtonian Fluid Mech 83:205–230, 1999), Munganga and Reddy (Math Models Methods Appl Sci 12:1177–1203, 2002) and Munganga et al. (J Non-Newtonian fluid Mech 92:135–150, 2000) that the particle number N _p must be less than 35/2. Exact solutions are calculated for Couette and Poiseuille flows. Solutions of Poiseuille flows are shown to be more accurate than those of Couette flow when a time perturbation is considered.     

Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations ${(-\Delta )^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}}$ with an isolated singularity, where ${\sigma\in (0,1)}$ . We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved by Caffarelli et al. (Comm Pure Appl Math 42:271–297, 1989).     

Free Boundary Problems for a Lotka–Volterra Competition System

In this paper we investigate two free boundary problems for a Lotka–Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time \(t\) goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of Guo and Wu (J Dyn Differ Equ 24:873–895, 2012).     

Mean Flow Structures Inside the Human Upper Airway

The lattice Boltzmann method (LBM) was used to conduct a direct numerical simulation study of the airflow inside an idealised human upper airway. Results from both a modest resolution (18 million control volumes, 320 Gb data set) and an extreme resolution (148 million control volumes, 800 Gb data set) LBM simulation were compared to those from experimental results (Johnstone, A.: Hot wire measurements in an oropharyngeal pathway. M.Sc. Thesis, Queen’s University, Kingston, ON, Canada, 2002; Johnstone et al., Expt Fluids 37(5): 673–689, 2004). A coarse resolution simulation (2.4 million control volumes, 105 Gb data set) was used to record the entire time-varying flow field; the nature of the mean structures in the three-dimensional flow field was studied using this data set. For the mean statistics, the LBM calculations yield better results than do the Reynolds averaged Navier–Stokes methods (Ball et al., Comput Fluids, 2007); the LBM reproduces significant detail of experimentally observed flow features. The flow is three-dimensional, obviously, and the interrogation of the mean flow structure is found to be unsteady so that sagittal plane and time-integrated measurements alone are insufficient to verify the accuracy of computational predictions of this flow.     

On Growth of Sobolev Norms in Linear Schr?dinger Equations with Time Dependent Gevrey Potential

We improve Delort??s method to show that solutions of linear Schr?dinger equations with a time dependent Gevrey potential on the torus, have at most logarithmically growing Sobolev norms. In particular, it contains the result of Wang (Commun Partial Differ Equ 33:2164?C2179, 2008), which deals with analytic potentials in dimension 1.     

The Role of Noise in Finite Ensembles of Nanomagnetic Particles

The dynamics of finitely many nanomagnetic particles are described by the stochastic Landau–Lifshitz–Gilbert equation. We show that the system relaxes exponentially quickly to the unique invariant measure which is described by a Boltzmann distribution. We present two approaches to verify this result. The first uses the general theory of (Meyn and Tweedy, Adv Appl Prob 24:542–574, 1992; Meyn and Tweedy, Adv Appl Prob 25:487–517 1993; Meyn and Tweedy, Adv Appl Prob 25:518–548, 1993) for Markov chains, which involves the concepts of a Lyapunov structure, and irreducibility of transition probabilities; we show exponential convergence in a supremum topology, but lack explicit rates. The second approach shows exponential ergodicity in a weaker L ² topology, with an explicit rate of convergence of the Arrhenius type law. Then, we discuss two implicit discretizations to approximate transition functions at both finite and infinite times: the first scheme is shown to inherit the geometric ‘unit-length’ property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with a rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at an optimal rate for all finite times. We use a general result of Shardlow and Stuart (Siam J Numer Anal 37(4):1120–1137 2000) to conclude convergence to the invariant measure of the limiting problem for both discretizations.     

A phenomenological model for microstructure-dependent inelasticity in shape-memory alloys

Degradation in shape-memory alloy response is a crucial concern for a variety of innovative applications. Under cyclic loadings, these materials generally experience permanent inelastic deformations. The onset of plasticization is known to be very sensitive to the microstructure of the polycrystalline specimen. Moving from recent experimental findings (Malard et al. in Funct Mater Lett 2:45–54, 2009; Acta Mater 59:1542–1556, 2011), we present a phenomenological model for permanent inelastic effects in shape-memory alloys taking into account the polycrystalline microstructure. In particular, the mechanical response under cyclic loadings is investigated in connection with the mean crystal grain size. Formulated within the variational frame of generalized standard materials, the model consists in an extension of the model in Auricchio et al. (Int J Plast 23:207–226, 2007) to the case of microstructure-dependent parameters. The mathematical setting is discussed and numerical simulations showing the capability of the model to reproduce experiments are presented.