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).     

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.     

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.     

Higher Integrability of Solutions to Generalized Stokes System Under Perfect Slip Boundary Conditions

We prove an L ^q theory result for generalized Stokes system in a \({\mathcal{C}^{2,1}}\) domain complemented with the perfect slip boundary conditions and under Φ-growth conditions. Since the interior regularity was obtained in Diening and Kaplický (Manu Math 141:336–361, 2013), a regularity up to the boundary is an aim of this paper. In order to get the main result, we use Calderón–Zygmund theory and the method developed in Caffarelli and Peral (Ann Math 130:189–213, 1989). We obtain higher integrability of the first gradient of a solution.     

Quasi-linear versus potential-based formulations of force–flux relations and the GENERIC for irreversible processes: comparisons and examples

An essential part in modeling out-of-equilibrium dynamics is the formulation of irreversible dynamics. In the latter, the major task consists in specifying the relations between thermodynamic forces and fluxes. In the literature, mainly two distinct approaches are used for the specification of force–flux relations. On the one hand, quasi-linear relations are employed, which are based on the physics of transport processes and fluctuation–dissipation theorems (de Groot and Mazur in Non-equilibrium thermodynamics, North Holland, Amsterdam, 1962, Lifshitz and Pitaevskii in Physical kinetics. Volume 10, Landau and Lifshitz series on theoretical physics, Pergamon Press, Oxford, 1981). On the other hand, force–flux relations are also often represented in potential form with the help of a dissipation potential (?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997). We address the question of how these two approaches are related. The main result of this presentation states that the class of models formulated by quasi-linear relations is larger than what can be described in a potential-based formulation. While the relation between the two methods is shown in general terms, it is demonstrated also with the help of three examples. The finding that quasi-linear force–flux relations are more general than dissipation-based ones also has ramifications for the general equation for non-equilibrium reversible–irreversible coupling (GENERIC: e.g., Grmela and Öttinger in Phys Rev E 56:6620–6632, 6633–6655, 1997, Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005). This framework has been formulated and used in two different forms, namely a quasi-linear (Öttinger and Grmela in Phys Rev E 56:6633–6655, 1997, Öttinger in Beyond equilibrium thermodynamics, Wiley Interscience Publishers, Hoboken, 2005) and a dissipation potential–based (Grmela in Adv Chem Eng 39:75–129, 2010, Grmela in J Non-Newton Fluid Mech 165:980–986, 2010, Mielke in Continuum Mech Therm 23:233–256, 2011) form, respectively, relating the irreversible evolution to the entropy gradient. It is found that also in the case of GENERIC, the quasi-linear representation encompasses a wider class of phenomena as compared to the dissipation-based formulation. Furthermore, it is found that a potential exists for the irreversible part of the GENERIC if and only if one does for the underlying force–flux relations.     

Weak Solutions of the Navier–Stokes Equations for Compressible Flows in a Half-Space with No-Slip Boundary Conditions

We consider the Navier–Stokes equations for the motion of compressible, viscous flows in a half-space ${\mathbb{R}^n_+,}$ n =  2,  3, with the no-slip boundary conditions. We prove the existence of a global weak solution when the initial data are close to a static equilibrium. The density of the weak solution is uniformly bounded and does not contain a vacuum, the velocity is Hölder continuous in (x, t) and the material acceleration is weakly differentiable. The weak solutions of this type were introduced by D. Hoff in Arch Ration Mech Anal 114(1):15–46, (1991), Commun Pure and Appl Math 55(11):1365–1407, (2002) for the initial-boundary value problem in ${\Omega = \mathbb{R}^n}$ and for the problem in ${\Omega = \mathbb{R}^n_+}$ with the Navier boundary conditions.     

Kirchhoff plates with viscous boundary dissipation

In this paper, we confine our attention to Kirchhoff thin plates in presence of boundary viscoelastic dissipative mechanisms, in order to investigate the well-posedness and the asymptotic behavior within the minimal state approach, following the guidelines proposed in Deseri et al. (Arch Rational Mech Anal 181:43–96, 2006) [see also Fabrizio et al. (Arch Rational Mech Anal 198:189–232, 2010)].     

Local Controllability to Trajectories of the Magnetohydrodynamic Equations

In this paper we prove the local controllability to trajectories of the three dimensional magnetohydrodynamic equations by means of two internal controls, one in the velocity equations and the other in the magnetic field equations and both localized in an arbitrary small subset with not empty interior of the domain. This paper improves the previous results (Barbu et al. in Comm Pure Appl Math 56:732–783, 2003; Barbu et al. in Adv Differ Equ 10:481–504, 2005; Havârneanu et al. in Adv Differ Equ 11:893–929, 2006; Havârneanu, in SIAM J Control Optim 46:1802–1830, 2007) where the second control is not localized and it allows to deduce the local controllability to trajectories with boundary controls. The proof relies on the Carleman inequality for the Stokes system of Imanuvilov et al. (Carleman estimates for second order nonhomogeneous parabolic equations, preprint) to deal with the velocity equations and on a new Carleman inequality for a Dynamo-type equation to deal with the magnetic field equations.     

Dynamics and Control at Feedback Vertex Sets. I: Informative and Determining Nodes in Regulatory Networks

We consider systems of differential equations which model complex regulatory networks by a graph structure of dependencies. We show that the concepts of informative nodes (Mochizuki and Saito, J Theor Biol 266:323–335, 2010) and determining nodes (Foias and Temam, Math Comput 43:117–133, 1984) coincide with the notion of feedback vertex sets from graph theory. As a result we can determine the long-time dynamics of the entire network from observations on only a feedback vertex set. We also indicate how open loop control at a feedback vertex set, only, forces the remaining network to stably follow prescribed stable or unstable trajectories. We present three examples of biological networks which motivated this work: a specific gene regulatory network of ascidian cell differentiation (Imai et al., Science 312:1183–1187, 2006), a signal transduction network involving the epidermal growth factor in mammalian cells (Oda et al., Mol Syst Biol 1:1–17, 2005), and a mammalian gene regulatory network of circadian rhythms (Mirsky et al., Proc Natl Acad Sci USA 106:11107–11112, 2009). In each example the required observation set is much smaller than the entire network. For further details on biological aspects see the companion paper (Mochizuki et al., J Theor Biol, 2013, in press). The mathematical scope of our approach is not limited to biology. Therefore we also include many further examples to illustrate and discuss the broader mathematical aspects.     

Microscale Fracture Toughness of Bismuth Doped Copper Bicrystals Using Double Edge Notched Microtensile Tests

The availability of focused ion beam (FIB) milling, nanoindentation, and microelectromechanical systems (MEMS) based test platforms has enabled small-scale mechanical testing to become an increasingly popular approach for measuring material properties. While great emphasis has been placed on measuring plastic properties at the micro- and nanoscale [1, 2], an area that has received significantly less consideration is the measurement of fracture toughness. A technique for performing small-scale, in situ fracture toughness tests using double edge notched tensile (DENT) specimens has been developed and used to measure a nearly 40 % reduction in toughness associated with the addition of Bi to the grain boundary of a Cu bicrystal. That Bi embrittles Cu grain boundaries is well known [3–10], however, as shown herein, the DENT technique offers certain advantages over existing boundary fracture tests, especially when used with ductile materials.     

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).     

Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.     

Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension

We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. The long time behavior of solutions near equilibrium has been an intriguing question since the work of Beale (Commun Pure Appl Math 34(3):359–392, 1981). This paper is the third in a series of three (Guo in Local well-posedness of the viscous surface wave problem without surface tension, Anal PDE 2012, to appear; in Decay of viscous surface waves without surface tension in horizontally infinite domains, Preprint, 2011) that answers this question. Here we consider the case in which the free interface is horizontally periodic; we prove that the problem is globally well-posed and that solutions decay to equilibrium at an almost exponential rate. In particular, the free interface decays to a flat surface. Our framework contains several novel techniques, which include: (1) a priori estimates that utilize a “geometric” reformulation of the equations; (2) a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; (3) a localization procedure that is compatible with the energy method and allows for curved lower surface geometry. Our decay estimates lead to the construction of global-in-time solutions to the surface wave problem.     

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.     

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).     

Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that from the time series of two identical systems, one can simply extract value of the stability parameter which value can be treated as largest LLE. Unlike the method presented in part, one developed method (LLEDPT) can be applied to the dynamical systems of any type, continuous, with discontinuities, with time delay and others. The theoretical improvement shows simplicity of the method and its obvious physical background. The proofs for the method effectiveness are based on results of the simulations of the experiments for Duffing and Van der Pole oscillators. These results were compared with ones obtained with use of the Stefanski method (Stefanski in Chaos Soliton Fract 11(15):2443–2451, 2000; Chaos Soliton Fract 15:233–244, 2003; Chaos Soliton Fract 23:1651–1659, 2005; J Theor Appl Mech 46(3):665–678, 2008) and LLEDP method. LLEDPT can be used also as the criterion of stability of the control system, where desired behavior of controlled system is explicitly known (Balcerzak et al. in Mech Mech Eng 17(4):325–339, 2013). The next step of development of the method can be considered in direction that allows estimation of LLE from the real time series, systems with discontinuities, with time delay and others.     

The Rot-Div System in Exterior Domains

The goal of this paper is to reconsider the classical elliptic system rot v =  f, div v =  g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L _p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\) . As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\) , provided the boundary is merely in \({B^{2-1/3}_{3,1}}\) . The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).     

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).     

Existence,Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in {\mathbb{R}^2} with Measure-Valued Initial Data

We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic–elliptic Patlak–Keller–Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption \({\max_{x \in \mathbb{R}^2} \mu (\{x\}) < 8 \pi}\) . This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier–Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.