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.     

Traveling Wave Solutions for a Predator?CPrey System With Sigmoidal Response Function

We study the existence of traveling wave solutions for a diffusive predator?Cprey system. The system considered in this paper is governed by a Sigmoidal response function which in some applications is more realistic than the Holling type I, II responses, and more general than a simplified form of the Holling type III response considered before. Our method is an improvement to the original method introduced in the work of Dunbar (J Math Biol 17:11?C32, 1983; SIAM J Appl Math 46:1057?C1078, 1986). A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in Dunbar (1983, 1986). The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski??s theorem which is much simpler than the extended version in Dunbar??s work.     

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.     

Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem

Using the variational method, Chenciner and Montgomery (Ann Math 152:881–901, 2000) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver numerically found a similar periodic solution called “super-eight” in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the scaling technique established by Tanaka (Ann Inst H Poincaré Anal Non Linéaire 10:215–238, 1993), (Proc Am Math Soc 122:275–284, 1994) and investigate the asymptotic behavior of a binary collision.     

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.     

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.     

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.     

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.     

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).     

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).     

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.     

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.     

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 Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent $a\le -1$ a ≤ ? 1 .     

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).     

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.     

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).     

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).     

Concerning the Existence of Classical Solutions to the Stokes System. On the Minimal Assumptions Problem

In this notes we consider the stationary Stokes system in a bounded, connected, three-dimensional smooth domain, with homogeneous Dirichlet boundary condition. Proofs also apply to the n-dimensional case, and to other boundary conditions, like Navier-slip ones. We say here that a solution is classical if all derivatives appearing in the equations are continuous up to the boundary. It is well known, for long time, that solutions of the Stokes system are classical if the external forces belong to the H?lder space \({C^{0,\; \lambda}(\bar{\Omega})}\) . It is also well known that, in general, solutions are not classical in the presence of continuous external forces. Hence, a very challenging problem is to find Banach spaces, strictly containing the H?lder spaces \({C^{0,\; \lambda}(\bar{\Omega})}\) such that solutions to the Stokes problem corresponding to forces in the above space are classical. We prove this result for external forces in a suitable functional space, denoted \({{\rm C}_*(\bar{\Omega})}\) , introduced in references Beirão da Veiga (On the solutions in the large of the two-dimensional flow of a non-viscous incompressible fluid, 1982) and Beirão da Veiga (J Differ Equ 54(3):373–389, 1984) in connection with the Euler equations.     

Wave fronts in second-order elasticity determined by perturbation method applied to the eikonal equation

In Marasco and Romano (Math Comput Model 49(7–8)1504–1518, 2009), Marasco (Math Comput Model 49(7–8):1644–1652, 2009; Int J Eng Sci 47(4):499–511, 2009), we have proposed a perturbation method to determine the speed and the amplitude of the acceleration waves in a second-order elastic body. In this paper, using the above results, we apply a perturbation procedure to analyze the evolution of the wave front of an acceleration wave in the same class of elastic materials. In particular, a second-order approximate solution of the eikonal equation is determined introducing a suitable system of coordinates. The general results are applied to an infinitesimal deformation, and the analytical solution of the eikonal equation is compared with the exact numerical one.