Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

We prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized Korteweg-de Vries equations in the subcritical case (this includes in particular the KdV equation and the modified KdV equation). This result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation, which has its own interest. The proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized KdV equation. Accepted December 1, 2000?Published online April 3, 2001     

Robust Stability Test for Dynamic Systems with Short Delays by Using Padé Approximation

The paper presents a simple approach to testifying the asymptotic stability and interval stability (robust stability against the change of system parameters in given intervals) for linear dynamic systems involving short time delays. The stability analysis starts with the study of the characteristic roots of a transcendental equation having exponential functions. By means of the Padé approximation to the exponential functions, the transcendental characteristic equation is approximated as an algebraic equation. Then, the test of asymptotic stability and interval stability of the system is completed in a very simple way. The stability analysis of a vibration system with short time delays in the feedback paths of displacement and velocity, taken as an example, is given in detail. The analysis and numerical examples indicate that the approach gives excellent accuracy for linear dynamic systems with short time delays.     

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

An Iteration Method for Integral Equations Arising from Axisymmetric Loading Problems

Let the concentrated forces and the centers of pressure with unknown density functions x(ξ)and y(ξ)respectively be distributed along the axis z outside the solid,then one can reduce anaxismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations.An iteration method for solving such equations is duscussed.A lemma equivalent to E.Rakotch’scontractive mapping theorem and a theorem concerning the convergent proof of the iteration methodare presented.     

Comparative Studies of Three Approaches for GDQ Computation of Incompressible Navier–Stokes Equations in Primitive Variable Form

Three numerical approaches for solving the incompressible Navier–Stokes equations in primitive variable form are proposed and compared in this work. All these approaches are based on the SIMPLE strategy with GDQ discretization on a non-staggered grid. It was found that the satisfying of the continuity equation on the boundary is critical to obtain an accurate numerical solution. The proposed three approaches are to make sure that the continuity equation is satisfied on the boundary, and in the meantime, recommend the boundary condition for pressure correction equation. Through a test problem of two-dimensional driven cavity flow, the performance of three approaches is comparatively studied in terms of the efficiency and accuracy. For all three approaches, accurate numerical results can be obtained by using just a few grid points.     

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

In this paper, we consider the generalized Navier?CStokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier?CStokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier?CStokes equations by the more general operator (???) ^?? with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier?CStokes equation is local well-posed for a large set of the initial data in H ^?1+, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .     

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).     

An Existence Theorem for Generalized von Kármán Equations

Using techniques from formal asymptotic analysis, the first two authors have recently identified generalized von Kármán equations, which constitute a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, the remaining portion being free. In this paper, we establish an existence theorem for these equations. To this end, we first reduce them to a single equation, which generalizes a cubic operator equation introduced by M.S. Berger and P. Fife. We then directly solve this equation, notably by adapting a crucial compactness method due to J.-L. Lions. Résumé. En utilisant les techniques de l'analyse asymptotique formelle, les deux premiers auteurs ont récemment identifié des équations de von Kármán généralisées, qui constituent un modèle bi-dimensionnel de plaque non linéairement élastique dont une partie seulement de la face latérale est soumise à des conditions aux limites de von Kármán, la partie restante étant libre. Dans cet article, on établit un théorème d'existence pour ces équations. À cette fin, elles sont d'abord réduites à une seule équation, qui généralise une équation faisant intervenir un opérateur cubique, introduite par M.S. Berger et P. Fife. On résout ensuite directement cette équation, en adaptant notamment une méthode cruciale de compacité due à J.-L. Lions.     

Optimal Navier–Stokes Design of Compressor Impellers Using Evolutionary Computation

In the design of modern centrifugal compressor impellers, it is fundamental to account for three-dimensional effects and to use an optimization strategy that helps the designer to achieve the required objectives with the presence of constraints. In this paper, a fully three-dimensional optimization method is described that combines a CFD code and an evolutionary algorithm. The design scenario contemplated here involves the maximization of impeller peak efficiency with constraints on the impeller pressure ratio and operating range. The method is used to improve the performances of a baseline impeller of known characteristics. An optimal solution is proposed and compared to the original configuration.     

Exact Solution for the Compressible Flow Equations through a Medium with Triple-Porosity

This paper obtains the exact solution for the unsteady radial flow equations of the slightly compressible liquid through a medium with triple-poro-sity by using,the method of decomposition.This solution not only reveals the essential characters of the unsteady flow of liquid through a medium with multiple-porosity,but also comprises the existing primal results.     

A Spalart–allmaras Turbulence Model Implementation for High-order Discontinuous Galerkin Solution of the Reynolds-averaged Navier-stokes Equations

A novel and robust approach has been proposed for the high-order discontinuous Galerkin (DG) discretization of the Reynolds-averaged Navier-Stokes (RANS) equations with the turbulence model of Spalart-Allmaras (SA). The solution polynomials of the SA equation are reconstructed by the Hermite weighted essentially non-oscillatory (HWENO) scheme. Several practical techniques are suggested to simplify and extend a positivity-preserving limiter to further guarantee the positivity of SA working variable. The resulting positivity-preserving HWENO limiting method is compact and easy to implement on arbitrary meshes. Typical turbulent flows are conducted to assess the accuracy and robustness of the present method. Numerical experiments demonstrate that with the increasing grid or order resolution, the limited results of the working variable are getting closer to the unlimited ones. And the most obvious improvement with proposed method is on the computation of the working variable field in wake regions.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Traveling Waves of a Singularly Perturbed System of Integral–Differential Equations Arising from Neuronal Networks

A nonlinear nonlocal model arising from synaptically coupled neuronal networks with two integral terms is considered. The existence and stability of several traveling wave solutions are established by using ideas in differential equations and functional analysis. Steady-state solutions of some inhomogeneous integral–differential equations are also investigated. We consider several types of kernel functions: (I) positive functions, such as and , where ρ>0 is a constant; (II) nonnegative kernels with compact supports, for examples, (i) 1$$" align="middle" border="0"> , and (ii) {\pi\over 2}$$" align="middle" border="0"> ; (III) Mexican hat type kernel functions, such as and , where A>B>0 and a>b>0 are constants.Dedicated to Professor Yulin Zhou and Professor Boling Guo on the Occassions of their birthdays.     

A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable Weak Solution to the Navier–Stokes Equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x ₀, t ₀). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x ₀, t ₀), intersected with the exterior of a certain space-time paraboloid with vertex at point (x ₀, t ₀). We make no special assumptions on the solution in the interior of the paraboloid.     

Solution of the Incompressible Unsteady Navier–Stokes Equations Only in Terms of the Velocity Components

A method which uses only the velocity components as primitive variables is described for solution of the incompressible unsteady Navier–Stokes equations. The method involves the multiplication of the primitive variable-based Navier–Stokes equations with the unit normal vector of finite volume elements and the integration of the resulting equations along the boundaries of four-node quadrilateral finite volume elements. Therefore, the pressure term is eliminated from the governing equations and any difficulty associated with pressure or vorticity boundary conditions is avoided. The equations are discretized on four-node quadrilateral finite volume elements by using the second-order-accurate central finite differences with the mid-point integral rule in space and the first-order-accurate backward finite differences in time. The resulting system of algebraic equations is solved in coupled form using a direct solver. As a test case, an impulsively accelerated lid-driven cavity flow in a square enclosure is solved in order to verify the accuracy of the present method.     

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

In this paper, we first introduce a novel generalized derivative and obtain the generalized first-order Taylor expansion of the nonsmooth functions. Then we derive the generalized Euler–Lagrange equation for the nonsmooth calculus of variations and solve this equation by using Chebyshev pseudospectral method, approximately. Finally, the optimal solutions of some problems in the nonsmooth calculus of variations are approximated.