A Rigorous Derivation of the Coupling of a Kinetic Equation and Burgers’ Equation

In this article, we investigate the kinetic/fluid coupling on a toy model, which we obtain rigorously from a hydrodynamical limit. The idea is that at the level of the full kinetic model, the coupling is obvious. We then investigate the coupling obtained when passing into the limit. We show that, especially in presence of a shock stuck on the interface, the coupling involves a kinetic layer known as the Milne problem. Due to this layer, the limit process is quite delicate and some blow-up techniques are needed to ensure its strong convergence.     

Front Solutions of Richards’ Equation

The design of a Compressed Air Energy Storage (CAES) plant requires knowledge of the pressure and temperature variations within the reservoir, for expected sets of plant operation. In the current work, a closed form approximate analytical solution for the pressure variations, in porous media reservoirs, was derived for conditions of steady periodic isothermal radial gas flow. Two different expressions for the pressure variation were obtained, one as an infinite series and the other as an integral, where the latter is the computationally preferred solution. In order to evaluate the model accuracy, a finite difference numerical solution of the full non-linear problem was developed. The accuracy of the analytical solution was confirmed through, both, error analysis and comparison against the numerical calculations. The analytical solution can be used to calculate the well pressure variations and the radius of the active region around the well. Examples of calculations are provided, and a parametric study is presented to demonstrate the sensitivity of the well pressure to pertinent parameters. The model could eventually yield improved CAES plant designs.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Stability of Navier-Stokes Equation(Ⅱ)

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Stability of Navier-Stokes Equation(Ⅰ)

ＳＴＡＢＩＬＩＴＹＯＦＮＡＶＩＥＲ－ＳＴＯＫＥＳＥＱＵＡＴＩＯＮ（Ⅰ）ＳｈｉＷｅｉ－ｈｕｉ（施惟慧）（ＳｈａｎｇｈａｉＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ）（ＲｅｃｅｉｖｅｄＤｅｃ．１０，１９９３；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉ－ｚａ...     

Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle ${\pi/q, q \geqq 4}$ . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Singular Limits of the Klein–Gordon Equation

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, ${\hbar\to 0}$ , we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, ${\hbar\to 0, c=\hbar^{-\alpha}\to \infty}$ for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.     

The Calculation of Semi-circular Corrugated Tube-An Application of the General Solution of Ring Shell

In this paper,the deformation and stress distribution in semi-circularcorrugated tube under axial force are calculated by means of the generalsolutions of circular ring shell given in previous paper[1].     

Asymptotics of the Solutions of the Random Schrödinger Equation

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transform \({\hat\zeta_\epsilon(t,\xi)}\) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit of \({\hat\zeta_\epsilon(t,\xi)}\) has the form \({\hat\zeta_0(\xi){\rm exp}(iB_\kappa(t,\xi))}\) where B _κ (t, ξ) is a fractional Brownian motion.     

Dust‐ion‐acoustic precursor of a shock wave

The effect of charged dust particles on the structure of the plasma precursor of a strong shock wave is studied. The conditions of formation of a weak discontinuity front are obtained. It is shown that resonant modes can occur in which the concentration of dust particles in the neighborhood of the front increases. In the case of positively charged particles of dust, the formation of a localized compaction region in the form of a soliton bunch is possible and the dependence of the amplitude of the soliton on shockwave velocity is nonmonotonic. In the case of negatively charged particles of dust, a rarefaction wave is formed. The indicated phenomena can substantially affect the concentration of the neutral component in a slightly ionized plasma.     

Calculation of Layered Shells by the Pseudogeometrical–Nonlinearity Method

The problem of determining the stress—strain state of a multilayered shell is solved. It is assumed that the layer material is nonlinearly elastic and the strain—displacement relations are nonlinear. The displacements are expanded in terms of the functions of transverse coordinate that contain unknown parameters. The governing equations are derived with the use of the Lagrange variational principle. A technique for minimizing the energy functional is proposed. An example of a three–layered beam is considered, calculation results are compared with the exact solution, and the specific features of the approach proposed are analyzed.     

Existence and Uniqueness of Solutions¶of an Asymptotic Equation¶Arising from a Variational Wave Equation¶with General Data

We establish here the global existence and uniqueness of admissible (both dissipative and conservative) weak solutions to a canonical asymptotic equation () for weakly nonlinear solutions of a class of nonlinear variational wave equations with any L ²(ℝ) initial datum. We use the method of Young measures and mollification techniques. Accepted April 25, 2000?Published online November 16, 2000     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 2. Re‐Evaluation of the Salt Dome Flow Problem

Case 5, Level 1 of the international HYDROCOIN groundwater flow modeling project is an example of idealized flow over a salt dome. The groundwater flow is strongly coupled to solute transport since density variations in this example are large (20%).Several independent teams simulated this problem using different models. Results obtained by different codes can be contradictory. We develop a new numerical model based on the mixed hybrid finite elements approximation for flow, which provides a good approximation of the velocity, and the discontinuous finite elements approximation to solve the advection equation, which gives a good approximation of concentration even when the dispersion tensor is very small. We use the new numerical model to simulate the salt dome flow problem.In this paper we study the effect of molecular diffusion and we compare linear and nonlinear dispersion equations. We show the importance of the discretization of the boundary condition on the extent of recirculation and the final salt distribution. We study also the salt dome flow problem with a more realistic dispersion (very small dispersion tensor). Our results are different to prior works with regard to the magnitude of recirculation and the final concentration distribution. In all cases, we obtain recirculation in the lower part of the domain, even for only dispersive fluxes at the boundary. When the dispersion tensor becomes very small, the magnitude of recirculation is small. Swept forward displacement could be reproduced by using finite difference method to compute the dispersive fluxes instead of mixed hybrid finite elements.     

Superfluous Order and the Proper Solution of the Maxwell Equation

In this paper it is proved that Maxwell equation is equivalent to a fourth order equation.Under a certain condition,its general solution is given by     

Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Calculation∗

ABSTRACT

The semi-analytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied in this paper. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semi-analytical method. It characterizes the difference in errors between a general finite difference method and the semi-analytical method. A method for improving the accuracy of the semi-analytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.