Stability of weak vector variational inequality

In this paper, a key assumption is introduced by virtue of a parametric gap function. Then, by using the key assumption, sufficient conditions of the continuity and Hausdorff continuity of a solution set map for a parametric weak vector variational inequality are obtained in Banach spaces with the objective space being finite-dimensional.     

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation.     

The ultra weak variational formulation for the modified mild-slope equation

This paper is devoted to evaluate a new alternative numerical method to capture accurately the diffraction-refraction process of waves in coastal areas. The modified mild-slope equation (MMSE) has been used to predict the water wave transformation when waves approach the shoreline. We perform numerical simulations in order to illustrate the efficiency of the ultra weak variational formulation (UWVF) method in comparison with the finite elements method (FEM). The UWVF method uses plane wave solutions on each element and has been shown to reduce the computational complexity at high wave numbers. We also present an alternative method to seek the angle of attack of the wave front on the domain boundary and show that the UWVF method reproduces effectively the numerical experimental data. We compare the FEM and the UWVF method for the MMSE and show that the UWVF method solves some of the difficulties that arise when the FEM is applied.     

Solution representations for a wave equation with weak dissipation

We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+t v_t=0, x∈?ⁿ, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain L_p–L_q estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L₂ energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of μand that μ=2 is critical. Copyright © 2004 John Wiley & Sons, Ltd.     

Kinetic equation for spin waves in a weak alternating magnetic field

Theoretical and Mathematical Physics -     

Lower semicontinuity of the solution map to a parametric vector variational inequality

This paper is concerned with the study of solution stability of a parametric vector variational inequality, where mappings may not be strongly monotone. Under some requirements that the operator of a unperturbed problem is monotone or it satisfies degree conditions then we show that the solution map of a parametric vector variational inequality is lower semicontinuous.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

A variational approach to a shape design problem for the wave equation

The problem of determining the optimal damping set for the stabilization of the wave equation may be not well-posed. By means of a vector variational reformulation and use of gradient Young measures, we present a general methodology to relax this kind of problems. From the optimal Young measure associated with the relaxed problem, we obtain information concerning minimizing sequences for the original problem as well as continuity properties of the relaxed cost function. To cite this article: A. Münch et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

高阶变形的Novikov方程弱解的全局存在性

研究了一类高阶变形的Novikov方程全局弱解的存在性,在初值满足条件u0∈H2,p,p 4时,通过黏性逼近的方法得到了高阶变形Novikov方程全局弱解的存在性.     

Gap functions and error bounds for weak vector variational inequalities

In this article, by using the image space analysis, a gap function for weak vector variational inequalities is obtained. Its lower semicontinuity is also discussed. Then, these results are applied to obtain the error bounds for weak vector variational inequalities. These bounds provide effective estimated distances between a feasible point and the solution set of the weak vector variational inequalities.     

Connectedness structure of the solution sets of vector variational inequalities

By a scalarization method and properties of semi-algebraic sets, it is proved that both the Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. Consequences of the results for vector optimization problems are discussed in details. The results of this paper solve in the affirmative some open questions for the case of general problems without requiring monotonicity of the operators involved.     

一个修正Novikov方程弱解的全局存在性

主要研究了一个修正的Novikov方程,并给出了当初值u0(x)满足一定条件时,方程弱解的全局存在性,推广了Novikov方程的相关结果.     

Numerical solution of a nonlinear wave equation in polar coordinates

We study a finite element scheme and a difference scheme for the radial solutions of a nonlinear Klein-Gordon equation.     

On the nonuniqueness of weak solution of the Euler equation

Weak solution of the Euler equations is defined as an L²-vector field satisfying the integral relations expressing the mass and momentum balance. Their general nature has been quite unclear. In this work an example of a weak solution on a 2-dimensional torus is constructed that is identically zero outside a finite time interval. This example is simpler and more transparent than the previous example of V. Scheffer (J. Geom. Anal. 3(4), 1993, pp. 343–401). © 1997 John Wiley & Sons, Inc.     

Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions

We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high‐order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth‐order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016