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In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the $N$-particle Newtonian system. For $L^1\cap L^\infty$ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the non-local non-linear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.

The paper deals with the existence and uniqueness of classical solutions of the homogeneous Neumann problem for a class of parabolic–hyperbolic system of partial differential equations in n dimensions. The problem arises from a model of the diffusion of N species of radioactive isotopes of the same element. Copyright © 2014 John Wiley & Sons, Ltd.

Following a suggestion from A. Thiaville and J. Miltat, whose work and experiments are about ferromagnetic thin layers and nanowires, we study in this paper the behaviour of the Landau–Lifschitz equation in a straight ferromagnetic wire. As the diameter of the domain and the exchange coefficient in the equation simultaneously tend to zero, we perform an asymptotic expansion to precise the solution for well-prepared initial conditions and are led to consider 2D exterior problems. Copyright © 2008 John Wiley & Sons, Ltd.

The quasi-static evolution of an elastoplastic body with a multi-surface constitutive law of linear kinematic hardening type allows the modelling of curved stress–strain relations. It generalizes classical small-strain elastoplasticity from one to various plastic phases. This paper presents the mathematical models and proves existence and uniqueness of the solution of the corresponding initial-boundary value problem. The analysis involves an explicit estimate for the effective ellipticity constant. Copyright © 2004 John Wiley & Sons, Ltd.

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

This paper is concerned with the question of necessary and sufficient conditions to find a vector field V ∈ Γ(T M) solving the equation divV = Φ under inhomogenous boundary conditions V|ϑM = Z|∈ Γ(TM). An existence and regularity result is given for an arbitrary Riemannian manifold with boundary, M. The proof is based on the Hodge theory of differential forms.

In this paper, we extend the basic Exp-function method to nonlinear lattice differential equations for constructing multi-wave and rational solutions for the first time. We consider a differential-difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.

This paper proves a trace regularity theorem for the solutions of general linear partial differential equations with smooth coefficients. Our result shows that by imposing additional microlocal smoothness along certain directions. the trace of the solution on a codimension one hypersurface will be just as regular as the solution itself. The proof is based on the Hoermander-Nirenberg pseudo-differential cut-off technique and a "fattening" lemma, together with standard energy estimates.

In this paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.

The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations (LBDIEs). The equivalence between the Dirichlet BVP and the corresponding LBDIE system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorization method we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces., 31 pages, Math. Meth. Appl. Sci., 2016, 21p