Convergence to Global Equilibrium for Spatially Inhomogeneous Kinetic Models of Non-Micro-Reversible Processes

We study the long-time asymptotics of linear kinetic models with periodic boundary conditions or in a rectangular box with specular reflection boundary conditions. An entropy dissipation approach is used to prove decay to the global equilibrium under some additional assumptions on the equilibrium distribution of the mass preserving scattering operator. We prove convergence at an algebraic rate depending on the smoothness of the solution. This result is compared to the optimal result derived by spectral methods in a simple one dimensional example.     

Extrapolation method for Fredholm integral equations with non-smooth kernels

Summary This note analyses the methods of extrapolation from certain approximate solutions of integral equations whose kernels have lower degree smoothness. We show that in order to generate a global superconvergent approximation the extrapolation procedure must be applied to the iterated collocation solution rather than to the usual Nyström solution.     

倒向随机微分方程解的光滑性

该文讨论了倒向随机微分方程Y_t=ξ+∫^T_t{g(s,Y_s,Z_s)}ds-∫^T_t{Z_s}dW_s 解在Malliavin微分意义下的光滑性.对任意的n讨论其解在Malliavin 意义下n 阶可微性,并且证明它是一个线性倒向随机微分方程的解,从而说明BSDE解的光滑性.     

THE QUADRATIC SPECHT TRIANGLE

We propose a class of 12 degrees of freedom triangular plate bending elements with quadratic rate of convergence.They may be viewed as the second order Specht triangle,while the Specht triangle is one of the best first order plate bending element.The convergence result is proved under minimal smoothness assumption on the solution.Numerical results for both the smooth solution and nonsmmoth solution confirm the theoretical prediction.     

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results. Received November 29, 1996 / Revised version received April 25, 1997     

Lösung von Differentialgleichungen mit Splinefunktionen. Eine Störungstheorie

Summary In this paper non-linear splines (depending onn+1 parameters) are used to patch up the solution of an initial value problem in intervals of stepsizeh. The elements of the solution are fixed byq smoothness conditions andd conditions derived from the differential equation in an appropriate setup. The feasibility of the method can be connected to that of the polynomial spline method by a perturbation type argument. Thus the question of convergence forh0 is closely connected to the linear (polynomial) case.A new elementary prove is given for divergence of the polynomial splines ifq is larger thand+1, as was done by Mülthei [4] with other techniques.A byproduct is an extention of the famous result for polynomial interpolation by Runge on equidistant grids that interpolation of a given function by splines of too high smoothness can cause divergence forh0.

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 entstanden     

Densities of One-Dimensional Backward SDEs

This work is devoted to the study of the existence and smoothness of the marginal densities of the solution of one-dimensional backward stochastic differential equations. Under monotonicity conditions of a function of the coefficients, we obtain that the smoothness properties of the forward process influencing the backward equation, transfer to the densities of the solution. Once established these conditions, we apply the result to study the tail behavior of the solution process. Mathematics Subject Classification (2000) 60H10.Fabio Antonelli: The first author was partially supported by the MIUR COFIN grant 2000.Arturo Kohatsu-Higa: The second author was partially supported by grants BFM2003-03324 and BFM 2003-04294. The authors wish to thank the referee for his/her comments.     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

On solvability of stationary problems of natural thermal convection of a high-viscosity fluid

In this paper, investigation of stationary problems of natural thermal convection in the Boussinesq approximation with irregular boundary data is carried out. Existence and uniqueness theorems for a weak solution of such problems are proved. The smoothness of a weak solution depending on the smoothness of the initial data and the smoothness of the boundary of the domain where the problem is considered is investigated.     

A local regularity result for Neumann parabolic problems with nonsmooth data

In this work we analyze the relations between two different concepts of solution of the Neumann problem for a second order parabolic equation: the usual notions of weak solution and those of transposition solution, which allow well-posedness of problems with measure data. We give a regularity result for the transposition solution and we prove that, under smoothness assumptions for the principal part of the operator, the local regularity of the transposition solution is the same as that of the usual weak solution. As an interesting particular case, we present a rigorous proof of local continuity of the solution for a convection–diffusion problem with pointwise source term.     

Fixed mesh approximation of ordinary differential equations with impulses

When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite. So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates. Received March 15, 1996 / Revised version received January 3, 1997     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

Local solvability and loss of smoothness of the Navier–Stokes–Maxwell equations with large initial data

The existence of a local-in-time unique solution and loss of smoothness of a full Magneto-Hydro-Dynamics (MHD) system are considered for periodic initial data. The result is proven using Fujita–Kato’s method in ${?}^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwell’s equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.     

多孔介质中可压缩流体力学方程组经典解的整体存在性与破裂现象

利用拟线性双曲型方程组极值原理,改进了HSIAO Ling和D.Serre得到的关于多孔介质中可压缩流体力学方程组解的存在性结果,给出了其Cauchy问题的一个关于经典解整体存在和破裂的完整结果．这些结果说明强耗散有助于“小”解的光滑性．     

A parameter choice strategy for (iterated) tikhonov regularization of iii-posed problems leading to superconvergence with optimal rates

For ordinary and iterated Tikhonov regularization of linear ill-posed problems, we propose a parameter choice strategy that leads to optimal (super-) convergence rates for certain linear functionals of the regularized solution. It is not necessary to know the smoothness index of the exact solution; approximate knowledge of the smoothness index for the linear functional suffices     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

Solvability of the stationary heat convection problem for a high-viscosity fluid

We study the stationary heat convection problem in the Boussinesq approximation. We derive a priori estimates for its solution. We prove existence and uniqueness theorems for a weak solution of the problem and analyze the smoothness of a weak solution for raised smoothness of the problem data. We consider the two- and three-dimensional cases.     

A saddle point variational formulation for projection–regularized parameter identification

Summary. This paper is concerned with the ill-posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection. As the theory has shown, the ansatz functions for the parameter have to be sufficiently smooth. In this paper we show that these – for a practical implementation unrealistic – smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation. We prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level goes to zero in the case of noisy data, as well as convergence rates under additional smoothness conditions. Received August 4, 2000 / Revised version received March 21, 2001 / Published online October 17, 2001     

On smoothness of a fundamental solution to a second order hyperbolic equation

We consider the problem of constructing a fundamental solution to a second order hyperbolic linear equation with variable coefficients depending on the space variable x ∈ ? ⁿ . Under the assumption of high but finite smoothness of the coefficients, we write out the structure of a fundamental solution, establish smoothness of the coefficients of the expansion of its singular part, and characterize smoothness of the regular part.     

On the domain dependence of solutions to the compressible Navier–Stokes equations of a barotropic fluid

We prove a general compactness result for the solution set of the compressible Navier–Stokes equations with respect to the variation of the underlying spatial domain. Among various corollaries, we then prove a general existence theorem for the system in question with no restrictions on smoothness of the spatial domain. Copyright © 2002 John Wiley & Sons, Ltd.