The Construction of an Evolution System in the Hyperbolic Case and Applications

We study the Cauchy problem for abstract linear and quasi–linear non–autonomous evolution equations of hyperbolic type using semigroup theory. Under weak differentiability assumptions on the time regularity of the coefficients we prove well–posedness and regularity of a solution. The abstract results are illustrated by their application to a series of equations of mathematical physics.     

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler–Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh.     

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II. Abstract splitting

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size, and under a condition of zero momentum on the nonlinearity. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for rounded numerical schemes avoiding at each step possible high frequency energy drift. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.     

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.     

Large time step maximum norm regularity of L-stable difference methods for parabolic equations

We establish almost sharp maximum norm regularity properties with large time steps for L-stable finite difference methods for linear second-order parabolic equations with spatially variable coefficients. The regularity properties for first and second spatial differences of the numerical solution mimic those of the continuous problem, with logarithmic factors in second differences. The regularity results for the inhomogeneous problem imply that the uniform rate of convergence of the numerical solution and its differences is controlled only by the maximum norm of the local truncation error.     

Determining the Source Term in an Abstract Parabolic Problem From a Time Integral of the Solution

We consider the problem of reconstruction of the source term in an abstract parabolic system. The supplementary information, which is necessary to determine the solution of the system together with the unknown part of the source term, is given by the knowledge of a time integral of the solution with respect to a general Borel measure in the time interval. A theorem of existence and uniqueness of a solution, which is also of maximal regularity type, is proved. Certain particular cases are treated, together with the fact that, in some circumstances, the problem enjoys a sort of Fredholm property.     

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. The associated differential Riccati equation is studied from the point of view of semigroup theory; it is shown to have a classical, explicitly represented solution for very general final data; weighted H?lder regularity results for the optimal pair are deduced. Accepted 10 December 1997     

Markov selections for the 3D stochastic Navier–Stokes equations

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.      

On Abstract Third-Order Differential Equation and Its Applications

Acta Mathematica Sinica, English Series - In this paper, we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution. We...     

Multilevel particle-partition of unity method

This paper is concerned with the particle-partition of unity method, a meshfree generalization of the finite element method. We present the fundamental construction principles and abstract approximation properties of the resulting function spaces V ^PU. Moreover, we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular. The presented construction not only yields optimal convergence rates globally independently of the regularity of the solution, our method shows a super-convergence near the singular points of the solution.     

A Note on the Existence of Nonsmooth Nonconvex Optimization Problems

Sufficient conditions for the existence of a solution to an abstract optimization problem in Banach spaces are given, which do not rely on convexity, regularity properties or a straightforward coerciveness assumption. Applications to sparsity-constrained optimization and to problems from mechanics are provided.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

Counterexamples on L_p-maximal regularity

We establish a transference result for -maximal regularity for the abstract Cauchy problem on Banach space. From this result we deduce counterexamples to -maximal regularity In particular we obtain an operator B without any -maximal regularity although it admits bounded imaginary powers with for all . We also derive an operator which satisfies -maximal regularity on bounded intervals [0, T[ but not on the half line Received March 5, 1997; in final form October 10, 1997     

An abstract Nash–Moser theorem with parameters and applications to PDEs

We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.     

Integration by parts formula and applications to equations with jumps

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes arising as the solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bichteler et?al. (Stochastics Monographs, vol 2. Gordon & Breach, New York, 1987) and Bismut (Z Wahrsch Verw Gebiete 63(2):147?C235, 1983) fails.     

Necessary and sufficient conditions for the regularity and stability of solutions for some partial neutral functional differential equations with infinite delay

In this paper we give necessary and sufficient conditions for the regularity and stability of solutions for some partial neutral functional differential equations with infinite delay. We establish also a generalization and extension of the characterization of the infinitesimal generator of the solution semigroup. To illustrate our abstract results, we study the stability of the neutral Lotka-Volterra model with diffusion.     

Necessary conditions of nondominated solutions in multicriteria decision making

A partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay. By means of abstract regularity results, global existence and uniqueness of a strict solution are obtained; moreover a characterization of the infinitesimal generator of the solution operator is given.     

A second order backward difference method with variable steps for a parabolic problem

The numerical solution of a parabolic problem is studied. The equation is discretized in time by means of a second order two step backward difference method with variable time step. A stability result is proved by the energy method under certain restrictions on the ratios of successive time steps. Error estimates are derived and applications are given to homogenous equations with initial data of low regularity.     

An <Emphasis Type="Italic">h</Emphasis>-<Emphasis Type="Italic">p</Emphasis> version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L ²-, H ¹- and L ^∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.