Conjugate Cauchy Problem for Parabolic Shilov Type Systems with Nonnegative Genus

The problem of solvability of the Gelfand–Shilov conjugate Cauchy problem for parabolic Shilov type systems with variable coefficients of bounded smoothness and nonnegative genus in the spaces S _β is studied by constructing the fundamental solution of this problem and analyzing its basic properties.     

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C^∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C^∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.     

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ +N ^?2 ln² N) ln(j +1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.     

Maximal smoothness of the anti-analytic part of a trigonometric null series

We proved recently (C. R. Acad. Sci. Paris, Ser. I 336 (2003) 475–478) that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may belong to L² on the circle. Here we prove that it can even be C^∞, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a ‘new quasi-analyticity’. To cite this article: G. Kozma, A. Olevski??, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers

A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients in a square domain is considered. Dirichlet conditions are specified on two sides orthogonal to the flow, and Neumann conditions are set on the other two sides. The right-hand side and the boundary functions are assumed to be sufficiently smooth, which ensures the required smoothness of the desired solution in the domain, except for neighborhoods of the corner points. Only zero-order compatibility conditions are assumed to hold at the corner points. The problem is solved numerically by applying an inhomogeneous monotone difference scheme on a rectangular piecewise uniform Shishkin mesh. The inhomogeneity of the scheme lies in that the approximating difference equations are not identical at different grid nodes but depend on the perturbation parameter. Under the assumptions made, the numerical solution is proved to converge ?-uniformly to the exact solution in a discrete uniform metric at an O(N ^?3/2ln² N) rate, where N is the number of grid nodes in each coordinate direction.     

Cauchy problem for fractional diffusion equations

We consider an evolution equation with the regularized fractional derivative of an order α∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.     

The principal part of a block map

A block map is a map f{0, 1}ⁿ {0, 1} for some n ? 1. Block maps can be represented by polynomials with coefficients in Z₂. The notion of the principal part of a block map is introduced. It is used to obtain some conditions under which block maps which are linear in the first variable but not linear in all variables are irreducible with respect to composition.     

The behavior of solutions of semilinear elliptic equations of second order of the form Lu = e u in the infinite cylinder

We consider a semilinear elliptic equation of second order with variable coefficients of the form Lu = e ^u in the semi-infinite cylinder whose solution satisfies a homogeneous Neumann condition on the lateral surface of the cylinder.     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Sharp estimates for a class of hyperbolic pseudo-differential equations

In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L ^p ? L^p, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R¹⁺ⁿ with n ≤ 4 (total dimension ≤ 5), we give a complete list of L ^p ? L^p properties. In particular, this contains the very important case R¹⁺³.     

Locally adaptive fitting of semiparametric models to nonstationary time series

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function μ(·) and a p-dimensional function θ(·)=(θ⁽¹⁾(·),…,θ^(p)(·))′ that parametrizes the time-varying spectral density f_θ(·)(λ). Whereas the mean function is estimated by a usual kernel estimator, each component of θ(·) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of θ⁽ⁱ⁾(·), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback–Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of θ⁽ⁱ⁾(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.     

Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle

Here nonsmooth solutions of a differential equation are treated as solutions for which the compatibility conditions are not required to hold at the corner points of the domain and hence corner singularities can occur. In the present paper, we drop the compatibility conditions at three of the four vertices of a rectangle. At the remaining vertex, from which a characteristic (inclined) of the reduced equation issues, we impose compatibility conditions providing the C ^3,λ -smoothness of the desired solution in a neighborhood of that vertex as well as additional conditions leading to the smoothness of solutions of the reduced equation occurring in the regular component of the solution of the considered problem. Under our assumptions and for a sufficient smoothness of the coefficients of the equation and its right-hand side, we show that the classical five-point upwind approximation on a Shishkin piecewise uniform mesh preserves the accuracy specific for the smooth case; i.e., the mesh solution uniformly (with respect to a small parameter) converges in the L _∞ ^h -norm to the exact solution at the rate O(N ⁻¹ ln² N), where N is the number of mesh nodes in each of the coordinate directions.     

Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems

We present a local almost everywhere C ^1,α -regularity result for a general class of p-nonlinear non-diagonal parabolic systems. The main part of the considered systems depends on space-time variable, solution and symmetric part of the gradient of solution. To obtain our result, we adapt for the symmetric-gradient case techniques developed for the full-gradient case by Duzaar, Mingione and coauthors.     

Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektoräumen

This paper is a continuation of [6], in which I identified thec ^∞-complete bornological locally convex spaces (in short: 1cs) as the right ones for infinite dimensional analysis. Here I discuss smooth mappings between arbitrary 1cs, where a mapping is called smooth iff its compositions with smooth curves are smooth. The 1^st part is mainly devoted to prove the cartesian closedness of the category of (bornological,c ^∞-complete) 1cs together with the smooth mappings between them. In the 2^nd part I discuss the bornology of function spaces and furthermore demonstrate the smoothness of the differentiation process. Finally, in the 3^rd part, I compare this concept of smoothness with several others, discussed byKeller in [5], and show it to be the weakest that fulfills the chainrule.     

L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes

This paper is concerned with the finite volume element methods on quadrilateral mesh for second-order elliptic equation with variable coefficients. An error estimate in L ² norm is shown on the quadrilateral meshes consisting of h ²-parallelograms. Superconvergence of numerical solution is also derived in an average gradient norm on h ²-uniform quadrilateral meshes. Numerical examples confirm our theoretical conclusions.     

Dirichlet Problem for a Class of Quasilinear Elliptic Equations

In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved.     

Description of traces of functions in the Sobolev space with a Muckenhoupt weight

We characterize the trace of the Sobolev space W _p ^l (? ⁿ , γ) with 1 < p < ∞ and weight γ ∈ A _p ^loc (? ⁿ ) on a d-dimensional plane for 1 ≤ d < n. It turns out that for a function φ to be the trace of a function f ∈ W _p ^l (? ⁿ , γ), it is necessary and sufficient that φ belongs to a new Besov space of variable smoothness, $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ , constructed in this paper. The space $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ is compared with some earlier known Besov spaces of variable smoothness.