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## Sobolev Spaces (sobolev + space)
Kinds of Sobolev Spaces
## Selected Abstracts## Global well-posedness of the Cauchy problem for certain magnetohydrodynamic-, models MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010Yi DuAbstract This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics-, model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as ,,0, the MHD-, model reduces to the MHD equations, and the solutions of the MHD-, model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Convergence rates toward the travelling waves for a model system of the radiating gas MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Masataka NishikawaAbstract The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t),1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Determining the temperature from incomplete boundary data MATHEMATISCHE NACHRICHTEN, Issue 16 2007B. Tomas JohanssonAbstract An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L2 -space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The boundary element method with Lagrangian multipliers, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009Gabriel N. GaticaAbstract On open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi-optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment. The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] ## Refined mixed finite element method for the elasticity problem in a polygonal domain NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002M. FarhloulAbstract The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient , are obtained for , large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323,339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009 [source] ## Non-homogeneous Navier,Stokes systems with order-parameter-dependent stresses MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2010Helmut AbelsAbstract We consider the Navier,Stokes system with variable density and variable viscosity coupled to a transport equation for an order-parameter c. Moreover, an extra stress depending on c and ,c, which describes surface tension like effects, is included in the Navier,Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier,Stokes system is also a special case of our system. We prove short-time existence of strong solution in Lq -Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## On Helmholtz decompositions and their generalizations,An overview MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2010W. SprössigAbstract Helmholtz' theorem initiates a remarkable development in the theory of projection methods that are adapted to the numerical solution of equations in fluid dynamics and elasticity. There is a dense connection with Hodge-de Rham decompositions of smooth 1-forms. We give an overview of this type of decompositions and discuss their applications to vector, quaternionic and Clifford-valued boundary value problems in the corresponding Hilbert,Sobolev spaces. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Traces of Sobolev functions with one square integrable directional derivative MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2006M. GregorattiAbstract We consider the Sobolev spaces of square integrable functions v, from ,n or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ,,,,v. In these spaces we define closed trace operators on the boundaries ,Q and on the hyperplanes {r,, = z}, z , ,\{0}, which turn out to be possibly unbounded with respect to the usual L2 -norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L2, we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ,n and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004Taeko YamazakiAbstract We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## The diffraction in a class of unbounded domains connected through a hole MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2003Yu. V. ShestopalovAbstract In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi-infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov,Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Global existence of solutions for non-small data to non-linear spherically symmetric thermoviscoelasticity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003J. GawineckiAbstract We consider some initial,boundary value problems for non-linear equations of thermoviscoelasticity in the three-dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003Andreas RathsfeldAbstract In this paper, we consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, ,1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]3) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N,(2,r)/2). Note that, in contrast to well-known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón,Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## The mapping properties of the radiosity operator along an edge MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2002Olaf HansenAbstract In this article we study the radiosity operator along an edge between two adjacent half-planes. First we show that the radiosity operator is invertible in a whole scale of anisotropic Sobolev spaces. In the absence of any shadows we are able to derive regularity properties of the solution, which depend only on the angle between the half-planes, the reflectivity coefficients and the right-hand side. This work can be considered as a supplement to the article of Rathsfeld (Mathematical Methods in the Applied Sciences 1999; 22: 217,241). Copyright © 2002 John Wiley & Sons, Ltd. [source] ## On the Stokes system and on the biharmonic equation in the half-space: an approach via weighted Sobolev spaces MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2002Tahar Z. BoulmezaoudAbstract In this paper, we investigate the Stokes system and the biharmonic equation in a half-space of ,n. Our approach rests on the use of a family of weighted Sobolev spaces as a framework for describing the behaviour at infinity. A complete class of existence, uniqueness and regularity results for both the problems is proved. The proofs are mainly based on the principle of reflection. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients MATHEMATISCHE NACHRICHTEN, Issue 9 2009Vladimir KozlovAbstract We consider the Dirichlet problem for non-divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi-linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,,, , , [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The Cauchy problem for quasilinear SG-hyperbolic systems MATHEMATISCHE NACHRICHTEN, Issue 7 2007Marco CappielloAbstract We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) , [0, T ] × ,n and presenting a linear growth for |x | , ,. We prove well-posedness in the Schwartz space ,, (,n). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Approximate identities in variable Lp spaces MATHEMATISCHE NACHRICHTEN, Issue 3 2007D. Cruz-Uribe SFOAbstract We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable Lp spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable Lp spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Minimal regularity of the solution of some boundary value problems of Signorini's type in polygonal domains MATHEMATISCHE NACHRICHTEN, Issue 6 2005Denis MercierAbstract We study the regularity in Sobolev spaces of the solution of transmission problems in a polygonal domain of the plane, with unilateral boundary conditions of Signorini's type in a part of the boundary and Dirichlet or Neumann boundary conditions on the remainder part. We use a penalization method combined with an appropriated lifting argument to get uniform estimates of the approximated solutions in order to obtain some minimal regularity results for the exact solution. The same method allows us to consider problems with thin obstacles. It can be easily extended to 3D problems. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Estimates of difference norms for functions in anisotropic Sobolev spaces MATHEMATISCHE NACHRICHTEN, Issue 1 2004V. I. KolyadaAbstract We investigate the spaces of functions on ,n for which the generalized partial derivatives Dkf exist and belong to different Lorentz spaces L . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non-increasing rearrangements. These methods enable us to cover also the case when some of the pk's are equal to 1. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The finite Hilbert transform and weighted Sobolev spaces MATHEMATISCHE NACHRICHTEN, Issue 1 2004David ElliottAbstract The boundedness of the finite Hilbert transform operator on certain weighted Lp spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## General theory of domain decomposition: Indirect methods NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002Ismael HerreraAbstract According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz-Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best-known formulations of domain decomposition methods, such as those based on the application of Steklov-Poincaré operators. Trefftz-Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296,322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008 [source] ## Remarks on Duality in Graph Spaces of First-Order Linear Operators PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006Max JensenArticle first published online: 4 DEC 200Graph spaces provide a setting alternative to Sobolev spaces and BV spaces, which is suitable for the analysis of first-order linear boundary value problems such as Friedrichs systems. Besides investigations of the well-posedness of the continuous problem there is also an increasing interest in the error analysis of finite element methods within a graph space framework. In this text we elucidate various methods for an explicit representation of dual spaces of graph spaces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2009Yuri TrakhininWe study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc. [source] ## On closed boundary value problems for equations of mixed elliptic-hyperbolic type, COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2007Daniela LupoFor partial differential equations of mixed elliptic-hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet-conormal types. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a , b , c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. © 2006 Wiley Periodicals, Inc. [source] ## An inverse problem for the dynamical Lamé system with two sets of boundary data COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2003Oleg ImanuvilovWe prove uniqueness and a Hölder-type stability of reconstruction of all three time-independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements. In proofs we use Carleman-type estimates in Sobolev spaces of negative order. © 2003 Wiley Periodicals, Inc. [source] ## The attractor for a nonlinear reaction-diffusion system in an unbounded domain COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2001Messoud A. EfendievIn this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc. [source] |