The resolvent problem for the stokes system in exterior domains: An elementary approach

We consider the Stokes system with resolvent parameter in an exterior domain: under Dirichlet boundary conditions. Here Ω is a bounded domain with C² boundary, and [λ??\] ? [∞, 0], ν >0. Using the method of integral equations, we are able to construct solutions ( u , π) in L^p spaces. Our approach yields an integral representation of these solutions. By evaluating the corresponding integrals, we obtain L^p estimates that imply in particular that the Stokes operator on exterior domains generates an analytic semigroup in L^p.     

On the stationary flows of viscous,incompressible and heat-conducting fluids

We consider a boundary-value problem describing the motion of viscous, incompressible and heat-conducting fluids in a bounded domain in ?³. We admit non-homogeneous boundary conditions, the appearance of exterior forces and heat sources. Our aim is to prove the existence of a solution of the problem in Sobolev spaces.     

Green Integrals on Manifolds with Cracks

We prove the existence of a limit in H ^m (D)of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchyproblem in D with data on S, for an elliptic operator A of order m 1, whenever these solutions exist.This representation involves the sum of a series whose terms are iterationsof the double layer potential. A similar regularisation is constructed also for a mixed problem in D.     

Weight-monodromy conjecture for <Emphasis Type="Italic">p</Emphasis>-adically uniformized varieties

The aim of this paper is to prove the weight-monodromy conjecture (Delignes conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants. We also consider a p-adic analogue by using the weight spectral sequence of Mokrane.     

Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

Anisotropic equations in weighted Sobolev spaces of higher order

We consider the strongly nonlinear boundary value problem,

Strong solutions to the Stokes equations of a flow around a rotating body in weighted Lq spaces

We consider the motion of a fluid in the exterior of a rotating obstacle. This leads to a modified version of the Stokes system which we consider in the whole space ${\mathds R}^n$, n = 2 or n = 3 and in an exterior domain $D\subset {\mathds R}^3$. For every q ∈ (1, ∞) we prove existence of solutions and estimates in function spaces with weights taken from a subclass of the Muckenhoupt class A_q. Moreover, uniqueness is shown modulo a vector space of dimension 3. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain

We consider the 3-D Navier-Stokes equations in the half-space ₊³, or a bounded domain with smooth boundary, or else an exterior domain with smooth boundary. Some new sufficient conditions on pressure or the gradient of pressure for the regularity of weak solutions to the Navier-Stokes equations are obtained.Mathematics Subject Classification (2000):35B45, 35B65, 76D05     

Analysis of a kinematic dynamo model with FEM–BEM coupling

We consider a kinematic dynamo model in a bounded interior simply connected region Ω and in an insulating exterior region . In the so‐called direct problem, the magnetic field B and the electric field E are unknown and are driven by a given incompressible flow field w . After eliminating E , a vector and a scalar potential ansatz for B in the interior and exterior domains, respectively, are applied, leading to a coupled interface problem. We apply a finite element approach in the bounded interior domain Ω, whereas a symmetric boundary element approach in the unbounded exterior domain Ω^c is used. We present results on the well‐posedness of the continuous coupled variational formulation, prove the well‐posedness and stability of the semi‐discretized and fully discretized schemes, and provide quasi‐optimal error estimates for the fully discretized scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

Weighted Schauder estimates for Evolution Stokes Problem

Abstract We prove the solvability of the evolution Stokes problem in bounded or exterior domain Ω∈ Rⁿ under the assumption that the initial value of the velocity vector field v₀ belongs to C^s(Ω), (in particular, it can be only continuous). The solution is obtained in some weighted H?lder spaces. This result makes it possible to prove the local solvability of a nonlinear problem under the same assumption concerning v₀. Keywords: Stokes equations, Weighted norms Mathematics Subject Classification (2000): 35Q30, 76D03     

On the coupling of MIXED-FEM and BEM for an exterior Helmholtz problem in the plane

Summary This paper deals with an elliptic boundary value problem posed in the plane, with variable coefficients, but whose restriction to the exterior of a bounded domain reduces to a Helmholtz equation. We consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium, coupled with a boundary integral method applied to the Helmholtz equation in . We utilize suitable auxiliary problems, duality arguments, and Fredholm alternative to show that the resulting formulation of the problem is well posed. Then, we define a corresponding Galerkin scheme by using rotated Raviart-Thomas subspaces and spectral elements (on the interface). We show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates. Finally we illustrate our analysis with some results from computational experiments.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups

For a Hopf algebra , we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of and (2) an exterior extension of the dual algebra ^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on . The first differential complex is an analogue of the de Rham complex. When ^* is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator Q. We give a recursive relation that uniquely defines the operator Q. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra U _q(gl(N)).     

The stationary and non-stationary stokes system in exterior domains with non-zero divergence and non-zero boundary values

In an exterior domain Ω??ⁿ, n ? 2, we consider the generalized Stokes resolvent problem in L^q-space where the divergence g = div u and inhomogeneous boundary values u = ψ with zero flux ∫_?Ωψ·N do = 0 may be prescribed. A crucial step in our approach is to find and to analyse the right space for the divergence g. We prove existence, uniqueness and a priori estimates of the solution and get new results for the divergence problem. Further, we consider the non-stationary Stokes system with non-homogeneous divergence and boundary values and prove estimates of the solution in L⁵(0, T;L^q(Ω)) for 1 < s, q < ∞.     

Nonstationary Stokes system in Sobolev–Slobodetski spaces

We consider an initial-boundary value problem for the nonstationary Stokes system in a bounded domain $\Omega \subset \mathbb R ^3$ with slip boundary conditions. We prove the existence in the Hilbert–Sobolev–Slobodetski spaces with fractional derivatives. The proof is divided into two main steps. In the first step by applying the compatibility conditions an extension of initial data transforms the considered problem to a problem with vanishing initial data such that the right-hand sides data functions can be extended by zero on the negative half-axis of time in the above mentioned spaces. The problem with vanishing initial data is transformed to a functional equation by applying an appropriate partition of unity. The existence of solutions of the equation is proved by a fixed point theorem. We prove the existence of such solutions that $v\in H^{l+2,l/2+1}(\Omega \times (0,T)),\,\nabla p\in H^{l,l/2}(\Omega \times (0,T)),\,v$ —velocity, $p$ —pressure, $l\in \mathbb R _+\cup \{0\},\,l \ne [l]+\frac{1}{2}$ and the spaces are introduced by Slobodetski and used extensively by Lions–Magenes. We should underline that to show solvability of the Stokes system we need only solvability of the heat and the Poisson equations in $\mathbb R ^3$ and $\mathbb R _+^3$ . This is possible because the slip boundary conditions are considered.     

Regularity of a Boundary Point for Degenerate Parabolic Equations with Measurable Coefficients

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation with the p-Laplace operator p. We prove a sufficient condition for the regularity of a boundary point in terms of C _p-capacity.     

Traces of Sobolev functions on the Ahlfors sets of Carnot groups

We prove the converse of the trace theorem for the functions of the Sobolev spaces W _p ^l on a Carnot group on the regular closed subsets called Ahlfors d-sets (the direct trace theorem was obtained in one of our previous publications). The theorem generalizes Johnsson and Wallin’s results for Sobolev functions on the Euclidean space. As a consequence we give a theorem on the boundary values of Sobolev functions on a domain with smooth boundary in a two-step Carnot group. We consider an example of application of the theorems to solvability of the boundary value problem for one partial differential equation.     

The principle of penalized empirical risk in severely ill-posed problems

We study a standard method of regularization by projections of the linear inverse problem Y=Ax+, where is a white Gaussian noise, and A is a known compact operator. It is assumed that the eigenvalues of AA^* converge to zero with exponential decay. Such behavior of the spectrum is typical for inverse problems related to elliptic differential operators. As model example we consider recovering of unknown boundary conditions in the Dirichlet problem for the Laplace equation on the unit disk. By using the singular value decomposition of A, we construct a projection estimator of x. The bandwidth of this estimator is chosen by a data-driven procedure based on the principle of minimization of penalized empirical risk. We provide non–asymptotic upper bounds for the mean square risk of this method and we show, in particular, that this approach gives asymptotically minimax estimators in our model example.Mathematics Subject Classification (2000):Primary 62G05, 62G20; secondary 62C20