Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

A new method is given to establish L_q,r-estimates for solutions of the nonstationary Stokes problem. The method is based on estimates for heat potentials in these spaces, and it is not connected with investigation of the resolvent for the Stokes operator. It is expected that the method is applicable to a wide class of parabolic initial boundary-value problems. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarow POMI, Vol. 222, 1994, pp. 124–151.     

Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms

An L_q(L_p)-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class VMO_x, which, in particular, contains all measurable functions depending only on t. The method of proving simplifies the methods previously used in the case p=q.     

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] ⁿ transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ _μ and ρ _ν . It is shown that if there exist positive constants ? and M such that ? < ρ _ν < M, ? < ρ _ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)^?1 (α + β)^?1 > 1 and ρ _μ ∈ W ^1,α (Θ), ρ _ν ∈ W ^1,β ) (Θ), where W ^1,q denotes a Sobolev class, then the mapping T belongs to the class W ^1,p (Θ).     

Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

We extend here some existence and uniqueness results for the exterior Stokes problem in weighted Sobolev spaces. We also study the regularity of the solutions ( u ,π) and prove optimal a priori estimates for the solutions with ∇² u ,∇π∈L^p. The influence of some compatibility conditions on the behaviour at infinity of the solution is finally studied and leads to new asymptotic expansions. Copyright © 2000 John Wiley & Sons, Ltd.     

Lorentz spaces,with mixed norms

We study Lorentz spaces, with mixed or iterated norms. The usual Lorentz spaces enable us to clarify the concept of weak-type; the spaces with mixed norms enable us to clarify and to classify four extended notions of weak-type in product space. We give also an interpolation theorem, of the Marcinkiewicz type.     

Some notes on embedding for anisotropic Sobolev spaces

In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, \(W_{{\Lambda ^{p,q}}(w)}^{{r_1}, \cdots ,{r_n}}\) and \(W_X^{{r_1}, \cdots ,{r_n}}\), where Λ ^p,q (w) is the weighted Lorentz space and X is a rearrangement invariant space in ? ⁿ . The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of B _p weights.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted Lp's theory with 1<p<∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions.     

Estimates for functions in Sobolev spaces defined on unbounded domains

Given a function u belonging to a suitable Beppo–Levi or Sobolev space and an unbounded domain Ω in , we prove several Sobolev-type bounds involving the values of u on an infinite discrete subset A of Ω. These results improve the previous ones obtained by Madych and Potter [W.R. Madych, E.H. Potter, An estimate for multivariate interpolation, J. Approx. Theory 43 (1985) 132–139] and Madych [W.R. Madych, An estimate for multivariate interpolation II, J. Approx. Theory 142 (2006) 116–128].     

Kolmogorov widths in finite-dimensional spaces with mixed norms

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 43–52, January, 1994.     

Uniqueness for the Navier–Stokes equations and multipliers between Sobolev spaces

We re-prove various uniqueness theorems for the Navier–Stokes equations, stating the assumptions in terms of multipliers between Sobolev spaces instead of Lebesgue or Lorentz spaces.     

Estimates of low Sobolev norms of solutions for some nonlinear evolution equations

In this work we obtain results on the estimates of low Sobolev norms for solutions of some nonlinear evolution equations, in particular we apply our method for the complex modified Korteweg-de Vries type equation and Benjamin-Ono equation.     

Embeddings of anisotropic Sobolev spaces on unbounded domains

Summary Integral representations of junctions in the anisotropic Sobolev spaces on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain,     

Estimates of Solutions of the Stokes Equations in S. L. Sobolev Spaces with a Mixed Norm

Estimates of solutions of the evolutionary Stokes and Navier–Stokes equations in a bounded n-dimensional domain are obtained. By using explicit formulas, the structure of these solutions is analyzed in the case of a half-space. Bibliography: 12 titles.     

The embedding theorem for Sobolev spaces with mixed norm for limit exponents

This article is devoted to the investigation of some properties of Sobolev spaces with mixed norm.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 62–72, January, 1996.