Sobolev Slobodetskii Spaces Research Articles

Local solutions for nonhomogeneous Navier–Stokes equations with large flux

The local existence of solutions to nonhomogeneous Navier–Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev–Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the ‐axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that ‐coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to , the gradient of pressure to and the density to , where .

On Some Elliptic Boundary Value Problems in Conic Domains

A model elliptic pseudodifferential equation in a polyhedral cone is considered, and the situation when some of the parameters of the cone tend to their limiting values is investigated. In Sobolev–Slobodetskii spaces, a solution of the equation in the cone is constructed in the case of a special wave factorization of the elliptic symbol. It is shown that a limit solution of the boundary value problem with an additional integral condition can exist only under additional constraints on the boundary function.

On the Stokes System in Cylindrical Domains

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev–Slobodetskii spaces such that the velocity belongs to \(W_r^{2+\sigma ,1+\sigma /2}(\Omega ^T)\) and the gradient of pressure to \(W_r^{\sigma ,\sigma /2}(\Omega ^T)\), where \(r\in (1,\infty )\), \(\sigma \in (0,1)\), \(\Omega ^T=\Omega \times (0,T)\). These are special Besov spaces: \(B_{r,r}^{2+\sigma ,1+\sigma /2}(\Omega ^T)\) and \(B_{r,r}^{\sigma ,\sigma /2}(\Omega ^T)\), respectively. The existence is proved by the technique of regularizer.

Frames of solutions and discrete analysis of pseudo‐differential equations

We construct discrete analogs of multidimensional singular integral operators and study their invertibility. Moreover, we give a comparison between continual and discrete case. We give the theory of periodic Riemann problem also, because it is needed for studying invertibility of so‐called paired equations. For more general case of pseudo‐differential operators, we construct the solvability theory for discrete pseudo‐differential equations in discrete analogs of Sobolev–Slobodetskii spaces. Some comparison results for discrete and continuous solutions are given also in appropriate discrete normed spaces.

On Discrete Solutions for Elliptic Pseudo-Differential Equations

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

On Solvability Conditions for a Certain Conjugation Problem

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

On Some Multidimensional Limit Boundary Value Problems

We study solvability for a model elliptic pseudo-differential equation with an additional integral condition in Sobolev–Slobodetskii spaces in certain conical domains in Euclidean space. . Using the wave factorization for an elliptic symbol we construct a solution for this boundary value problem and study the behavior of the solution when parameters of cones tend to their extremal values. It was shown that for such a solvability we need certain additional condition besides presented integral condition.

On elliptic problems in domains with low dimensional boundaries

We study a solvability for some concrete boundary value problems in domains with a low dimensional boundary. Such boundaries are considered as a limit situation of a volume problem for which one or more parameters of a domain tend to their endpoint values. The solvability conditions in Sobolev-Slobodetskii spaces are given using the wave factorization concept for an alliptic symbol..

Boundary Value Problems for Elliptic Pseudodifferential Equations in a Multidimensional Cone

We consider model boundary value problems for elliptic pseudodifferential equations in multidimensional cones. A result on the unique solvability and representation of solutions of some boundary value problems in suitable Sobolev–Slobodetskii spaces is obtained. A priori estimates of solutions are given.

On Certain 3-Dimensional Limit Boundary Value Problems

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

On some pseudo-differential equations and transmutation operators

We describe some operators for solving model elliptic pseudo-differential equations in special canonical domains. It helps us to write a general solution of corresponding pseudo-differential equation in an explicit form. Moreover, knowing a general solution we can choose additional (possibly boundary) conditions to determine uniquely the solution. All considerations we give in Sobolev–Slobodetskii spaces.

Stationary compressible Navier–Stokes equations with inflow condition in domains with piecewise analytical boundaries

We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries. The mentioned situation creates a singularity which limits regularity at such points. We show the existence and uniqueness of regular solutions in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. The proofs require the language of suitable fractional Sobolev spaces. In other words our result is an example where application of fractional spaces is irreplaceable, although the subject is a classical system.

On some distributions associated to boundary value problems

We consider an elliptic pseudo-differential equation in a multidimensional cone in Sobolev–Slobodetskii space and describe its general solution for some cases. This description is based on some distributions supported on a conical surface. For certain conical surfaces we describe such distributions. It permits to use such distributions to refine a general solution.

Pseudo-differential equations and conical potentials: 2-dimensional case

We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.

To the theory of discrete boundary value problems

We consider discrete analogues of pseudo-differential operators and related discrete equations and boundary value problems. Existence and uniqueness results for special elliptic discrete boundary value problem and comparison for discrete and continuous solutions are given for certain smooth data in discrete Sobolev–Slobodetskii spaces.

On Some Discrete Potential Like Operators

Abstract We consider some discrete pseudo-differential equations in discrete Sobolev–Slobodetskii spaces. For a discrete half-space and certain values of an index of periodic factorization for an elliptic symbol we introduce additional potential-like unknowns and prove existence and uniqueness theorem in appropriate discrete Sobolev–Slobodetskii spaces.

Linearization of a Free Boundary Problem of Magnetohydrodynamics

We consider the linear conjugation problem for a magnetic field. Such a problem arises in the linearization of a free boundary problem with two fluids. We prove the unique solvability in the Sobolev–Slobodetskii spaces.

Pseudodifferential Operators and Equations of Variable Order

A new class of elliptic pseudodifferential equations of variable order is considered. Local Sobolev–Slobodetskii spaces are introduced and used to obtain theorems on the continuity of these pseudodifferential operators and describe the Fredholm properties of the corresponding pseudodifferential equations and boundary value problems.

Model Elliptic Boundary-Value Problems for Pseudodifferential Operators in Canonical Nonsmooth Domains

We consider a simplest elliptic pseudodifferential equation in a multi-dimensional cone (multi-dimensional angle) and describe all possible structures of its solutions related to the wave factorization of the elliptic symbol. Depending on the index of wave factorization, we consider various statements of well-posed boundary-value problems. The existence of solutions is studied in Sobolev–Slobodetskii spaces.

Steady transport equation in Sobolev–Slobodetskii spaces

We show existence of a regular solution in Sobolev-Slobodetskii spaces to stationary transport equation with inflow boundary condition in a bounded domain $\Omega \subset \mathbb{R}^2$. Our result is subject to quite general constraint on the shape of the boundary around the points where the characteristics become tangent to the boundary which applies in particular to piecewise analytical domains. Our result gives a new insight on the issue of boundary singularity for the inflow problem for stationary transport equation, solution of which is crucial for investigation of stationary compressible Navier-Stokes equations with inflow/outflow.