CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in ${D}_{\vartheta}$...............................................346. Existence of solutions of (3.14) in ${H}_{\mu}^{k}\left({D}_{\vartheta}\right)$............417. Green function................................................................528....

We consider the motion of a viscous compressible barotropic fluid in ${\mathbb{R}}^{3}$ bounded by a free surface which is under constant exterior pressure. For a given initial density, initial domain and initial velocity we prove the existence of local-in-time highly regular solutions. Next assuming that the initial density is sufficiently close to a constant, the initial pressure is sufficiently close to the external pressure, the initial velocity is sufficiently small and the external force vanishes we prove...

Existence and uniqueness of global regular special solutions to Navier-Stokes equations with boundary slip conditions in axially symmetric domains is proved. The proof of global existence relies on the global existence results for axially symmetric solutions which were obtained by Ladyzhenskaya and Yudovich-Ukhovskiĭ in 1968 who employed the problem for vorticity. In this paper the equations for vorticity also play a crucial role. Moreover, the boundary slip conditions imply appropriate boundary...

Using the Il'in integral representation of functions, imbedding theorems for weighted anisotropic Sobolev spaces in 𝔼ⁿ are proved. By the weight we assume a power function of the distance from an (n-2)-dimensional subspace passing through the domain considered.

The aim of this paper is to prove the existence of solutions to the Poisson equation in weighted Sobolev spaces, where the weight is the distance to some distinguished axis, raised to a negative power. Therefore we are looking for solutions which vanish sufficiently fast near the axis. Such a result is useful in the proof of the existence of global regular solutions to the Navier-Stokes equations which are close to axially symmetric solutions.

The existence of solutions to an initial-boundary value problem to the heat equation in a bounded domain in ℝ³ is proved. The domain contains an axis and the existence is proved in weighted anisotropic Sobolev spaces with weight equal to a negative power of the distance to the axis. Therefore we prove the existence of solutions which vanish sufficiently fast when approaching the axis. We restrict our considerations to the Dirichlet problem, but the Neumann and the third boundary value problems can...

The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, ${v\xb7n\u0305|}_{S}=0$, S = ∂Ω, in weighted Hilbert spaces is proved. It is assumed that Ω contains an axis L and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of L. Their existence in Ω follows by regularization.

The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v\xb7n\u0305|}_{S}=0$, S = ∂Ω in weighted ${L}_{p}$-Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

Global existence of axially symmetric solutions to the Navier-Stokes equations in a cylinder with the axis of symmetry removed is proved. The solutions satisfy the ideal slip conditions on the boundary. We underline that there is no restriction on the angular component of velocity. We obtain two kinds of existence results. First, under assumptions necessary for the existence of weak solutions, we prove that the velocity belongs to ${W}_{4/3}^{2,1}\left(\Omega \times (0,T)\right)$, so it satisfies the Serrin condition. Next, increasing regularity...

The existence and uniqueness of solutions to the Navier-Stokes equations in a cylinder Ω and with boundary slip conditions is proved. Assuming that the azimuthal derivative of cylindrical coordinates and azimuthal coordinate of the initial velocity and the external force are sufficiently small we prove long time existence of regular solutions such that the velocity belongs to ${W}_{5/2}^{2,1}\left(\Omega \times (0,T)\right)$ and the gradient of the pressure to ${L}_{5/2}\left(\Omega \times (0,T)\right)$. We prove the existence of solutions without any restrictions on the lengths of the...

We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat conducting fluid. The inequality is essential in proving the global existence of solutions.

We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat concluding capillary fluid. The inequality is essential in proving the global existence of solutions.

The motion of a viscous compressible heat conducting fluid in a domain in ℝ³ bounded by a free surface is considered. We prove local existence and uniqueness of solutions in Sobolev-Slobodetskiĭ spaces in two cases: with surface tension and without it.

We prove the existence of weak solutions to the Navier-Stokes equations describing the motion of a fluid in a Y-shaped domain.

We derive an inequality for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. This inequality is crucial to proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskiĭ spaces and close to an equilibrium state.

We derive inequalities for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. The inequalities are crucial in proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskii space and close to an equilibrium state.

We examine the regularity of solutions of a certain parabolic system in the weighted Sobolev space ${W}_{2,\mu}^{2,1}$, where the weight is of the form ${r}^{\mu}$, r is the distance from a distinguished axis and μ ∈ (0,1).

We consider the Poisson equation with the Dirichlet and the Neumann boundary conditions in weighted Sobolev spaces. The weight is a positive power of the distance to a distinguished plane. We prove the existence of solutions in a suitably defined weighted space.

Download Results (CSV)