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The existence and uniqueness of solutions of an initial-boundary value problem for the second order linear parabolic system which appears in free boundary problems for compressible Navier-Stokes equations are proved. Moreover, an estimate in anisotropic Sobolev-Slobodetskii spaces with noninteger derivatives is found. The existence and regularity of solutions are shown by means of a regularizer in the same way as in the case of general parabolic systems. However, the problem must be considered separately...

We examine the nonstationary Stokes system in a bounded domain with the boundary slip conditions. We assume that there exists a line which crosses the domain and that the data belong to Sobolev spaces with weights equal to some powers of the distance to the line. Then the existence of solutions in Sobolev spaces with the corresponding weights is proved.

We prove the local existence of solutions for equations of motion of a viscous compressible barotropic fluid in a domain bounded by a free surface. The solutions are shown to exist in exactly those function spaces where global solutions were found in our previous papers [14, 15].

We consider the nonstationary Stokes system with slip boundary conditions in a bounded domain which contains some distinguished axis. We assume that the data functions belong to weighted Sobolev spaces with the weight equal to some power function of the distance to the axis. The aim is to prove the existence of solutions in corresponding weighted Sobolev spaces. The proof is divided into three parts. In the first, the existence in 2d in weighted spaces near the axis is shown. In the second, we show...

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f\in L_{2,-\mu }$, $v₀\in H_{-\mu }¹$, μ ∈ (0,1). We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along the axis...

Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I={\sum }_{i=1}^2\left(\right||{\partial }_{x₃}^i{v\left(0\right)\left|\right|}_{L₂\left(\Omega \right)}+\left|\right|{\partial }_{x₃}^i{f\left|\right|}_{L₂\left(\Omega \times \right(0,T\left)\right)})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by the Leray-Schauder...

Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_p^{2,1}\left(\Omega \times (0,T)\right)$ and the pressure q satisfies $\nabla q\in L_p\left(\Omega \times (0,T)\right)$ for p ≥ 7/3.

The existence of global regular axially symmetric solutions to Navier-Stokes equations in a bounded cylinder and for boundary slip conditions is proved. Next, stability of these solutions is shown.