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On an initial-boundary value problem for the parabolic system which appears in free boundary problems for compressible Navier-Stokes equations

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...

Existence of solutions to the nonstationary Stokes system in H - μ 2 , 1 , μ ∈ (0,1), in a domain with a distinguished axis. Part 1. Existence near the axis in 2d

W. M. Zajączkowski — 2007

Applicationes Mathematicae

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...

Existence of solutions to the nonstationary Stokes system in H - μ 2 , 1 , μ ∈ (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski — 2007

Applicationes Mathematicae

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 L 2 , - μ , v H - μ ¹ , μ ∈ (0,1). We prove an estimate of the velocity in the H - μ 2 , 1 norm and of the gradient of the pressure in the norm of L 2 , - μ . We apply the Fourier transform with respect to the variable along the axis...

Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski — 2005

Studia Mathematica

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 = i = 1 2 ( | | x i v ( 0 ) | | L ( Ω ) + | | x i f | | L ( Ω × ( 0 , T ) ) ) 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 existence of solutions for incompressible magnetohydrodynamic equations

Wisam AlameW. M. Zajączkowski — 2004

Applicationes Mathematicae

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 ( Ω × ( 0 , T ) ) and the pressure q satisfies q L p ( Ω × ( 0 , T ) ) for p ≥ 7/3.

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