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1
Content available remote

Stefan problem in a 2D case

100%
Mucha P.
Colloquium Mathematicum
|
2006
|
tom 105
|
nr 1
149-165
EN
The aim of this paper is to analyze the well posedness of the one-phase quasi-stationary Stefan problem with the Gibbs-Thomson correction in a two-dimensional domain which is a perturbation of the half plane. We show the existence of a unique regular solution for an arbitrary time interval, under suitable smallness assumptions on initial data. The existence is shown in the Besov-Slobodetskiĭ class with sharp regularity in the L₂-framework.
2
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On an estimate for the linearized compressible Navier-Stokes equations in the $L_{p}$-framework

100%
Mucha P.
Colloquium Mathematicum
|
2001
|
tom 87
|
nr 2
159-169
EN
An $L_{p}$-estimate with a constant independent of time for solutions of the linearized compressible Navier-Stokes system in the whole space (under the assumption that solutions have compact supports in space) is obtained.
3
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Global solutions, structure of initial data and the Navier-Stokes equations

100%
Mucha P.
Banach Center Publications
|
2008
|
tom 81
|
nr 1
277-286
EN
In this note we present a proof of existence of global in time regular (unique) solutions to the Navier-Stokes equations in an arbitrary three dimensional domain with a general boundary condition. The only restriction is that the L₂-norm of the initial datum is required to be sufficiently small. The magnitude of the rest of the norm is not restricted. Our considerations show the essential role played by the energy bound in proving global in time results for the Navier-Stokes equations.
4
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The Eulerian limit and the slip boundary conditions-admissible irregularity of the boundary

100%
Mucha P.
Banach Center Publications
|
2005
|
tom 70
|
nr 1
169-183
EN
We investigate the inviscid limit for the stationary Navier-Stokes equations in a two dimensional bounded domain with slip boundary conditions admitting nontrivial inflow across the boundary. We analyze admissible regularity of the boundary necessary to obtain convergence to a solution of the Euler system. The main result says that the boundary of the domain must be at least C²-piecewise smooth with possible interior angles between regular components less than π.
5
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On the structure of flows through pipe-like domains satisfying a geometrical constraint

100%
Mucha P.
Applicationes Mathematicae
|
2004
|
tom 31
|
nr 4
457-471
EN
We study solutions of the steady Navier-Stokes equations in a bounded 2D domain with the slip boundary conditions admitting flow across the boundary. We show conditions guaranteeing uniqueness of the solution. Next, we examine the structure of the solution considering an approximation given by a natural linearization. Suitable error estimates are also obtained.
6
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Stability of Constant Solutions to the Navier-Stokes System in ℝ³

100%
Mucha P.
Applicationes Mathematicae
|
2001
|
tom 28
|
nr 3
301-310
EN
The paper examines the initial value problem for the Navier-Stokes system of viscous incompressible fluids in the three-dimensional space. We prove stability of regular solutions which tend to constant flows sufficiently fast. We show that a perturbation of a regular solution is bounded in $W^{2,1}_r(ℝ³×[k,k+1])$ for k ∈ ℕ. The result is obtained under the assumption of smallness of the L₂-norm of the perturbing initial data. We do not assume smallness of the $W^{2-2/r}_r(ℝ³)$-norm of the perturbing initial data or smallness of the $L_r$-norm of the perturbing force.
7
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Asymptotic behavior of a steady flow in a two-dimensional pipe

100%
Mucha P.
Studia Mathematica
|
2003
|
tom 158
|
nr 1
39-58
EN
The paper investigates the asymptotic behavior of a steady flow of an incompressible viscous fluid in a two-dimensional infinite pipe with slip boundary conditions and large flux. The convergence of the solutions to data at infinities is examined. The technique enables computing optimal factors of exponential decay at the outlet and inlet of the pipe which are unsymmetric for nonzero fluxes of the flow. As a corollary, the asymptotic structure of the solutions is obtained. The results show strong dependence on the magnitude of the Reynolds number.
8
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Inviscid limit for the 2-D stationary Euler system with arbitrary force in simply connected domains

64%
Glass O., Mucha P.
Applicationes Mathematicae
|
2008
|
tom 35
|
nr 1
49-67
EN
We study the convergence in the vanishing viscosity limit of the stationary incompressible Navier-Stokes equation towards the stationary Euler equation, in the presence of an arbitrary force term. This requires that the fluid is allowed to pass through some open part of the boundary.
9
Content available remote

Almost classical solutions of static Stefan type problems involving crystalline curvature

64%
Mucha P., Rybka P.
Banach Center Publications
|
2009
|
tom 86
|
nr 1
223-234
EN
In this note we analyze equilibria of static Stefan type problems with crystalline/singular weighted mean curvature in the plane. Our main goal is to improve the meaning of variational solutions so that their properties allow us to call them almost classical solutions. The idea of our approach is based on a new definition of a composition of multivalued functions.
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