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Global free boundary problem for incompressible magnetohydrodynamics

100%
Kacprzyk P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We consider the motion of incompressible mhd in a domain bounded a free surface. In the external domain there exists an electromagnetic field generated by some currents which keeps the mhd flow in the bounded domain. On the free surface transmission conditions for the electromagnetic fields are imposed. For sufficiently small initial velocity and vanishing external force the global existence is proved. The L₂-approach is used. This helps us to treat the transmission conditions.
2
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Local free boundary problem for incompressible magnetohydrodynamics

100%
Kacprzyk P.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We consider the motion of an incompressible magnetohydrodynamic (mhd) fluid in a domain bounded by a free surface. In the external domain there exists an electromagnetic field generated by some currents which keeps the mhd flow in the bounded domain. Then on the free surface transmission conditions for electromagnetic fields are imposed. In this paper we prove existence of local regular solutions by the method of successive approximations. The L₂ approach is used. This helps us to treat the transmission conditions.
3
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Global attractor for the Navier-Stokes equations in a cylindrical pipe

95%
Kacprzyk P.
Annales Polonici Mathematici
|
2010
|
tom 97
|
nr 3
201-217
EN
Global existence of regular special solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has already been shown. In this paper we prove the existence of the global attractor for the Navier-Stokes equations and convergence of the solution to a stationary solution.
4
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Global existence for the inflow-outflow problem for the Navier-Stokes equations in a cylinder

95%
Kacprzyk P.
Applicationes Mathematicae
|
2009
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tom 36
|
nr 2
195-212
EN
Global existence of regular solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. To prove the long time existence we need smallness of derivatives, with respect to the variable along the axis of the cylinder, of the external force and of the initial velocity in L₂-norms. Moreover, we need smallness of derivatives of inflow and outflow with respect to tangent directions to the boundary and with respect to time in some norms. The global existence is proved step by step using the existence on the time interval [0,T], with T sufficiently large.
5
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Regularity properties of the attractor to the Navier-Stokes equations

95%
Kacprzyk P.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 3
341-351
EN
Existence of a global attractor for the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has been shown already. In this paper we prove the higher regularity of the attractor.
6
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Free boundary problem for the equations of magnetohydrodynamic incompressible viscous fluid

95%
Kacprzyk P.
Applicationes Mathematicae
|
2007
|
tom 34
|
nr 1
75-95
EN
The existence of a global motion of magnetohydrodynamic fluid in a domain bounded by a free surface and under the external electrodynamic field is proved. The motion is such that the velocity and magnetic field are small in the H³-space.
7
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Global regular nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

95%
Kacprzyk P.
Applicationes Mathematicae
|
2007
|
tom 34
|
nr 3
289-307
EN
Global existence of regular solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. Global existence is proved in two steps. First, by the Leray-Schauder fixed point theorem we prove local existence with large existence time. Next, the local solution is prolonged step by step. The existence is proved without any restrictions on the magnitudes of the inflow, outflow, external force and initial velocity.
8
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Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid

95%
Kacprzyk P.
Applicationes Mathematicae
|
2004
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tom 31
|
nr 1
69-77
EN
Almost global in time existence of solutions for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surfaced is proved. In the exterior domain we have an electromagnetic field which is generated by some currents which are located on a fixed boundary. We prove that a solution exists for t ∈ (0,T), where T > 0 is large if the data are small.
9
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On global regular solutions to the Navier-Stokes equations with heat convection

95%
Kacprzyk P.
Annales Polonici Mathematici
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2013
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tom 108
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nr 2
155-184
EN
Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on $||f(t)||_{L₂(Ω)}$, $||f_{,x₃}(t)||_{L₂(Ω)}$ we continue the local solutions step by step up to a global one.
10
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Local existence of solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid

95%
Kacprzyk P.
Applicationes Mathematicae
|
2003
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tom 30
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nr 4
461-488
EN
Local existence of solutions is proved for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surface. In the exterior domain we have an electromagnetic field which is generated by some currents located on a fixed boundary. First by the Galerkin method and regularization techniques the existence of solutions of the linarized equations is proved; next by the method of successive aproximations the local existence is shown for the nonlinear problem.
11
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Local existence of solutions of the free boundary problem for the equations of a magnetohydrodynamic compressible fluid

95%
Kacprzyk P.
Applicationes Mathematicae
|
2004
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tom 31
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nr 2
209-227
EN
Local existence of solutions for the equations describing the motion of a magnetohydrodynamic compressible fluid in a domain bounded by a free surface is proved. In the exterior domain we have an electromagnetic field which is generated by some currents located on a fixed boundary. First by the Galerkin method and regularization techniques the existence of solutions of the linearized equations is proved, next by the method of successive aproximations local existence to the nonlinear problem is shown.
12
Content available remote

Global existence of solutions of the free boundary problem for the equations of magnetohydrodynamic compressible fluid

95%
Kacprzyk P.
Banach Center Publications
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2005
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tom 70
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nr 1
105-129
EN
Global existence of solutions for equations describing a motion of magnetohydrodynamic compresible fluid in a domain bounded by a free surface is proved. In the exterior domain we have an electromagnetic field which is generated by some currents located on a fixed boundary. We have proved that the domain occupied by the fluid remains close to the initial domain for all time.
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