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1
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Global special regular solutions to the Navier-Stokes equations in axially symmetric domains under boundary slip conditions

100%
Zajączkowski W.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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 conditions for vorticity, which is of crucial importance to this paper. Finally, we prove the existence of solutions which remain close to the axially symmetric solutions for all time. The existence of axially symmetric solutions was proved in weighted Sobolev spaces with the weight equal to a power of the distance to the axis of symmetry because in such spaces a global estimate for vorticity could be obtained. Therefore in this paper similar weighted Sobolev spaces are also used.
2
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Solvability of the heat equation in weighted Sobolev spaces

94%
Zajączkowski W.
Applicationes Mathematicae
|
2011
|
tom 38
|
nr 2
147-171
EN
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 be treated in the same way. The proof of the existence is split into the following steps. First by an appropriate extension of initial data the initial-boundary value problem is reduced to an elliptic problem with a fixed t ∈ ℝ. Applying the regularizer technique it is considered locally. The most difficult part is to show the existence in weighted spaces near the axis, because the existence in neighbourhoods located at a positive distance from the axis is well known. In a neighbourhood of a point where the axis meets the boundary, the elliptic problem considered is transformed to a problem near an interior point of the axis by an appropriate reflection. Using cutoff functions the problem near the axis is considered in ℝ³ with sufficiently fast decreasing functions as |x| → ∞. Then by applying the Fourier-Laplace transform we are able to show an appropriate estimate in weighted spaces and to prove local in space existence. The result of this paper is necessary to show the existence of global regular solutions to the Navier-Stokes equations which are close to axially symmetric solutions.
3
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Solvability of the Poisson equation in weighted Sobolev spaces

94%
Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 3
325-339
EN
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.
4
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Existence of solutions to the (rot,div)-system in $L_p$-weighted spaces

94%
Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 2
127-142
EN
The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, $v·n̅|_S = 0$, 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.
5
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Existence of solutions to the (rot,div)-system in L₂-weighted spaces

94%
Zajączkowski W.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 1
83-106
EN
The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, $v·n̅|_S = 0$, 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.
6
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On imbedding theorems for weighted anisotropic Sobolev spaces

94%
Zajączkowski W.
Applicationes Mathematicae
|
2002
|
tom 29
|
nr 1
51-63
EN
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.
7
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Global existence of axially symmetric solutions to Navier-Stokes equations with large angular component of velocity

94%
Zajączkowski W.
Colloquium Mathematicum
|
2004
|
tom 100
|
nr 2
243-263
EN
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 $W_{4/3}^{2,1}(Ω × (0,T))$, so it satisfies the Serrin condition. Next, increasing regularity of the external force and initial data we prove existence of solutions (by the Leray-Schauder fixed point theorem) such that $v ∈ W_{r}^{2,1}(Ω × (0,T))$ with r > 4/3, and we prove their uniqueness.
8
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Global regular solutions to the Navier-Stokes equations in a cylinder

94%
Zajączkowski W.
Banach Center Publications
|
2006
|
tom 74
|
nr 1
235-255
EN
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 $W_{5/2}^{2,1}(Ω × (0,T))$ and the gradient of the pressure to $L_{5/2}(Ω × (0,T))$. We prove the existence of solutions without any restrictions on the lengths of the initial velocity and the external force.
9
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On local motion of a general compressible viscous heat conducting fluid bounded by a free surface

60%
Zadrzyńska E., Zajączkowski W.
Annales Polonici Mathematici
|
1994
|
tom 59
|
nr 2
133-170
EN
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.
10
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On a differential inequality for a viscous compressible heat conducting capillary fluid bounded by a free surface

60%
Zadrzyńska E., Zajączkowski W.
Annales Polonici Mathematici
|
1996-1997
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tom 65
|
nr 1
23-53
EN
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.
11
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Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity

60%
Pawłow I., Zajączkowski W.
Annales Polonici Mathematici
|
2010
|
tom 98
|
nr 1
1-21
EN
The long-time behaviour of a unique regular solution to the Cahn-Hilliard system coupled with viscoelasticity is studied. The system arises as a model of the phase separation process in a binary deformable alloy. It is proved that for a sufficiently regular initial data the trajectory of the solution converges to the ω-limit set of these data. Moreover, it is shown that every element of the ω-limit set is a solution of the corresponding stationary problem.
12
Content available remote

Long time existence of regular solutions to 3d Navier-Stokes equations coupled with heat convection

60%
Socała J., Zajączkowski W.
Applicationes Mathematicae
|
2012
|
tom 39
|
nr 2
231-242
EN
We prove long time existence of regular solutions to the Navier-Stokes equations coupled with the heat equation. We consider the system in a non-axially symmetric cylinder, with the slip boundary conditions for the Navier-Stokes equations, and the Neumann condition for the heat equation. The long time existence is possible because the derivatives, with respect to the variable along the axis of the cylinder, of the initial velocity, initial temperature and external force are assumed to be sufficiently small in the L₂ norms. We prove the existence of solutions such that the velocity and temperature belong to $W_σ^{2,1}(Ω × (0,T))$, where σ > 5/3. The existence is proved by using the Leray-Schauder fixed point theorem.
13
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Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids

60%
Pawłow I., Zajączkowski W.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 2
313-334
EN
The paper is concerned with the existence of measure-valued solutions to the Cahn-Hilliard system coupled with elasticity. The system under consideration is anisotropic and heterogeneous in the sense of admitting the elasticity and gradient energy tensors dependent on the order parameter. Such dependences introduce additional nonlinearities to the model for which the existence of weak solutions is not known so far.
14
Content available remote

Long time estimate of solutions to 3d Navier-Stokes equations coupled with heat convection

60%
Socała J., Zajączkowski W.
Applicationes Mathematicae
|
2012
|
tom 39
|
nr 1
23-41
EN
We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in ℝ³. The domain is a cylinder along the x₃ axis. The aim of this paper is to show long time estimates without assuming smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of the L₂ norms of the x₃-derivatives of these three quantities.
15
Content available remote

Large time regular solutions to the MHD equations in cylindrical domains

60%
Alame W., Zajączkowski W.
Applicationes Mathematicae
|
2011
|
tom 38
|
nr 2
117-146
EN
We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in $W₂^{2,1}$ without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.
16
Content available remote

Solvability of the stationary Stokes system in spaces $H²_{-μ}$, μ ∈ (0,1)

60%
Zadrzyńska E., Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 1
13-38
EN
We consider the stationary Stokes system with slip boundary conditions in a bounded domain. Assuming that data functions belong to weighted Sobolev spaces with weights equal to some power of the distance to some distinguished axis, we prove the existence of solutions to the problem in appropriate weighted Sobolev spaces.
17
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Global existence of solutions to Navier-Stokes equations in cylindrical domains

60%
Nowakowski B., Zajączkowski W.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 2
169-182
EN
We prove the existence of global and regular solutions to the Navier-Stokes equations in cylindrical type domains under boundary slip conditions, where coordinates are chosen so that the x₃-axis is parallel to the axis of the cylinder. Regular solutions have already been obtained on the interval [0,T], where T > 0 is large, on the assumption that the L₂-norms of the third component of the force field, of derivatives of the force field, and of the velocity field with respect to the direction of the axis of the cylinder are small. In this paper we continue the solution to all times.
18
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Existence of solutions to the Poisson equation in $L_p$-weighted spaces

60%
Rencławowicz J., Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 1
1-12
EN
We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of $L_p$, p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.
19
Content available remote

Global attractor for Navier-Stokes equations in cylindrical domains

60%
Nowakowski B., Zajączkowski W.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 2
183-194
EN
Global and regular solutions of the Navier-Stokes system in cylindrical domains have already been obtained under the assumption of smallness of (1) the derivative of the velocity field with respect to the variable along the axis of cylinder, (2) the derivative of force field with respect to the variable along the axis of the cylinder and (3) the projection of the force field on the axis of the cylinder restricted to the part of the boundary perpendicular to the axis of the cylinder. With the same assumptions we prove in this paper the existence of a global attractor for the Navier-Stokes equations and convergence of solutions to the stationary solutions for the large viscosity coefficient.
20
Content available remote

Existence of solutions to the Poisson equation in L₂-weighted spaces

60%
Rencławowicz J., Zajączkowski W.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 3
309-323
EN
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.
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