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Noncharacteristic mixed problems for hyperbolic systems of the first order

100%
Zadrzyńska E.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction....................................................................................................................................................5 2. Notations and preliminaries .........................................................................................................................11  2.1. Function spaces and spaces of distributions............................................................................................11  2.2. Perturbations of linear operators.............................................................................................................20  2.3. Properties of matrices..............................................................................................................................24 3. A boundary value problem for a linear hyperbolic system of the first order in a halfspace...........................26  3.1. Assumptions.............................................................................................................................................27  3.2. Construction of a symmetrizer..................................................................................................................31  3.3. An estimate of a solution of the boundary value problem (3.1)-(3.2) in $L²_{η}-norm$............................57  3.4. An estimate of a solution of the problem formally adjoint to (3.1)-(3.2) in $L²_{-η}-norm$.......................85  3.5. Existence and uniqueness of solution of the boundary value problem (3.1)-(3.2)....................................91  3.6. An estimate of a solution of the Cauchy problem for system (3.1) in $H^{s}_{η}$-norm...........................95 4. A mixed problem for a system of linear hyperbolic equations of the first order in a halfspace....................101  4.1. A mixed problem with nonzero initial condition........................................................................................101  4.2. A mixed problem with zero initial condition..............................................................................................127 5. A mixed problem for a system of linear hyperbolic equations of the first order in a bounded domain.........128 6. A mixed problem for a nonlinear system of hyperbolic equations of the first order.....................................141 References....................................................................................................................................................145
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Free boundary problems for nonstationary Navier-Stokes equations

100%
Zadrzyńska E.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The results concerning free boundary problems for both incompressible and compressible Navier-Stokes equations are reviewed in this paper.
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On some free boundary problems for Navier-Stokes equations

95%
Zadrzyńska E.
Banach Center Publications
|
2005
|
tom 70
|
nr 1
279-294
EN
In this survey we report on existence results for some free boundary problems for equations describing motions of both incompressible and compressible viscous fluids. We also present ways of controlling free boundaries in two cases: a) when the free boundary is governed by surface tension, b) when surface tension does not occur.
4
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On nonstationary motion of a fixed mass of a general viscous compressible heat conducting capillary fluid bounded by a free boundary

95%
Zadrzyńska E.
Applicationes Mathematicae
|
1998-1999
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tom 25
|
nr 4
489-511
EN
The motion of a fixed mass of a viscous compressible heat conducting capillary fluid is examined. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global-in-time solution which is close to the constant state for any moment of time. Moreover, we present an analogous result for the case of a barotropic viscous compressible fluid.
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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.
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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.
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On nonstationary motion of a fixed mass of a viscous compressible barotropic fluid bounded by a free boundary

60%
Zadrzyńska E., Zajączkowski W. M.
Colloquium Mathematicum
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1999
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tom 79
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nr 2
283-310
8
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Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting fluids

60%
Zadrzyńska E., Zajączkowski W. M.
Applicationes Mathematicae
|
1998-1999
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tom 25
|
nr 2
179-220
EN
The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems by the Galerkin method and by regularization techniques. Next, we use the method of successive approximations to prove the local existence for the main problem.
9
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Solvability of the stationary Stokes system in spaces $H²_{-μ}$, μ ∈ (0,1)

60%
Zadrzyńska E., Zajączkowski W.
Applicationes Mathematicae
|
2010
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tom 37
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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.
10
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Some linear parabolic system in Besov spaces

60%
Zadrzyńska E., Zajączkowski W.
Banach Center Publications
|
2008
|
tom 81
|
nr 1
567-612
EN
We study the solvability in anisotropic Besov spaces $B_{p,q}^{σ/2,σ}(Ω^T)$, σ ∈ ℝ₊, p,q ∈ (1,∞) of an initial-boundary value problem for the linear parabolic system which arises in the study of the compressible Navier-Stokes system with boundary slip conditions. The proof of existence of a unique solution in $B_{p,q}^{σ/2 + 1,σ + 2}(Ω^T)$ is divided into three steps: 1° First the existence of solutions to the problem with vanishing initial conditions is proved by applying the Paley-Littlewood decomposition and some ideas of Triebel. All considerations in this step are performed on the Fourier transform of the solution. 2° Applying the regularizer technique the existence is proved in a~bounded domain. 3° The problem with nonvanishing initial data is solved by an appropriate extension of initial data.
11
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On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

60%
Zadrzyńska E., Zajączkowski W.
Annales Polonici Mathematici
|
1996
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tom 63
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nr 3
199-221
EN
We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.
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On nonstationary motion of a fixed mass of a general fluid bounded by a free surface

60%
Zadrzyńska E., Zajączkowski W.
Banach Center Publications
|
2003
|
tom 60
|
nr 1
253-266
EN
In the paper the motion of a fixed mass of a viscous compressible heat conducting fluid is considered. Assuming that the initial data are sufficiently close to an equilibrium state and the external force, the heat sources and the heat flow through the boundary vanish, we prove the existence of a global in time solution which is close to the equilibrium state for any moment of time.
13
Content available remote

On some inequalities for solutions of equations describing the motion of a viscous compressible heat-conducting capillary fluid bounded by a free surface

60%
Zadrzyńska E., Zajączkowski W.
Applicationes Mathematicae
|
2001
|
tom 28
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nr 1
31-53
EN
We derive inequalities for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. The inequalities are crucial in proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskii space and close to an equilibrium state.
14
Content available remote

On an inequality for a free boundary problem for equations of a viscous compressible heat-conducting capillary fluid

60%
Zadrzyńska E., Zajączkowski W.
Applicationes Mathematicae
|
2002
|
tom 29
|
nr 4
399-438
EN
We derive an inequality for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. This inequality is crucial to proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskiĭ spaces and close to an equilibrium state.
15
Content available remote

On a differential inequality for equations of a viscous compressible heat conducting fluid bounded by a free surface

60%
Zadrzyńska E., Zajączkowski W.
Annales Polonici Mathematici
|
1995
|
tom 61
|
nr 2
141-188
EN
We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat conducting fluid. The inequality is essential in proving the global existence of solutions.
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