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1

Convolution of radius functions on ℝ³

100%

Holly K.

Annales Polonici Mathematici

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1994-1995

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tom 60

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nr 1

1-32

EN

We reduce the convolution of radius functions to that of 1-variable functions. Then we present formulas for computing convolutions of an abstract radius function on ℝ³ with various integral kernels - given by elementary or discontinuous functions. We also prove a theorem on the asymptotic behaviour of a convolution at infinity. Lastly, we deduce some estimates which enable us to find the asymptotics of the velocity and pressure of a fluid (described by the Navier-Stokes equations) in the boundary layer.

2

Some application of the implicit function theorem to the stationary Navier-Stokes equations

100%

Holly K.

Annales Polonici Mathematici

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1991

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tom 54

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nr 2

93-109

EN

We prove that - in the case of typical external forces - the set of stationary solutions of the Navier-Stokes equations is the limit of the (full) sequence of sets of solutions of the appropriate Galerkin equations, in the sense of the Hausdorff metric (for every inner approximation of the space of velocities). Then the uniqueness of the N-S equations is equivalent to the uniqueness of almost every of these Galerkin equations.

3

Nonlinear orthogonal projection

63%

Dudek E., Holly K.

Annales Polonici Mathematici

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1994

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tom 59

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nr 1

1-31

EN

We discuss some properties of an orthogonal projection onto a subset of a Euclidean space. The special stress is laid on projection's regularity and characterization of the interior of its domain.

4

Applications of the Carathéodory theorem to PDEs

63%

Holly K., Orewczyk J.

Annales Polonici Mathematici

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2000

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tom 73

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nr 1

1-27

EN

We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE $ẋ {(*)\over=} ℱ(t,x)$ for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases "dom ℱ is open" and "dom ℱ = ℐ × Ω for some Ω ⊂ X". We show how to use the theorems mentioned above to get approximate solutions of a nonlinear parabolic PDE and exact solutions of a linear evolution PDE with distribution data.

Ograniczanie wyników

4 Annales Polonici Mathematici

4 Holly K.

1 Dudek E.

1 Orewczyk J.

1 2000

2 1994

1 1991

Convolution of radius functions on ℝ³

Some application of the implicit function theorem to the stationary Navier-Stokes equations

Nonlinear orthogonal projection

Applications of the Carathéodory theorem to PDEs