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On the existence for the Cauchy-Neumann problem for the Stokes system in the L p -framework

Piotr MuchaWojciech Zajączkowski — 2000

Studia Mathematica

The existence for the Cauchy-Neumann problem for the Stokes system in a bounded domain Ω 3 is proved in a class such that the velocity belongs to W r 2 , 1 ( Ω × ( 0 , T ) ) , where r > 3. The proof is divided into three steps. First, the existence of solutions is proved in a half-space for vanishing initial data by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing initial data is considered....

On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

Piotr MuchaWojciech Zajączkowski — 2000

Applicationes Mathematicae

The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that u W r 2 , 1 ( Ω ˜ T ) with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the L p -approach the Lagrangian coordinates must be used. We are looking...

On the structure of flows through pipe-like domains satisfying a geometrical constraint

Piotr Bogusław Mucha — 2004

Applicationes Mathematicae

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.

Stability of Constant Solutions to the Navier-Stokes System in ℝ³

Piotr Bogusław Mucha — 2001

Applicationes Mathematicae

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 r 2 , 1 ( ³ × [ 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 r 2 - 2 / r ( ³ ) -norm of the perturbing initial data or smallness of the...

Stefan problem in a 2D case

Piotr Bogusław Mucha — 2006

Colloquium Mathematicae

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.

Asymptotic behavior of a steady flow in a two-dimensional pipe

Piotr Bogusław Mucha — 2003

Studia Mathematica

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...

The Eulerian limit and the slip boundary conditions-admissible irregularity of the boundary

Piotr Bogusław Mucha — 2005

Banach Center Publications

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.

Global solutions, structure of initial data and the Navier-Stokes equations

Piotr Bogusław Mucha — 2008

Banach Center Publications

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.

Problème de Stokes et système de Navier-Stokes incompressible à densité variable dans le demi-espace

Raphaël DanchinPiotr Bogusław Mucha

Séminaire Équations aux dérivées partielles

On s’intéresse à la résolution du système de Navier-Stokes incompressible dans le demi-espace + n : = n - 1 × ] 0 , [ en dimension n 3 . On considère des données initiales à régularité . On établit que si la densité initiale est proche d’une constante strictement positive dans L W ˙ 1 , n et si la vitesse initiale est petite par rapport à la viscosité dans l’espace de Besov homogène B ˙ n , 1 0 alors le système de Navier-Stokes admet une unique solution globale. La démonstration repose sur de nouvelles estimations de régularité maximale pour...

Almost classical solutions of static Stefan type problems involving crystalline curvature

Piotr Bogusław MuchaPiotr Rybka — 2009

Banach Center Publications

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.

On the existence for the Dirichlet problem for the compressible linearized Navier-Stokes system in the L p -framework

Piotr Boguslaw MuchaWojciech Zajączkowski — 2002

Annales Polonici Mathematici

The existence of solutions to the Dirichlet problem for the compressible linearized Navier-Stokes system is proved in a class such that the velocity vector belongs to W r 2 , 1 with r > 3. The proof is done in two steps. First the existence for local problems with constant coefficients is proved by applying the Fourier transform. Next by applying the regularizer technique the existence in a bounded domain is shown.

On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

Piotr Boguslaw MuchaWojciech Zajączkowski — 2002

Annales Polonici Mathematici

The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the L p -framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the L p -approach, because the velocity belongs to W r 2 , 1 with r > 3.

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