Displaying similar documents to “On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain”

Stokes equations in asymptotically flat layers

Helmut Abels (2005)

Banach Center Publications

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We study the generalized Stokes resolvent equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Ω = n - 1 × ( - 1 , 1 ) . Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. We discuss the results on unique solvability of the generalized Stokes resolvent equations as well as the existence of a bounded H -calculus for the associated Stokes operator...

Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

Thierry Gallay (2012)

Journées Équations aux dérivées partielles

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We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as t . This result was...

On the Stokes and Navier-Stokes flows in a perturbed half-space

Takayuki Kubo, Yoshihiro Shibata (2005)

Banach Center Publications

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We give the L p - L q estimate for the Stokes semigroup in a perturbed half-space and some global in time existence theorems for small solutions to the Navier-Stokes equation.

On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem

Hugo Beirão da Veiga (2009)

Journal of the European Mathematical Society

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We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case p 2 . Actually, we are interested in proving regularity results in L q ( Ω ) spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous...

Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski (2005)

Studia Mathematica

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Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity I = i = 1 2 ( | | x i v ( 0 ) | | L ( Ω ) + | | x i f | | L ( Ω × ( 0 , T ) ) ) is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by...

Global regular solutions to the Navier-Stokes equations in a cylinder

Wojciech M. Zajączkowski (2006)

Banach Center Publications

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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...

Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition

Reinhard Farwig, Hideo Kozono, Hermann Sohr (2008)

Banach Center Publications

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Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition | | u | | L s ( 0 , T ; L q ( Ω ) ) < where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties...

Global existence of axially symmetric solutions to Navier-Stokes equations with large angular component of velocity

Wojciech M. Zajączkowski (2004)

Colloquium Mathematicae

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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...

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

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

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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 the Stokes equation with Neumann boundary condition

Yoshihiro Shibata, Senjo Shimizu (2005)

Banach Center Publications

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In this paper, we study the nonstationary Stokes equation with Neumann boundary condition in a bounded or an exterior domain in ℝⁿ, which is the linearized model problem of the free boundary value problem. Mainly, we prove L p - L q estimates for the semigroup of the Stokes operator. Comparing with the non-slip boundary condition case, we have the better decay estimate for the gradient of the semigroup in the exterior domain case because of the null force at the boundary.

On global regular solutions to the Navier-Stokes equations with heat convection

Piotr Kacprzyk (2013)

Annales Polonici Mathematici

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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 ) | | L ( Ω ) , | | f , x ( t ) | | L ( Ω ) we continue the local solutions step by step up to a global one. ...

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

Piotr Bogusław Mucha (2001)

Applicationes Mathematicae

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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...

A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition

H. Morimoto, H. Fujita (2001)

Mathematica Bohemica

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We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain Ω under the general outflow condition. Let T be a 2-dimensional straight channel × ( - 1 , 1 ) . We suppose that Ω { x 1 < 0 } is bounded and that Ω { x 1 > - 1 } = T { x 1 > - 1 } . Let V be a Poiseuille flow in T and μ the flux of V . We look for a solution which tends to V as x 1 . Assuming that the domain and the boundary data are symmetric with respect to the x 1 -axis, and that the axis intersects every component of the boundary, we have shown...

Some linear parabolic system in Besov spaces

Ewa Zadrzyńska, Wojciech M. Zajączkowski (2008)

Banach Center Publications

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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....

A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf (2008)

Banach Center Publications

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In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if l i m s u p R 0 1 / R Q R ( x , t ) | c u r l u × u / | u | | ² d x d t ε * for a sufficiently small ε * > 0 .

Long time existence of solutions to 2d Navier-Stokes equations with heat convection

Jolanta Socała, Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

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Global existence of regular solutions to the Navier-Stokes equations for (v,p) coupled with the heat convection equation for θ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence is proved by the Leray-Schauder fixed point theorem. We prove the existence of solutions such that v , θ W s 2 , 1 ( Ω T ) , p L s ( Ω T ) , s>2.

Long-time behavior for 2D non-autonomous g-Navier-Stokes equations

Cung The Anh, Dao Trong Quyet (2012)

Annales Polonici Mathematici

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We study the first initial boundary value problem for the 2D non-autonomous g-Navier-Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback σ -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force...

Maximal regularity and viscous incompressible flows with free interface

Senjo Shimizu (2008)

Banach Center Publications

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We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time L p - L q maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.

Self-improving bounds for the Navier-Stokes equations

Jean-Yves Chemin, Fabrice Planchon (2012)

Bulletin de la Société Mathématique de France

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We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to - 1 . Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.

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

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

Similarity:

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.

Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains

Serguei A. Nazarov, Maria Specovius-Neugebauer (2008)

Banach Center Publications

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This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + div u = 0, and a Neumann boundary condition for the pressure is added. Using parameter dependent Sobolev norms, for bounded domains and for sufficiently smooth data we prove H 5 / 2 - δ convergence for the velocity part and H 3 / 2 - δ convergence for the pressure to the solution of the Stokes problem,...

A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations

Jiří Neustupa (2014)

Mathematica Bohemica

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We deal with a suitable weak solution ( 𝐯 , p ) to the Navier-Stokes equations in a domain Ω 3 . We refine the criterion for the local regularity of this solution at the point ( 𝐟 x 0 , t 0 ) , which uses the L 3 -norm of 𝐯 and the L 3 / 2 -norm of p in a shrinking backward parabolic neighbourhood of ( 𝐱 0 , t 0 ) . The refinement consists in the fact that only the values of 𝐯 , respectively p , in the exterior of a space-time paraboloid with vertex at ( 𝐱 0 , t 0 ) , respectively in a ”small” subset of this exterior, are considered. The consequence...

A continuity property for the inverse of Mañé's projection

Zdeněk Skalák (1998)

Applications of Mathematics

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Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d F ( X ) , and P 0 an orthogonal projection in H of rank greater than or equal to 2 d F ( X ) + 1 . For every δ > 0 , there exists an orthogonal projection P in H of the same rank as P 0 , which is injective when restricted to X and such that P - P 0 < δ . This result follows from Mañé’s paper. Thus the inverse ( P | X ) - 1 of the restricted mapping P | X X P X is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse...