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

Takayuki Kubo; Yoshihiro Shibata

Displaying similar documents to “On the Stokes and Navier-Stokes flows in a perturbed half-space”

Asymptotic stability of Navier-Stokes flow past an obstacle

Piotr Biler, Marco Cannone, Grzegorz Karch (2004)

Banach Center Publications

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Results on the asymptotic stability of solutions of the exterior Navier-Stokes problem in ℝ³ are proved in the framework of weak $L^{p}$ spaces.

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 $t \to \infty$ . This result was...

On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain

Rieko Shimada, Norikazu Yamaguchi (2008)

Banach Center Publications

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This paper deals with a nonstationary problem for the Navier-Stokes equations with a free slip boundary condition in an exterior domain. We obtain a global in time unique solvability theorem and temporal asymptotic behavior of the global strong solution when the initial velocity is sufficiently small in the sense of Lⁿ (n is dimension). The proof is based on the contraction mapping principle with the aid of $L^{p} - L^{q}$ estimates for the Stokes semigroup associated with a linearized problem, which...

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 an estimate for the linearized compressible Navier-Stokes equations in the $L_{p}$ -framework

Piotr Bogusław Mucha (2001)

Colloquium Mathematicae

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An $L_{p}$ -estimate with a constant independent of time for solutions of the linearized compressible Navier-Stokes system in the whole space (under the assumption that solutions have compact supports in space) is obtained.

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...

$L^{p}$ -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

Geissert, M., Hieber, M.

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A global existence result for the compressible Navier-Stokes-Poisson equations in three and higher dimensions

Zhensheng Gao, Zhong Tan (2012)

Annales Polonici Mathematici

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The paper is dedicated to the global well-posedness of the barotropic compressible Navier-Stokes-Poisson system in the whole space $ℝ^{N}$ with N ≥ 3. The global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces. The initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations. The proof relies on a uniform estimate for a mixed hyperbolic/parabolic linear system with a convection term. ...

On the stability of compressible Navier-Stokes-Korteweg equations

Tong Tang, Hongjun Gao (2014)

Annales Polonici Mathematici

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We consider the compressible Navier-Stokes-Korteweg (N-S-K) equations. Through a remarkable identity, we reveal a relationship between the quantum hydrodynamic system and capillary fluids. Using some interesting inequalities from quantum fluids theory, we prove the stability of weak solutions for the N-S-K equations in the periodic domain $Ω =^{N}$ , when N=2,3.

Some remarks on variants of the Navier-Stokes equations

R. H. Dyer, D. E. Edmunds (1971)

Colloquium Mathematicae

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Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity

Alexis Vasseur (2009)

Applications of Mathematics

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In this short note we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation and the behavior of the direction of the velocity $u / | u |$ . It is shown that the control of $Div (u / | u |)$ in a suitable $L_{t}^{p} (L_{x}^{q})$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based...

Higher order estimates in further dimensions for the solutions of Navier-Stokes equations

Michael Wiegner (2003)

Banach Center Publications

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Uniqueness of weak solutions to a Keller-Segel-Navier-Stokes model with a logistic source

Miaochao Chen, Shengqi Lu, Qilin Liu (2022)

Applications of Mathematics

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We prove a uniqueness result of weak solutions to the $n D$ $(n \geq 3)$ Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.

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 Qualitative Behavior of the Solutions to the 2-D Navier-Stokes Equation

M. Pulvirenti (2008)

Bollettino dell'Unione Matematica Italiana

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This talk, based on a research in collaboration with E. Caglioti and F.Rousset, deals with a modified version of the two-dimensional Navier-Stokes equation wich preserves energy and momentum of inertia. Such a new equation is motivated by the occurrence of different dissipation time scales. It is also related to the gradient flow structure of the 2-D Navier-Stokes equation. The hope is to understand intermediate asymptotics.

The Stokes system in the incompressible case-revisited

Rainer Picard (2008)

Banach Center Publications

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The classical Stokes system is reconsidered and reformulated in a functional analytical setting allowing for low regularity of the data and the boundary. In fact the underlying domain can be any non-empty open subset Ω of ℝ³. A suitable solution concept and a corresponding solution theory is developed.

On the $(x, t)$ asymptotic properties of solutions of the Navier-Stokes equations in the half-space.

Crispo, F., Maremonti, P. (2004)

Zapiski Nauchnykh Seminarov POMI

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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 $p \geq 2$ . 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...

Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $ℝ^{n}$

Reinhard Farwig, Hermann Sohr (2009)

Czechoslovak Mathematical Journal

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For a bounded domain $Ω \subset ℝ^{n}$ , $n \geq 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $- Δ u + u \cdot \nabla u + \nabla p = f$ , $div u = k$ , $u_{|_{\partial Ω}} = g$ with $u \in L^{q}$ , $q \geq n$ , and very general data classes for $f$ , $k$ , $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of...