Long time existence of regular solutions to 3d Navier-Stokes equations coupled with heat convection

Jolanta Socała; Wojciech M. Zajączkowski

Displaying similar documents to “Long time existence of regular solutions to 3d Navier-Stokes equations coupled with heat convection”

Long time estimate of solutions to 3d Navier-Stokes equations coupled with heat convection

Jolanta Socała, Wojciech M. Zajączkowski (2012)

Applicationes Mathematicae

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We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in ℝ³. The domain is a cylinder along the x₃ axis. The aim of this paper is to show long time estimates without assuming smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of the L₂ norms of the x₃-derivatives of these three quantities. ...

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 ${| | f (t) | |}_{L ₂ (Ω)}$ , $| | f_{, x ₃} (t) {| |}_{L ₂ (Ω)}$ we continue the local solutions step by step up to a global one. ...

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 $v, θ \in W_{s}^{2, 1} (Ω^{T})$ , $\nabla p \in L_{s} (Ω^{T})$ , s>2.

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 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.

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.

Regular solutions of the Navier-Stokes system.

Chelkak, S., Koshelev, A., Oganesyan, L. (1997)

Memoirs on Differential Equations and Mathematical Physics

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Global existence for the inflow-outflow problem for the Navier-Stokes equations in a cylinder

Piotr Kacprzyk (2009)

Applicationes Mathematicae

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Global existence of regular solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. To prove the long time existence we need smallness of derivatives, with respect to the variable along the axis of the cylinder, of the external force and of the initial velocity in L₂-norms. Moreover, we need smallness of derivatives of inflow and outflow with respect to tangent directions to the boundary...

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.

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...

Formal passage from kinetic theory to incompressible Navier–Stokes equations for a mixture of gases

Marzia Bisi, Laurent Desvillettes (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in...

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. ...

Some remarks on variants of the Navier-Stokes equations

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

Colloquium Mathematicae

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On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting capillary fluid.

Zadrzyńska, E., Zaja̧czkowski, W.M. (1996)

Journal of Applied Analysis

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Global attractor for the Navier-Stokes equations in a cylindrical pipe

Piotr Kacprzyk (2010)

Annales Polonici Mathematici

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Global existence of regular special solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has already been shown. In this paper we prove the existence of the global attractor for the Navier-Stokes equations and convergence of the solution to a stationary solution.

Spectral Numerical Study of a Problem Governed by Navier-Stokes Equations, Influence of Rayleigh and Prandtl Numbers

E. El Guarmah, A. Cheddadi (2010)

Mathematical Modelling of Natural Phenomena

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We present in this work a numerical study of a problem governed by Navier-Stokes equations and heat equation. The mathematical problem under consideration is obtained by modelling the natural convection of an incompressible fluid, in laminar flow between two horizontal concentric coaxial cylinders, the temperature of the inner cylinder is supposed to be greater than the outer one. The numerical simulation of the flow is carried out by ...

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...

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.

The Navier-Stokes equation with inhomogeneous boundary conditions based on vorticity

Jiří Neustupa, Patrick Penel (2008)

Banach Center Publications

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We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.