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A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum

Jens Frehse, Sonja Goj, Josef Málek (2005)

Applications of Mathematics

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ρ i of the fluids and their velocity fields u ( i ) are prescribed at infinity: ρ i | ∞ = ρ i ∞ > 0 , u ( i ) | ∞ = 0 . Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ρ i ≡ ρ i ∞ , u ( i ) ≡ 0 , i = 1 , 2 .

A uniqueness theorem for the approximable solutions of the stationary Navier-Stokes equations

Giovanni Prouse (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

It is proved that there can exist at most one solution of the homogeneous Dirichlet problem for the stationary Navier-Stokes equations in 3-dimensional space which is approximable by a given consistent and regular approximation scheme.

A uniqueness theorem for viscous flows on exterior domains with summability assumptions on the gradient of pressure.

Giovanni P. Galdi, Paolo Maremonti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questa Nota si fornisce un teorema di unicità per soluzioni regolari delle equazioni di Navier-Stokes in domini esterni. Tale teorema non richiede che le velocità tendano ad un prefissato limite all'infinito, mentre il gradiente di pressione è supposto essere di q -ma potenza sommabile nel cilindro spazio-temporale ( q ∈ ( 1 , ∞ ) ) . Questo risultato non può essere ulteriormente generalizzato al caso q = ∞ , a causa di noti controesempi.

A zoology of boundary layers.

David Gérard-Varet, Emmanuel Grenier (2002)

RACSAM

In meteorology and magnetohydrodynamics many different boundary layers appear. Some of them are already mathematically well known, like Ekman or Hartmann layers. Others remain unstudied, and can be much more complex. The aim of this paper is to give a simple and unified presentation of the main boundary layers, and to propose a simple method to derive their sizes and equations.

About the decay of surface waves on viscous fluids without surface tension

Gerhard Ströhmer (2003)

Banach Center Publications

We study the decay of the motions of a viscous fluid subject to gravity without surface tension with a free boundary at the top. We show that the solutions of the linearization about the equilibrium state decay, but not exponentially in a uniform manner. We also discuss the consequences of this for the non-linear equations.

Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

Simone Deparis, Miguel Angel Fernández, Luca Formaggia (2003)

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

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of the structure....

Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions

Simone Deparis, Miguel Angel Fernández, Luca Formaggia (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of...

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

Xinlong Feng, Zhifeng Weng, Hehu Xie (2014)

Applications of Mathematics

This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error...

Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class L ∞ ( 0 , T , L 3 ( Ω ) 3 )

Zdeněk Skalák (2003)

Applications of Mathematics

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution 𝐮 belongs to L ∞ ( 0 , T , L 3 ( Ω ) 3 ) , then the set of all possible singular points of 𝐮 in Ω is at most finite at every time t 0 ∈ ( 0 , T ) .

All-at-once preconditioning in PDE-constrained optimization

Tyrone Rees, Martin Stoll, Andy Wathen (2010)

Kybernetika

The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound...

Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid

Piotr Kacprzyk (2004)

Applicationes Mathematicae

Almost global in time existence of solutions for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surfaced is proved. In the exterior domain we have an electromagnetic field which is generated by some currents which are located on a fixed boundary. We prove that a solution exists for t ∈ (0,T), where T > 0 is large if the data are small.

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