The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions

Kučera, Petr

doi:10.3934/dcdss.2010.3.325

Cited by 20 publications

(26 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for small data or short time) were obtained e.g. in [24] and in [25] for stationary and for time dependent case, respectively. Global existence analysis is, however, an open problem because (1.2) does not prevent backward flow through the boundary and thus an uncontrolled amount of kinetic energy can be brought into the domain.…”

Section: Introductionmentioning

confidence: 99%

On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities

Lanzendörfer

Stebel

2011

Appl Math

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities

Lanzendörfer

Stebel

2011

Appl Math

View full text Add to dashboard Cite

“…In [16]- [18], Kračmar & Neustupa prescribed an additional condition on the output (which bounds the kinetic energy of an eventual backward flow) and formulated steady and evolutionary Navier-Stokes problems by means of appropriate variational inequalities. In [20], Kučera & Skalák proved the local-in-time existence of a strong solution of the nonsteady Navier-Stokes problem with boundary condition (1.6) on the part of the boundary. In this paper, we study the same problem and we prove the global-in-time existence and uniqueness of a strong solution in a small neighbourhood of another known solution.…”

Section: Introductionmentioning

confidence: 99%

“…Due to this fact, the question of the global in time existence of a weak solution of this problem is still open. Some qualitative properties of the Navier-Stokes equations with these boundary conditions are studied in [16,17,18,20]. In [16]- [18], Kračmar & Neustupa prescribed an additional condition on the output (which bounds the kinetic energy of an eventual backward flow) and formulated steady and evolutionary Navier-Stokes problems by means of appropriate variational inequalities.…”

Section: Introductionmentioning

confidence: 99%

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

Beneš

Kučera

2015

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Abstract:In this paper we consider the system of the non-steady Navier-Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y , respectively, to be the space of "possible" solutions of this problem and the space of its data. We define the operator N : X → Y and formulate our problem in terms of operator equations. Let u ∈ X and G P u : X → Y be the Frechet derivative of N at u. We prove that G P u is one-to-one and onto Y . Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. Next result proved in the Appendix of this paper is W 2,2 -regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.

show abstract

“…(10) As a matter of fact, the results of [66] cover more general situations than that described by problem (1.240). Actually, the domain V need not be "pipe-like" with cuts orthogonal to the axis in each outlet, whereas the boundary conditions include the case when P is prescribed as (suitable) function of space and time.…”

Section: Lemma 18 There Exists a Constantmentioning

confidence: 98%

“…In fact, in the paper [66], Kučera and Skalák show, by means of a fixed-point argument, existence of solutions such that (with…”

Section: 43mentioning

confidence: 99%

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

Galdi

Oberwolfach Seminars

View full text Add to dashboard Cite

IntroductionBlood flow per se is a very complicated subject. Thus, it is not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and challenging.The role of mathematics in the investigation of blood flow properties -as in the most part of applied sciences -is twofold and is directed toward the accomplishment of the following objectives. The first one, of a more theoretical nature, is the validation of the models proposed by engineers, and consists in securing conditions under which the governing equations possess the fundamental requirements of well-posedness, such as existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more applied character, is to prove that these models give a satisfactory interpretation of the observed phenomena. In general, both tasks present serious difficulties in that they require the study of several different, and frequently combined, topics that include, among others, Navier-Stokes equations, non-Newtonian fluid models, nonlinear elasticity, fluid-structure interaction and multi-phase flow. It must be added that some of these topics are still at the beginning of a systematic mathematical research, whereas some others are in a continuous growth.As a matter of fact, the initiation or the methodical investigation of several of the above research areas was just motivated by problems arising in blood flow. Moreover, blood flow can also pose challenging questions in "classical" topics, questions that, in the past, happened to receive little or no attention at all. A typical example is provided by the problem of the flow of a Navier-Stokes liquid in an unbounded piping system, under a given time-periodic flow-rate, that has been "discovered" only in 2005, thanks to the work of H. Beirão da Veiga; see [7]. G.P. GaldiIt seemed to me hopeless to present and to describe in this article all relevant aspects of the mathematical analysis related to blood flow, even at an introductory level. Therefore, I preferred to concentrate on some, of the many, topics which are at the foundation of this analysis, and to point out directions for future research. More specifically, I focused on three different subjects which are the content of as many separate chapters.The first chapter deals with the study of some fundamental properties of the flow of a Navier-Stokes liquid in a piping system, which can be either unbounded or bounded. Here, I have concentrated the analysis mostly on steady-state and time-periodic motions, and on their attainability. There are several reasons for this choice. On the one hand, because these types of motions are the most "elementary" to occur in the arterial and venuous system, and, on the other hand, because the initial-boundary value problem in an unbounded piping system has been investigated in full detail in the most recent article [86], to which I refer the interested reader.The second chapter is dedicated to the mathematical analysis of certain non-Newtonian...

show abstract

The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions

Cited by 20 publications

References 4 publications

On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities

On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

Contact Info

Product

Resources

About