Attractors and dynamical systems generated by initial-boundary problems for the equations of motion of viscoelastic fluids

Karazeeva, N. A.; Oskolkov, A. P.

doi:10.1007/bf01099349

Cited by 14 publications

(37 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem of existence of a finite dimensional global attractor and estimates for the number of determining modes on the global attractor of Kelvin-Voight fluids of order L ≥ 1 is established in [27]. In [45] the existence of a finite dimensional global attractor is established for 1D GBBM equation under periodic boundary conditions.…”

Section: The Inviscid Limitmentioning

confidence: 99%

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

Kalantarov

Titi

2009

Chin. Ann. Math. Ser. B

145

150

View full text Add to dashboard Cite

Abstract. We investigate the long-term dynamics of the three-dimensional NavierStokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, we also show that the weak solutions of the Navier-StokesVoight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0. MSC Classification: 37L30, 35Q35, 35Q30, 35B40

show abstract

Section: The Inviscid Limitmentioning

confidence: 99%

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

Kalantarov

Titi

2009

Chin. Ann. Math. Ser. B

145

150

View full text Add to dashboard Cite

show abstract

“…(0.1) with time T-periodic external force f; 3. the principle of linearized stability and related topics; 4. attractor theory. These problems are examined in [17,18,21,[25][26][27][28][29][30][31][32][33][34], and in our paper we give a review of the results. All the results of our paper are also valid for the "Oskolkov equations" [35][36][37] ()~ and for the equations of the motion of the Kelvin-Voight fluids of order L = 1, 2,... with "memory of length T " [31,33,34] L v, -vAv -xAv, + VkVxh --~ fltAut + gradp = f, div v = 0, …”

Section: (O!)mentioning

confidence: 99%

Nonlocal problems for the equations of motion of Kelvin-Voight fluids

Oskolkov¹

1995

J Math Sci

Self Cite

View full text Add to dashboard Cite

show abstract

“…We mention here some of the papers which contain surveys on this subject, different approaches, constructions, and methods of investigation: [1], [8], [10]- [16].…”

mentioning

confidence: 99%

“…For L = M and under additional conditions for coefficients {λ l }, ν and {ae m } (see [8]) the equation of the fluid motion has the following form:…”

mentioning

confidence: 99%

See 1 more Smart Citation

The topological degree method for equations of the Navier‐Stokes type

Dmitrienko

Zvyagin

1997

Abstract and Applied Analysis

View full text Add to dashboard Cite

Abstract. We obtain results of existence of weak solutions in the Hopf sense of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps A − g, where A is invertible and g is A-condensing, is used.Various problems for the Navier-Stokes equations describing the motion of the Newton fluid, and its generalizations for nonlinearly-viscous and viscoelastic fluids, have been developed in many papers. We mention here some of the papers which contain surveys on this subject, different approaches, constructions, and methods of investigation:Here we consider the problem of the existence of weak solutions, in the Hopf sense, of the initial-boundary value problem for equations of the NavierStokes type. These equations include the ones describing the movement of nonlinear-viscous and viscous-elastic fluids. We reduce the above problem to an evolution equation in the space of functionals, and then to the equivalent operator equation. The method of this paper consists of constructing operator equations which approximate the original ones, and then investigating their solvability by means of infinite-dimensional degree theory. As we know, the Galerkin-Faedo method or iteration methods have already been used instead of the degree theory for the classical Navier-Stokes equations and for some their generalizations (see, for example, 2 V. T. DMITRIENKO AND V. G. ZVYAGINThis paper consists of four sections.In the first section we introduce the main notations and notions, set up the problem of weak solutions of the initial-boundary value problem for generalized Navier-Stokes equations, and formulate our main results of existence and uniqueness of weak solutions.In the second section the problem of weak solutions is reduced to the investigation of an equivalent operator equation. Then we construct the approximating equations and investigate the properties of the operators involved.In the third section a priori estimates of solutions of approximating equations are established and a proposition on the existence of solutions of such equations is obtained.In the last section the possibility of the limit procedure in the sequence of solutions of approximating equations is established. We present two different approaches to proven the convergence and, as a corollary, we get propositions for the existence of weak solutions of the initial-boundary value problem for some cases of the generalized Navier-Stokes equations. We consider the uniqueness of solutions for dimension n = 2 as well.It should be noted that our interest in this problem arose when Professor P. E. Sobolevskii posed to one of the authors the question of the applicability of topological methods to the initial-boundary value problems in hydrodynamics. The authors are grateful to P. E. Sobolevskii , and Yu. A. Agranovich for discussions on some problems in hydrodynamics.

show abstract

Attractors and dynamical systems generated by initial-boundary problems for the equations of motion of viscoelastic fluids

Cited by 14 publications

References 10 publications

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

Nonlocal problems for the equations of motion of Kelvin-Voight fluids

The topological degree method for equations of the Navier‐Stokes type

Contact Info

Product

Resources

About