Abstract:Abstract. We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n = 2, 3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity u suitably coupled with a convective … Show more
“…Our proof of the existence of the global attractor slightly differs from the one in [33]. While here we obtain the optimal regularity in one step (also as in [21,Section 6]), the method of [33] requires a bootstrapping procedure which implies first the regularity in V α . Here, we also wanted to rephrase those results into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α.…”
Section: Lemma 33mentioning
confidence: 96%
“…We emphasize that all the results proven in this article are also valid in a more general setting, when Ω is a compact Riemannian manifold with or without boundary and in the presence of other boundary conditions. We refer the reader to [21,26] for more details. System (2.1)-(2.3) can be rewritten as an abstract evolution equation of the form…”
Section: Mathematical Setting Formentioning
confidence: 99%
“…In recent years, many regularized equations have been proposed for the purpose of direct numerical simulations of turbulent incompressible flows modeled by the Navier-Stokes (NSE) equations [26] (for models of two-phase incompressible fluid flows, we refer the reader to [21] for further discussion and results). One such regularized model is the Navier-Stokes-Voigt (NSV) equations introduced by Oskolkov [41] as a model for the motion of a linear, viscoelastic, incompressible fluid.…”
Section: Introductionmentioning
confidence: 99%
“…While the first issue (a) has already been treated in [21,Section 6] to some extent as a special case of a general family of regularized models, in this contribution we also wanted to rephrase the results associated with the 3D NSV into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α > 0. One feature of this analysis is that we are able to obtain the optimal regularity of the global attractor in one step as opposed to employing a complicated bootstrapping procedure as in [33].…”
Section: Introductionmentioning
confidence: 99%
“…We emphasize that the former procedure has proved quite useful in the treatment of other regularized models for the 3D NSE (cf. [21,Section 6]). …”
ABSTRACT. We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [33].
“…Our proof of the existence of the global attractor slightly differs from the one in [33]. While here we obtain the optimal regularity in one step (also as in [21,Section 6]), the method of [33] requires a bootstrapping procedure which implies first the regularity in V α . Here, we also wanted to rephrase those results into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α.…”
Section: Lemma 33mentioning
confidence: 96%
“…We emphasize that all the results proven in this article are also valid in a more general setting, when Ω is a compact Riemannian manifold with or without boundary and in the presence of other boundary conditions. We refer the reader to [21,26] for more details. System (2.1)-(2.3) can be rewritten as an abstract evolution equation of the form…”
Section: Mathematical Setting Formentioning
confidence: 99%
“…In recent years, many regularized equations have been proposed for the purpose of direct numerical simulations of turbulent incompressible flows modeled by the Navier-Stokes (NSE) equations [26] (for models of two-phase incompressible fluid flows, we refer the reader to [21] for further discussion and results). One such regularized model is the Navier-Stokes-Voigt (NSV) equations introduced by Oskolkov [41] as a model for the motion of a linear, viscoelastic, incompressible fluid.…”
Section: Introductionmentioning
confidence: 99%
“…While the first issue (a) has already been treated in [21,Section 6] to some extent as a special case of a general family of regularized models, in this contribution we also wanted to rephrase the results associated with the 3D NSV into α-dependent spaces, in order to simplify the handling of the dependence on the parameter α > 0. One feature of this analysis is that we are able to obtain the optimal regularity of the global attractor in one step as opposed to employing a complicated bootstrapping procedure as in [33].…”
Section: Introductionmentioning
confidence: 99%
“…We emphasize that the former procedure has proved quite useful in the treatment of other regularized models for the 3D NSE (cf. [21,Section 6]). …”
ABSTRACT. We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [33].
Communicated by H. Nussenzveig LopesThis paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned.
Communicated by: W. Sprößig MSC Classification: 35K55; 35Q35; 76D05Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, for higher-order Navier-Stokes-Cahn-Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.
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