On uniqueness and time regularity of flows of power-law like non-Newtonian fluids

Abstract

Large class of non-Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three-dimensional domain is established for subcritical ps, i.e. for p>11 / 5. Our method works for 'all' physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.Keywords: non-Newtonian fluids; time regularity; uniqueness; exponential attractor The problem and the main resultsLet us consider an incompressible, homogeneous fluid, which is characterized by Newton's rheological lawwhere S is the constitutively determined part of the Cauchy stress tensor, Dv the symmetric velocity gradient. One should assume that the viscosity >0 is a constant, we obtain the so-called Navier-Stokes equations (NSE). There is no doubt that, at least in certain regimes, this is a reasonable model, and the NSEs belong to the most studied systems of mathematical physics.On the other hand, there is both theoretical and experimental evidence suggesting that the viscous forces can be effective function of other physical quantities; the pressure, density, electric field and last, but not the least, the shear rate |Dv|. We refer to the survey article [1] for the detailed discussion of both the physical background and the state of art of the available mathematical results.In this paper, we focus on qualitative properties of flows of such fluids, more precisely, on the uniqueness, regularity and also large-time behavior of weak solutions to the following system of equations:that are supposed to be satisfied in a space-time cylinder Q := I× , where I ⊂ R denotes an open bounded time interval and ⊂ R 3 is an open bounded set. The first equation in (1) represents the balance of linear momentum driven by the given external body forces f , while the second one can be considered as the balance of mass for homogeneous incompressible fluid. The internal properties of the fluid are given by the constitutively determined part of the Cauchy stress tensor S. The classical examples of the fluids that can be treated are described by the so-called power-law model, where S is given by the relation S = S(Dv):= 0 |Dv| p−2 Dv, and its non-degenerate variant, the so-called Ladyzhenskaya model, where S takes the form S := 0 (1+|Dv| 2 ) (p−2)/2 Dv.From the mathematical point of view, for power-law like fluid one formally obtains the L p -estimates on ∇v, which (for p>2) are a certain remedy to the well-known lack of (the information about) the regularity of solutions of the original Navier-Stokes problem,

On the dimension of the attractor for a perturbed 3d Ladyzhenskaya model

“…If ≥ 12 /5, one can test the equation by time derivative to obtain ∈ L ∞ (0 T ; W 1 ), see [2,8]. The same result can be actually deduced for arbitrary > 11 /5, using spaces of fractional time regularity [3]. Alternatively, it is possible to improve spatial regularity (formally: test the equation by ∆ ); this is viable for > 9 /4 in the Dirichlet setting, while one can consider ≥ 11 /5 in the periodic case, see [8].…”

“…Recall that for Ω ⊂ R 3 open, bounded and sufficiently smooth, we are allowed to define an equivalent norm · on W 3 2 0 (Ω) (and naturally also on V 3 ) induced by the scalar product…”

“…in (0 T ), where the first term expresses duality between (V ∩ V 3 ) * and V ∩ V 3 . We also want to attain the initial condition, i.e.…”

“…Indeed, the energy equality provides us with both the dissipativity and compactness of the set of weak solutions, implying the existence of a compact global attractor, using the framework of [1]. Moreover, if > 11 /5, additional time regularity of solutions established in [3] yields not only the unique continuation property, but also existence of a finite-dimensional exponential attractor.…”

“…The aim of this paper is to corroborate this fact by a more rigorous argument. We study the Ladyzhenskaya model, perturbed by a higher order artificial dissipation of the form ε∆ 3 We establish the existence, uniqueness, and dissipativity of solutions; then we prove the existence of attractor and finally, estimate its dimension explicitly in terms of the data of the equation. In all the analysis, we are careful to employ dependence on ε as little as possible.…”