Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

Abstract

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.1991 Mathematics Subject Classification. 35B40, 35B41, 35Q35.

On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient

“…We also observe that (40) and (41) would just as well work with ϕ ∈ L 1+1/δ (Ω ) for some small δ > 0, whence…”

“…The number β 0 is then called the Darcy coefficient and β 1 the Forchheimer coefficient (see [40]). We will not investigate such a digression for the difference from Assumption 2.3 is minimal, at least in terms of the existence theory analysis.…”

A Regularity Criterion for the Navier-Stokes Equations with Damping in Terms of One Directional Derivative of the Velocity Field in BMO Space

“…As far as I know, Cai and Jiu [2] Markowich, Titi and Trabelsi [13] extended results in [2] and obtained existence and uniqueness of weak and strong solutions for a larger range of α , with initial data in 1 H . Kalantarov and Zelik [10] showed the uniqueness of solutions for 1 α ≥ with the Dirichlet boundary conditions and regular enough initial data in 2 H . Zhou [25] proved the existence and uniqueness of global strong solutions for 1 α ≥ and gave two regularity criteria as 0 1 α ≤ < .…”

“…(1)- (3) on (0, ) T which satisfies the energy equality (10). Assume that the condition (8) holds, then for any t T ≤ , we have …”

Nonlinear stability in microfluidic porous convection problems

“…407-415. Within the fields of porous media, convection with LTNE effects, and various other areas of partial differential equations, structural stability has recently been focussed on by Aulisa et al [4,5], Capone [7], Celebi et al [24], Chirita et al [25], Ciarletta et al [26], D'Amore [28], De Angelis and Renno [29], Gentile and Straughan [36], Hoang et al [43], Hoang and Ibragimov [41,42], Kalantarov and Zelik [44], Kandem [45], Kang and Park [46], Kelliher et al [48], Knops and Payne [49], Knops and Payne [50], Layton and Rebholz [53], Li et al [54], Lin and Payne [55][56][57], Liu [58,59], Liu et al [60,61], Ouyang and Yang [70], Passarella et al [71], Payne [72][73][74], Payne et al [75], Payne and Straughan [76][77][78][79][80][81], Rionero and Vergori [98], Salvadori and Visentin [99], Ugurlu [112], You et al [114].…”