Abstract
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with a nonlinearity of 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.
Highlights
We study the Brinkman-Forchheimer (BF) equations in the following form:∂tu − ∆u + f (u) + ∇p = g, div u = 0, u ∂Ω = 0, u t=0 = u0. (1.1)Here Ω ⊂ R3 is an open, bounded domain with C2 boundary ∂Ω, g = g(x) = (g1, g2, g3) is a given function, u = (u1, u2, u3) is the fluid velocity vector, p is the pressure and f is a given nonlinearity.The BF equations are used to describe the fluid flow in a saturated porous media, see [16, 21] and references therein
Where a ∈ R and b > 0 are the Darcy and Forchheimer coefficients respectively (the original Brinkman-Forchheimer model corresponds to the choice r = 2, more complicated nonlinear terms (r = 2) appear, e.g., in the theory of non-Newtonian fluids, see [20])
Note that the analogous equations are used in the study of tidal dynamics
Summary
We study the Brinkman-Forchheimer (BF) equations in the following form:. ∂tu − ∆u + f (u) + ∇p = g, div u = 0, u ∂Ω = 0, u t=0 = u0. Which claims that the solution w belongs to H2 if g ∈ L2 This result is straightforward for the case of periodic boundary conditions (it follows via the multiplication of the equation by ∆w and integrating by parts). We apply our maximal regularity result in order to establish the existence of smooth solutions for the so-called convective BF equations:. We obtain a number of a priori estimates for the solutions of the problem (1.1) assuming that the sufficiently regular solution (u, p) of this equation is given These estimates will be used in order to establish the existence and uniqueness of solution, their regularity, etc.
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