This work is focused on the analysis of non-linear flows of slightly compressible fluids in porous media not adequately described by Darcy's law. We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws. The non-linear Forchheimer equation is inverted to a non-linear Darcy equation with implicit permeability tensor depending on the pressure gradient. This results in a degenerate parabolic equation for the pressure. Two classes of boundary conditions are considered, given pressure and given total flux. In both cases they are allowed to be unbounded in time. The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed. The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters. Some numerical simulations are also included.

We study the generalized Forchheimer equations for slightly compressible fluids in porous media. The structural stability is established with respect to either the boundary data or the coefficients of the Forchheimer polynomials. An inhomogeneous Poincare-Sobolev inequality related to the non-linearity of the equation is used to study the asymptotic behavior of the solutions. Moreover, we prove a perturbed monotonicity property of the vector field associated with the resulting non-Darcy equation, where the correction is linear in the coefficients of the Forchheimer polynomials.

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Abstract. Let Λ be a complete metric space, and let {S λ (·) : λ ∈ Λ} be a parametrised family of semigroups with global attractors A λ . We assume that there exists a fixed bounded set D such that A λ ⊂ D for every λ ∈ Λ. By viewing the attractors as the limit as t → ∞ of the sets S λ (t)D, we give simple proofs of the equivalence of 'equi-attraction' to continuity (when this convergence is uniform in λ) and show that the attractors A λ are continuous in λ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise). Global attractorsThe global attractor of a dynamical system is the unique compact invariant set that attracts the trajectories starting in any bounded set at a uniform rate. [7]. Global attractors exist for many infinitedimensional models [18], with familiar low-dimensional ODE models such as the Lorenz equations providing a testing ground for the general theory [8].While upper semicontinuity with respect to perturbations is easy to prove, lower semicontinuity (and hence full continuity) is more delicate, requiring structural assumptions on the attractor or the assumption of a uniform attraction rate. However, Babin & Pilyugin [2] proved that the global attractor of a parametrised set of semigroups is continuous at a residual set of parameters, by taking advantage of the known upper semicontinuity and then using the fact that upper semicontinuous functions are continuous on a residual set.Here we reprove results on equi-attraction and residual continuity in a more direct way, which also serves to demonstrate more clearly why these results are true. Given equi-attraction the attractor is the uniform limit of a sequence of continuous functions, and hence continuous (the converse requires a generalised version of Dini's Theorem); more generally, it is the pointwise limit of a sequence of continuous functions, i.e. a 'Baire one' function, and therefore the set of continuity points forms a residual set.2000 Mathematics Subject Classification. Primary 35B41.

We investigate the global time existence of smooth solutions for the ShigesadaKawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global W 1,p -estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.