Luan Hoang

Analysis of generalized Forchheimer flows of compressible fluids in porous media

This work is focused on the analysis of nonlinear 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 generalized Forchheimer equation is inverted to a nonlinear 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.

Structural stability of generalized Forchheimer equations for compressible fluids in porous media

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. A weighted Poincare–Sobolev inequality related to the nonlinearity of the equation is used to study the asymptotic behaviour 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 explicit and Lipschitz continuous in the coefficients of the Forchheimer polynomials.

Gradient Estimates and Global Existence of Smooth Solutions to a Cross-Diffusion System

We investigate the global time existence of smooth solutions for the Shigesada--Kawasaki--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

Generalized Forchheimer Flows of Isentropic Gases

W^{1,p}

ASYMPTOTIC EXPANSION IN GEVREY SPACES FOR SOLUTIONS OF NAVIER-STOKES EQUATIONS

-estimates of Calderon--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.

Generalized Forchheimer flows in heterogeneous porous media

We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids in porous media. By using Muskat’s and Ward’s general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the

Fluid flows of mixed regimes in porous media

L^\infty

On the helicity in 3D-periodic Navier- Stokes equations I: the non-statistical case

L∞ and