arXiv subjects

Hermann Douanla

Publications and source records attributed to Hermann Douanla.

9 recordsLinked to original sources

arXiv2026

Nonlinear Darcy Laws for Generalized Newtonian Fluids with Orlicz Growth and Oscillating Navier Slip in Periodic Porous Media

We study the periodic homogenization of a stationary generalized Navier--Stokes system in a periodically perforated domain $Ω\subset\mathbb R^N$, $N\in\{2,3\}$, with a heterogeneous bulk stress of general Orlicz growth and an oscillating nonlinear Navier friction law on the pore walls, both at the Darcy scale. If the lower growth exponent at infinity satisfies $p^-_\infty>3N/(N+2)$ and the inertial exponent exceeds $κ_c:=1+N(2/p^-_\infty-1)_+$, inertia vanishes and, under strict monotonicity of the bulk stress, the whole family converges to the nonlinear Darcy law $U=\mathcal K(f-\nabla p_0)$, $\operatorname{div}U=0$, $U\cdotν=0$. The permeability map $\mathcal K$ is continuous, strictly monotone, obeys two-sided Orlicz bounds and incorporates the wall slip through a nonlinear cell problem, and the extended microscopic pressure converges strongly to $p_0$. The proof relies on a uniform Orlicz--Korn inequality, a Darcy-scaled two-scale trace theorem and a coupled bulk--surface Minty argument.

math.AP↗

arXiv2026

Homogenization of degenerate parabolic equations with nonlinear Robin reactions in perforated domains

We study periodic homogenization of a parabolic equation with linear, degenerate bulk terms in a perforated domain, with a monotone nonlinear Robin reaction of order $\varepsilon$ on the pore walls. The storage coefficient may vanish on sets of positive measure, and the diagonal diffusion weights are not bounded below but satisfy an inverse-integrability condition. Under positivity of the mean storage, a quantitative cell-detection estimate yields strong compactness of the states and of the scaled traces. The effective equation has positive storage, a positive-definite diffusion matrix given by a weighted cell problem, and a nonlinear volume reaction obtained by averaging over the pore surface. We also prove a strong weighted-gradient corrector and uniform-in-time convergence of the dissipated and stored energies and of the states in the storage-weighted $L^2$ norm.

math.AP↗

arXiv2026

Homogenization of degenerate parabolic Signorini problems with vanishing storage and oscillating surface fluxes

We study the homogenization of a unilateral quasilinear parabolic problem in a periodically perforated medium. The model combines two degeneracies: the coefficient of the time derivative is non-negative and may vanish, while the directional diffusion weights are positive almost everywhere but need not be bounded away from zero. The exterior boundary carries a Signorini condition, and the holes carry an unscaled zero-mean flux oscillating on the diffusive time scale. For any family of microscopic solutions we prove strong space--time $L^2$ compactness through a weighted detection inequality, which replaces the Aubin--Lions lemma when the storage vanishes, and we recover spatial $H^1$ regularity of the limit by a closed-range argument for the weighted two-scale gradients. The limit solves a coupled two-scale problem in Minty form, and microscopic existence holds under joint monotonicity. Under a strengthened condition, the limit decouples into a macroscopic Signorini inequality whose uniqueness follows from an exponential regularization, without time regularity of the limit, and the whole family converges. The surface datum enters both the effective flux and the exterior boundary condition.

math.AP↗

arXiv2026

Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains

We study normalized principal Steklov eigencouples on periodically perforated domains when both the bulk energy and the boundary normalization have general, possibly nonhomogeneous, Orlicz growth. The surface term on the hole boundaries carries the critical surface-to-volume scaling. For each fixed mass level, the constrained minima converge along the whole family, while selected minimizers, fluxes and Lagrange multipliers converge along subsequences to the corresponding objects of a homogenized problem. In the limit, the bulk operator is given by a convex cell problem, and the boundary term becomes a volume term weighted by the total surface density of the reference hole. The key tool is a strong approximation of the microscopic traces by cell averages, measured in the scaled surface modular.

math.AP↗

arXiv2018

Deterministic homogenization of variational inequalities with unilateral constraints

The article studies the reiterated homogenization of linear elliptic variational inequalities arising in problems with unilateral constrains. We assume that the coefficients of the equations satisfy and abstract hypothesis covering on each scale a large set of concrete deterministic behavior such as the periodic, the almost periodic, the convergence at infinity. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is of the same type as the problem with rapidly oscillating coefficients.

math-ph↗

arXiv2017

Reiterated homogenization of hyperbolic-parabolic equations in domains with tiny holes

This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of the same type as the problem with oscillating coefficients.

math.AP↗

arXiv2015

Homogenization of reaction-diffusion equations in fractured porous media

The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of convection-diffusion-reaction type.

math.AP↗

arXiv2012

Almost periodic homogenization of a generalized Ladyzhenskaya model for incompressible viscous flow

The paper deals with the existence and almost periodic homogenization of some model of generalized Navier-Stokes equations. We first establish an existence result for non-stationary Ladyzhenskaya equations with a given non constant density. The external force depends nonlinearly on the velocity. Next, to proceed with homogenization, as the density of the fluid is non constant, we combine the sigma-convergence method with some compactness arguments.

math.AP↗

arXiv2012

Homogenization of a nonlinear elliptic problem with large nonlinear potential

Homogenization is studied for a nonlinear elliptic boundary-value problem with a large nonlinear potential. More specifically we are interested in the asymptotic behavior of a sequence of p-Laplacians of the form $$ -\text{div}(a(\frac{x}{\varepsilon})|Du_\varepsilon|^{p-2}Du_\varepsilon) +\frac{1}{\varepsilon}V(\frac{x}{\varepsilon})|u_\varepsilon|^{p-2}u_\varepsilon=f. $$ It is shown that, under a centring condition on the potential $V$, there exists a two-scale homogenized system with solution $(u, u_1)$ such that the sequence $u_\varepsilon$ of solutions converges weakly to $u$ in $W^{1,p}$ and the gradients $D_x u_\varepsilon$ two-scale converges weakly to $D_x u+ D_y u_1$ in $L^p$, respectively. We characterize the limit system explicitly by means of two-scale convergence and a new convergence result.

math.AP↗