Darcy equation for random porous media

Many attempts have been made recently to deduce Darcy equations for flows through porous media. Much has been done for models having a periodic microstructure. This paper presents a mathematical analysis based on the more general and more realistic assumption that the microstructure of porous media is random and stochastically homogeneous. For this type of random domain we consider homogenization of the Poisson and Stokes equations supplemented by homogeneous Dirichlet boundary conditions on the random boundary. With this approach we get the Darcy equation in general as well as present details for the particular case of a checkerboard model of porous media. © 1996 John Wiley & Sons, Inc.     

Regularity results for the porous media equation

Sunto In questo lavoro si considera il problema di Cauchy per l'equazione di filtrazione ∂u/∂t=∂ ² ϕ(u)/∂x ² nella regioneR×(0,T],0<T<∞. Sotto opportune ipotesi sulla funzione ϕ(u) si determina una stima dell'incremento temporale della soluzione u(x, t) (intesa nel senso debole). Nel caso politropico (ϕ(u)=u^m), quando m>2 si trova in particolare un comportamento h?lderiano di u(x, t) rispetto a t con l'esponente1/(m−1); viene anche dimostrato che questo esponete è effettivamente assunto da una particolare soluzione, per cui la stima ottenuta è la migliore possibile.

---

---

Entrata in Redazione il 18 luglio 1978.     

Variational principle for the spectral exponent of polynomials of weighted composition operators

In this paper we derive a relationship between the Legendre-Fenchel transform of the spectral exponent of weighted composition operators acting in Lp-spaces and the Legendre-Fenchel transform obtained for their polynomials. We establish the variational principle for the spectral exponent of polynomials of weighted composition operators.     

On a random scaled porous media equation

It is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well-posed in L^∞. In the fast diffusion case we show existence in Lp.     

Uncertain seepage equation in fissured porous media

Fuzzy Optimization and Decision Making - Seepage equation in fissured porous media is a partial differential equation describing the variation of pressure of a given area over time. In traditional...     

A weighted dual porous medium equation applied to image restoration

In this paper, we discuss a weighted dual porous medium equation which could be applied to image restoration. After theoretical analysis based on nonlinear semigroup theory, we give experiments to show the efficiency of this novel model. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

The boundary value problems for the filtration equation in partially saturated porous media

In this paper we considered the first and the second boundary value problems for one-dimensional filtration in partially saturated porous media. The existence, uniqueness and regularity of weak solutions were obtained. We also studied the interface problems and discussed the asymptotic behavior of weak solutions.     

Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

The blow-up rate and uniqueness of large solution for a porous media logistic equation

In this paper we ascertain the exact blow-up rate of the large solutions of a class of sublinear elliptic problems of a logistic type related to the porous media equation, from which we can obtain the uniqueness of the solution. The weight function in front of the nonlinearity vanishes on the boundary of the underlying domain with a general decay rate which can be approximated by a distance function.     

On some local property of the shape of interfaces for the porous media equation

We consider the porous media equation with absorption for various conditions and prove that the shape of ist interface never becomes strongly upward convex. For this sake we derive an improperly posed estimate for solutions of the porous media equation for the non‐characteristic Cauchy problem (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)