In biological transport mechanisms such as insect respiration and renal filtration, fluid travels along a leaky channel allowing material exchange with systems exterior to the channel. The channels in these systems may undergo peristaltic pumping which is thought to enhance the material exchange. To date, little analytic work has been done to study the effect of pumping on material extraction across the channel walls. In this paper, we examine a fluid extraction model in which fluid flowing through a leaky channel is exchanged with fluid in a reservoir. The channel walls are allowed to contract and expand uniformly, simulating a pumping mechanism. In order to efficiently determine solutions of the model, we derive a formal power series solution for the Stokes equations in a finite channel with uniformly contracting/expanding permeable walls. This flow has been well studied in the case in which the normal velocity at the channel walls is proportional to the wall velocity. In contrast we do not assume flow that is proportional to the wall velocity, but flow that is driven by hydrostatic pressure, and we use Darcy’s law to close our system for normal wall velocity. We incorporate our flow solution into a model that tracks the material pressure exterior to the channel. We use this model to examine flux across the channel-reservoir barrier and demonstrate that pumping can either enhance or impede fluid extraction across channel walls. We find that associated with each set of physical flow and pumping parameters, there are optimal reservoir conditions that maximize the amount of material flowing from the channel into the reservoir.

For the viscous and heat-conductive fluids governed by the compressible NavierStokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the L^p - L^q estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L²-norm are obtained when the L¹-norm of the perturbation is bounded.

In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n⩽9. This result, that was only known to be true for n⩽4, is optimal: log(1/|x|^2) is a W^{1,2} singular stable solution for n⩾10. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n⩽9, stable solutions are bounded in terms only of their L^1 norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are W^{1,2} in every dimension and they are smooth in dimension n⩽9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.

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We consider a one-dimensional model biodegradation system consisting of two reactionadvection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (<i>pdf</i>) and the traveling front profiles. The travel time <i>pdf</i> does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles