Porous medium equation with nonlocal pressure

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$, which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when $N=1$ and $m>2$, and the asymptotic behavior of solutions when $N=1$. The cases $m = 1$ and $m = 2$ were rather well known.Comment: 24 pages, 2 figure

Comparison of Marine Spatial Planning Methods in Madagascar Demonstrates Value of Alternative Approaches

The Government of Madagascar plans to increase marine protected area coverage by over one million hectares. To assist this process, we compare four methods for marine spatial planning of Madagascar's west coast. Input data for each method was drawn from the same variables: fishing pressure, exposure to climate change, and biodiversity (habitats, species distributions, biological richness, and biodiversity value). The first method compares visual color classifications of primary variables, the second uses binary combinations of these variables to produce a categorical classification of management actions, the third is a target-based optimization using Marxan, and the fourth is conservation ranking with Zonation. We present results from each method, and compare the latter three approaches for spatial coverage, biodiversity representation, fishing cost and persistence probability. All results included large areas in the north, central, and southern parts of western Madagascar. Achieving 30% representation targets with Marxan required twice the fish catch loss than the categorical method. The categorical classification and Zonation do not consider targets for conservation features. However, when we reduced Marxan targets to 16.3%, matching the representation level of the “strict protection” class of the categorical result, the methods show similar catch losses. The management category portfolio has complete coverage, and presents several management recommendations including strict protection. Zonation produces rapid conservation rankings across large, diverse datasets. Marxan is useful for identifying strict protected areas that meet representation targets, and minimize exposure probabilities for conservation features at low economic cost. We show that methods based on Zonation and a simple combination of variables can produce results comparable to Marxan for species representation and catch losses, demonstrating the value of comparing alternative approaches during initial stages of the planning process. Choosing an appropriate approach ultimately depends on scientific and political factors including representation targets, likelihood of adoption, and persistence goals

Nonlocal generalized models for a confined plasma in a Tokamak

AbstractThe following model appears in plasma physics for a Tokamak configuration: −Δu + g(u) = 0, u ∈ V = H01(Ω) ⊗ R, ∫aΩ auan = I > 0, where I is a given positive constant, which is equivalent to find a fixed point u = F(u −g(u)) + ϕ0 where F is a compact operator on L2(Ω). According to Grad and Shafranov the nonlinearity g can depend on u∗ which is the generalized inverse of the distribution function m(t) = measx : u(x) > t = −vb u > t -vb (see [1]). But in these cases the map u → g(u) cannot be continuous on all the space v but only on a nonlinear nonclosed set v0. This implies that the standard direct method for fixed point cannot be applied to solve the preceding problem. Nevertheless, using the Galerkin method and a topological argument, we prove that there exists a solution u fixed point of u = F(u − g(u)) + ϕ0 under suitable assumptions on g.The model we treat covers a large new class of nonlinearities including relative rearrangment and monotone rearrangment. The resolution of the concrete model needs an extension of the strong continuity result of the relative rearrangement map made in [2] (see Theorem 1.1 below for the definition and result)

On a nonlocal problem for a confined plasma in a Tokamak

summary:The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms $u'_{\ast }(|u>u(x)|)$ and $|u>u(x)|$, which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem

Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of IRN. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain