95 research outputs found
Porous medium equation with nonlocal pressure
We provide a rather complete description of the results obtained so far on
the nonlinear diffusion equation , which describes a flow through a porous medium driven by a
nonlocal pressure. We consider constant parameters and , 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 and , and the
asymptotic behavior of solutions when . The cases and were
rather well known.Comment: 24 pages, 2 figure
Asymptotic analysis of an elastic rod with rounded ends
We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.Peer reviewe
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