40,076 research outputs found
Impermeability through a perforated domain for the incompressible 2D Euler equations
We study the asymptotic behavior of the motion of an ideal incompressible
fluid in a perforated domain. The porous medium is composed of inclusions of
size separated by distances and the fluid fills
the exterior.
If the inclusions are distributed on the unit square, the asymptotic behavior
depends on the limit of when
goes to zero. If , then the limit
motion is not perturbed by the porous medium, namely we recover the Euler
solution in the whole space. On the contrary, if
, then the fluid cannot penetrate the
porous region, namely the limit velocity verifies the Euler equations in the
exterior of an impermeable square.
If the inclusions are distributed on the unit segment then the behavior
depends on the geometry of the inclusion: it is determined by the limit of
where is related to the geometry of the lateral boundaries of the
obstacles. If , then the presence of holes is not felt at the limit, whereas an
impermeable wall appears if this limit is zero. Therefore, for a distribution
in one direction, the critical distance depends on the shape of the inclusions.
In particular it is equal to for balls
Small volume expansions for elliptic equations
This paper analyzes the influence of general, small volume, inclusions on the
trace at the domain's boundary of the solution to elliptic equations of the
form \nabla \cdot D^\eps \nabla u^\eps=0 or (-\Delta + q^\eps) u^\eps=0
with prescribed Neumann conditions. The theory is well-known when the
constitutive parameters in the elliptic equation assume the values of different
and smooth functions in the background and inside the inclusions. We generalize
the results to the case of arbitrary, and thus possibly rapid, fluctuations of
the parameters inside the inclusion and obtain expansions of the trace of the
solution at the domain's boundary up to an order \eps^{2d}, where is
dimension and \eps is the diameter of the inclusion. We construct inclusions
whose leading influence is of order at most \eps^{d+1} rather than the
expected \eps^d. We also compare the expansions for the diffusion and
Helmholtz equation and their relationship via the classical Liouville change of
variables.Comment: 42 page
Characterizations of safety in hybrid inclusions via barrier functions
This paper investigates characterizations of safety in terms of barrier functions for hybrid systems modeled by hybrid inclusions. After introducing an adequate definition of safety for hybrid inclusions, sufficient conditions using continuously differentiable as well as lower semicontinuous barrier functions are proposed. Furthermore, the lack of existence of autonomous and continuous barrier functions certifying safety, guides us to propose, inspired by converse Lyapunov theorems for only stability, nonautonomous barrier functions and conditions that are shown to be both necessary as well as sufficient, provided that mild regularity conditions on the system's dynamics holds
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