145 research outputs found
NonLocal Systems of Balance Laws in Several Space Dimensions with Applications to Laser Technolog
For a class of systems of nonlinear and nonlocal balance laws in several
space dimensions, we prove the local in time existence of solutions and their
continuous dependence on the initial datum. The choice of this class is
motivated by a new model devoted to the description of a metal plate being cut
by a laser beam. Using realistic parameters, solutions to this model obtained
through numerical integrations meet qualitative properties of real cuts.
Moreover, the class of equations considered comprises a model describing the
dynamics of solid particles along a conveyor belt
Isentropic Fluid Dynamics in a Curved Pipe
In this paper we study isentropic flow in a curved pipe. We focus on the
consequences of the geometry of the pipe on the dynamics of the flow. More
precisely, we present the solution of the general Cauchy problem for isentropic
fluid flow in an arbitrarily curved, piecewise smooth pipe. We consider initial
data in the subsonic regime, with small total variation about a stationary
solution. The proof relies on the front-tracking method and is based on [1]
On the Stability Functional for Conservation Laws
This note is devoted to the explicit construction of a functional defined on
all pairs of \L1 functions with small total variation, which is equivalent to
the \L1 distance and non increasing along the trajectories of a given system
of conservation laws. Two different constructions are provided, yielding an
extension of the original stability functional by Bressan, Liu and Yang.Comment: 26 page
Hyperbolic predators vs parabolic preys
We present a nonlinear predator-prey system consisting of a nonlocal
conservation law for predators coupled with a parabolic equation for preys. The
drift term in the predators' equation is a nonlocal function of the prey
density, so that the movement of predators can be directed towards region with
high prey density. Moreover, Lotka-Volterra type right hand sides describe the
feeding. A theorem ensuring existence, uniqueness, continuous dependence of
weak solutions and various stability estimates is proved, in any space
dimension. Numerical integrations show a few qualitative features of the
solutions.Comment: 35 pages, 7 figure
Non Local Conservation Laws in Bounded Domains
The well posedness for a class of non local systems of conservation laws in a
bounded domain is proved and various stability estimates are provided. This
construction is motivated by the modelling of crowd dynamics, which also leads
to define a non local operator adapted to the presence of a boundary. Numerical
integrations show that the resulting model provides qualitatively reasonable
solutions
Polynomial Profits in Renewable Resources Management
A system of renewal equations on a graph provides a framework to describe the
exploitation of a biological resource. In this context, we formulate an optimal
control problem, prove the existence of an optimal control and ensure that the
target cost function is polynomial in the control. In specific situations,
further information about the form of this dependence is obtained. As a
consequence, in some cases the optimal control is proved to be necessarily
bang--bang, in other cases the computations necessary to find the optimal
control are significantly reduced
Stability and Optimization in Structured Population Models on Graphs
We prove existence and uniqueness of solutions, continuous dependence from
the initial datum and stability with respect to the boundary condition in a
class of initial--boundary value problems for systems of balance laws. The
particular choice of the boundary condition allows to comprehend models with
very different structures. In particular, we consider a juvenile-adult model,
the problem of the optimal mating ratio and a model for the optimal management
of biological resources. The stability result obtained allows to tackle various
optimal management/control problems, providing sufficient conditions for the
existence of optimal choices/controls.Comment: 22 pages, 7 figure
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