400 research outputs found
An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions
The theory of weak solutions for nonlinear conservation laws is now well
developed in the case of scalar equations [3] and for one-dimensional
hyperbolic systems [1, 2]. For systems in several space dimensions, however,
even the global existence of solutions to the Cauchy problem remains a
challenging open question. In this note we construct a conterexample showing
that, even for a simple class of hyperbolic systems, in two space dimensions
the Cauchy problem can be ill posed.Comment: 12 pages, 5 figure
Hyperbolic systems of conservation laws in one space dimension
Aim of this paper is to review some basic ideas and recent developments in
the theory of strictly hyperbolic systems of conservation laws in one space
dimension. The main focus will be on the uniqueness and stability of entropy
weak solutions and on the convergence of vanishing viscosity approximations
Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation
The paper is concerned with conservative solutions to the nonlinear wave
equation . For an open dense set of
initial data, we prove that the solution is piecewise smooth in the
- plane, while the gradient can blow up along finitely many
characteristic curves. The analysis is based on a variable transformation
introduced in [7], which reduces the equation to a semilinear system with
smooth coefficients, followed by an application of Thom's transversality
theorem.Comment: 25 page
Existence of optima and equilibria for traffic flow on networks
This paper is concerned with a conservation law model of traffic flow on a
network of roads, where each driver chooses his own departure time in order to
minimize the sum of a departure cost and an arrival cost. The model includes
various groups of drivers, with different origins and destinations and having
different cost functions. Under a natural set of assumptions, two main results
are proved: (i) the existence of a globally optimal solution, minimizing the
sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium
solution, where no driver can lower his own cost by changing his departure time
or the route taken to reach destination. In the case of Nash solutions, all
departure rates are uniformly bounded and have compact support.Comment: 22 pages, 5 figure
Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization
The paper is concerned with patchy vector fields, a class of discontinuous,
piecewise smooth vector fields that were introduced in AB to study feedback
stabilization problems. We prove the stability of the corresponding solution
set w.r.t. a wide class of impulsive perturbations. These results yield the
robusteness of patchy feedback controls in the presence of measurement errors
and external disturbances.Comment: 22 page
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