Alberto Bressan

Global dissipative solutions of the Camassa-Holm equation

This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data

Global solutions of the Hunter-Saxton equation

\bar u\in H^1 ({\mathbb R})

The unique limit of the Glimm scheme

, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

The semigroup generated by 2 × 2 conservation laws

We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equation modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.

Impulsive control systems without commutativity assumptions

We introduce the definitions of a standard Riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws. For a class including general 2×2 systems, it is proved that the solutions obtained by a wavefront tracking algorithm or by the Glimm scheme are precisely the semigroup trajectories. In particular, these solutions are unique and depend Lipschitz continuously on the initial data in the L1 norm.

Flows on networks: recent results and perspectives

Consider the Cauchy problem for a strictly hyperbolic 2×2 system of conservation laws in one space dimension: {ie1-01} assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. This paper develops a new algorithm, based on wave-front tracking, which yields a Cauchy sequence of approximate solutions, converging to a unique limit depending continuously on the initial data. The solutions that we obtain constitute a semigroup S, defined on a set {ie1-02} of integrable functions with small total variation. For some Lipschitz constant L, we have the estimate {ie1-03}

A Generic Classification of Time-Optimal Planar Stabilizing Feedbacks

AbstractThis paper is concerned with optimal control problems for an impulsive system of the form

Total Blow-Up versus Single Point Blow-Up

\dot x(t) = f(t, x, u) + \sum\limits_{i = 1}^m {g_i } (t, x, u)\dot u_i ,u(t) \in U,