Andrea Terracina

Conservation laws with discontinuous flux

We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.

CONVERGENCE OF A RELAXATION APPROXIMATION TO A BOUNDARY VALUE PROBLEM FOR CONSERVATION LAWS

We propose a semilinear relaxation approximation to the unique entropy solutions of an initial boundary value problem for a scalar conservation law. Without any restriction on the initial{boundary data or on the ux function, we prove uniform a priori estimates and convergence of that approximation as the relaxation parameter tends to zero.

A free boundary problem for scalar conservation laws

We investigate a class of free boundary problems for scalar conservation laws, which is suggested by a model of ion etching. First we give an entropy formulation of the problem; then we prove existence, uniqueness, continuous dependence, and comparison properties of solutions for a large class of initial data.

Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem

We discuss some qualitative aspects of a forward-backward parabolic problem that has been introduced in [L. C. Evans and M. Portilheiro, Math. Models Methods Appl. Sci., 14 (2004), pp. 1599–1620], [C. Mascia, A. Terracina, and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in Asymptotic Analysis and Singularities, Mathematical Society of Japan, Tokyo, 2007, pp. 451–478] and further analyzed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. This problem arises in models of phase transition in which two stable phases are separated by an interface. In particular, we consider here the problem of the extension in time of the solution constructed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. We analyze the regularity of the solution u defined in a domain

Two-phase entropy solutions of forward-backward parabolic problems with unstable phase

\mathbb{R}\times(0,T)

Radon measure-valued solutions of first order scalar conservation laws

and give an estimate, depending on the initial datum, of the number of convex regions of the function

A Riemann Solver Approach for Conservation Laws with Discontinuous Flux

u(\cdot...

Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

This paper study the two--phase problem for the forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are both stable. Here we consider the situation in which the unstable phase is taken in account, obtaining not trivial solution of the problem. It is interesting to note that such solutions are given by solving generalized Abels equations.

Two-phase entropy solutions of a forward-backward parabolic equation

Abstract We study nonnegative solutions of the Cauchy problem { ∂ t ⁡ u + ∂ x ⁡ [ φ ⁢ ( u ) ] = 0 in ⁢ ℝ × ( 0 , T ) , u = u 0 ≥ 0 in ⁢ ℝ × { 0 } , \left\{\begin{aligned} &\displaystyle\partial_{t}u+\partial_{x}[\varphi(u)]=0&% &\displaystyle\phantom{}\text{in }\mathbb{R}\times(0,T),\\ &\displaystyle u=u_{0}\geq 0&&\displaystyle\phantom{}\text{in }\mathbb{R}% \times\{0\},\end{aligned}\right. where u 0 {u_{0}} is a Radon measure and φ : [ 0 , ∞ ) ↦ ℝ {\varphi\colon[0,\infty)\mapsto\mathbb{R}} is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u 0 {u_{0}} is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ ⁢ ( u ) = u {\varphi(u)=u} this happens for all times). In the latter case, we describe the evolution of the singular parts.

Large-time behavior for conservation laws with source in a bounded domain

1 Dipartimento di Matematica e Applicazioni Universita di Milano-Bicocca, Via R. Cozzi 53 20125 Milano, Italy mauro.garavello@unimib.it 2 Istituto per le Applicazioni del Calcolo ”M. Picone” C. N. R. Viale del Policlinico, 137 00161 Roma, Italy r.natalini@iac.cnr.it 3 Istituto per le Applicazioni del Calcolo ”M. Picone” C. N. R. Viale del Policlinico, 137 00161 Roma, Italy b.piccoli@iac.cnr.it 4 Dipartimento di Matematica Universita di Roma ”La Sapienza” Piazzale Aldo Moro, 5 00185 Roma, Italy terracin@mat.uniroma1.it