# Conservation laws

Thursday, June 29, 2017 - 9:00am - 9:50am

Chi-Wang Shu (Brown University)

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature.

Friday, July 31, 2009 - 9:00am - 9:50am

Rinaldo Colombo (Università di Brescia)

Given a model based on a conservation law, we study how the solution

depends from the initial/boundary datum, from the flow and from

various constraints. With this tool, several control problems can be

addressed and the existence of an optimal control can be

proved. Models describing escape dynamics of pedestrians, traffic at

toll gates, open canals management and fluid flow in gas pipelines

fall within this framework. In particular, a necessary condition for

optimality is obtained, which applies to a supply chain model.

depends from the initial/boundary datum, from the flow and from

various constraints. With this tool, several control problems can be

addressed and the existence of an optimal control can be

proved. Models describing escape dynamics of pedestrians, traffic at

toll gates, open canals management and fluid flow in gas pipelines

fall within this framework. In particular, a necessary condition for

optimality is obtained, which applies to a supply chain model.

Thursday, July 30, 2009 - 2:00pm - 2:50pm

Mauro Garavello (Università del Piemonte Orientale Amedeo Avogadro)

In this talk we consider a conservation law (or a system of conservation

laws) on a network consisting in a finite number of arcs and vertices.

This setting is justified by various applications, such as car traffic,

gas pipelines, data networks, supply chains, blood circulation and so on.

The key point in the extension of conservation laws on networks is to

define solutions at vertices. Indeed, it is sufficient to define

solutions only for Riemann problems at vertices, i.e. Cauchy problems

laws) on a network consisting in a finite number of arcs and vertices.

This setting is justified by various applications, such as car traffic,

gas pipelines, data networks, supply chains, blood circulation and so on.

The key point in the extension of conservation laws on networks is to

define solutions at vertices. Indeed, it is sufficient to define

solutions only for Riemann problems at vertices, i.e. Cauchy problems

Monday, July 27, 2009 - 2:00pm - 2:50pm

Helge Kristian Jenssen (The Pennsylvania State University)

Motivated by the problem of symmetric collapsing gas-dynamical shocks we

present a scalar toy model that captures blowup of focusing waves. This model

is simple enough to allow for explicit calculations, and we study some solutions in

detail. As a scalar model it does not describe reflection of waves and this necessitates

a new concept of weak solutions.

Returning to gas-dynamics we review a part of the extensive literature on

compressible flow with symmetry, and collapsing shocks in particular.

present a scalar toy model that captures blowup of focusing waves. This model

is simple enough to allow for explicit calculations, and we study some solutions in

detail. As a scalar model it does not describe reflection of waves and this necessitates

a new concept of weak solutions.

Returning to gas-dynamics we review a part of the extensive literature on

compressible flow with symmetry, and collapsing shocks in particular.

Monday, July 27, 2009 - 11:00am - 11:50am

John Hunter (University of California)

The Kriess-Sakamoto theory for the well-posedness of hyperbolic IBVPs and the Majda theory for shock-wave stability apply under the assumption that a suitable Lopatinski condition holds uniformly. The failure of uniformity is associated with the presence of surface waves on the boundary or discontinuity. We will derive asymptotic equations for genuine nonlinear surface waves that decay exponentially away from the surface, such as Rayleigh waves in elasticity or surface waves on a tangential discontinuity in MHD.

Saturday, July 25, 2009 - 11:00am - 11:50am

Laura Valentina Spinolo (Scuola Normale Superiore)

The talk will be based on a joint work with L. Ambrosio, G. Crippa and A.

Figalli. First, some new well-posedness results for continuity and

transport equations with weakly differentiable velocity fields will be

discussed. These results can be applied to the analysis of a 2 x 2 system

of conservation laws in one space dimension known as the chromatography

system, leading to global existence and uniqueness results for suitable

classes of entropy admissible solutions.

Figalli. First, some new well-posedness results for continuity and

transport equations with weakly differentiable velocity fields will be

discussed. These results can be applied to the analysis of a 2 x 2 system

of conservation laws in one space dimension known as the chromatography

system, leading to global existence and uniqueness results for suitable

classes of entropy admissible solutions.

Wednesday, July 22, 2009 - 9:00am - 9:50am

Denis Serre (École Normale Supérieure de Lyon)

In his celebrated thesis, S. Kawashima gave a framework for the analysis of the Cauchy problem for nonlinear viscous systems of conservation laws. Some assumptions are quite natural, while other ones are mysterious and require cumbersome calculations. We clarify the situation, by introducing a set of assumption which is natural and very easy to verify. We derive an existence and uniqueness result of strong solutions, in a slightly larger class than the one known before.

Saturday, July 18, 2009 - 9:00am - 10:15am

Gui-Qiang Chen (Northwestern University)

Same abstract as lecture 1.