Lagrangian representation for conservation laws

Prof. Stefano Bianchini, SISSA, Trieste

The use of characteristics to represent smooth solutions in quasilinear first order systems is textbook classic, and it is also well known that after the so called gradient catastrophe, this representation fails because the solution becomes discontinuous and there is no canonical way to continue the representation. For linear transport equations, instead, even if the solution is very weak (say a measure) and the vector field is only locally integrable, a representation in terms of superposition of characteristics is the base of important progresses regarding uniqueness and existence of a more regular flow.

In this talk I will show how the method of characteristic can be extended to scalar equations and hyperbolic systems of conservation laws, yielding a new representation of the solution, the Lagrangian representation. I will address in particular the following points:

• questions where the Lagrangian representation arises naturally;
• the Lagrangian representation as a continuous wave tracing;
• fine description of $L^\infty$-solutions for scalar equations and $BV$-solutions for systems.