Thisyear 's lecture focused on first-order hyperbolicsystemsknown as " scalar conservation laws ". Aconsiderableliterature exists on this subject from the work of P. Lax, O. Oleinik and the importanttheory of S. N. Kruzkhovto recentdevelopments. The lecturedeveloped some aspects ofrecent and ongoing workcarried out in collaboration with P. E. Souganidis. Our research project is tocompletelyrevise the knowntheory with many extensions on the type of solutions, theirdefinition,data regularity, infinite behavior and boundary conditions. In addition to the classical boundary conditions, we are introducing new ones (saturated or conservative solutions, etc.). In this lecturesummary, we will confine ourselves to giving a few examples of theresults stated anddemonstrated in the lecture.
There's no need to go intodetail about the importance of conservation laws for applications. Suffice it to say that, in addition to the intrinsicinterest of these questions, we believe it is useful to bring thetheory of scalar conservation laws to a level ofunderstanding comparable to that which exists for the first-order Hamilton-Jacobi equations (achievedthrough thetheory of viscosity solutions). All the more so as the links between the two classes ofequations are close since, at least in dimension 1 of space, we can formally pass from one to the other by simpleintegration orderivation. Moreover , thetheory of Mean Field Games also leads to thestudy of classes of hyperbolicsystemsfor which ourresults are relevant.