[Mini Courses] (Mini-course) An Introduction to Stochastic Scalar Conservation Laws (1)

This mini-course will contain the following: 
1) Deterministic scalar conservation laws. The example of the Burgers equation and the non-existence of a global smooth solution. Definition of weak solution. Non-uniqueness. The concept of entropy solution. Young measures. Measure-valued solutions. Existence and $L^1$ stability of bounded entropy solutions. Kinetic formulation. Kinetic defect measure. The concept of kinetic solution. Example of the Burgers equation.  2) Stochastic conservation laws. A primer in Stochastic integration. It\^o's formula. Kinetic formulation. Generalized kinetic solution. Energy estimate. Estimate of the kinetic measures. Estimate of the Young measures. Doubling of variables. L1 contraction. Overview of invariant measures of stochastic conservation laws. 

References: 
[DPZ] G. Da Prato, J. Zabczyk. ``Stochastic Equations in Infinite Dimensions''. Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge. 2nd Edition 2014.
[DV] A. Debussche, J. Vovelle. Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259 (2010) 1014-1042. 
[DV2] A. Debussche, J. Vovelle. Invariant measures of scalar first-order conservation laws with stochastic  forcing. Probab. Theory Relat. Fields (2015) 163, 575--611. 
[Kr] S.N. Kruzhkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123) (1970) 228-255. 
[Pe] B. Perthame. ``Kinetic Formulation of Conservation Laws''. Oxford Lecture Ser. Math. Appl. Vol. 21, Oxford University Press, Oxford, 2002.
[FLMNZ] H. Frid, Y. Li, D. Marroquin, J.F. Nariyoshi and Z. Zeng. The Strong Trace Property and  the Neumann Problem for Stochastic Conservation Laws. Stochastic Partial Differential Equations: Analysis and Computations, Published online, 2021.