Year:    2022

CAM-Net Digest, Vol. 19 (2022), Iss. 21 : p. 5

Abstract

Time: 2022-10-31 19:00 — 21:00
Venue: 腾讯会议
Speaker: Hermano Frid
Affiliation: Instituto de Matematica Pura e Aplicada-IMPA, Brazil
Tencent Meeting ID: 463 9509 7151
Meeting Password: 1234 https://meeting.tencent.com/dm/RpAX62WD3oM6

Abstract:
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. Itô's formula. Kinetic formulation. Generalized kinetic solution. Energy estimate. Estimate of the kinetic measures. Estimate of the Young measures. Doubling of variables. $L^1$ 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.