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Transport equation and Cauchy problem for non-smooth vector fields. (English) Zbl 1159.35041

Dacorogna, Bernard (ed.) et al., Calculus of variations and nonlinear partial differential equations. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, June 27–July 2, 2005. With a historical overview by Elvira Mascolo. Berlin: Springer (ISBN 978-3-540-75913-3/pbk). Lecture Notes in Mathematics 1927, 1-42 (2008).
In these lectures the author presents the well-posedness theory of the Cauchy problem for the conservative continuity equation and for the corresponding transport equation. The main assumption is the spatial Sobolev (this corresponds to the DiPerna-Lions theory) or BV regularity of the coefficients. Relation with the ODE generated by the field of coefficients is studied. Applications to special systems of conservation laws are also given.
For the entire collection see [Zbl 1126.35004].

MSC:

35L65 Hyperbolic conservation laws
35F10 Initial value problems for linear first-order PDEs
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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