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I'm looking for a source, like a textbook or something else that be used as a reference in a paper, for the theory of linear transport equations.

I want to have a proof of the well-known result that shows that the solution of these equations is given by the initial conditions composed with the solution of the characteristic system of ODEs.

All the results I need are for example perfectly represented in chapter 2 of here. But unfortunately, that are just online lecture notes that I don't want to cite in a paper.

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  • $\begingroup$ The Cauchy Problem in Kinetic Theory by Robert T. Glassy published in SIAM covers a lot about the Vlasov equation in the linear and nonlinear cases. $\endgroup$ – Dayton Jan 16 at 22:07
  • $\begingroup$ Thanks for the answer. I haven't found the basic theorems of chapter 2 in there and I think this book is already too advanced. To clarify I'm not interested in Vlasov equations, but just the existence theory for linear transport equations with variable coefficients $\endgroup$ – taylor123123 Jan 16 at 22:29
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Maybe you are looking for a PDE book which addresses the method of characteristics for (linear) scalar first-order equations in $n$ variables. There are few of them that could be used:

  1. R. Courant, D. Hilbert, Methods of Mathematical Physics: Partial Differential Equations, Sec. 2.2, Wiley, 1962. doi:10.1002/9783527617234

  2. L.C. Evans, Partial Differential Equations, 2nd ed., Sec. 2.1, AMS, 2010. 10.1090/gsm/019

  3. F. John, Partial Differential Equations, 4th ed., Sec 1.5, Springer, 1982. 10.1007/978-1-4684-0059-5

etc. See also references therein.

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