Geodesic
Limit Passage in the Construction of a Geometric Solution: The Case of a Rarefaction Wave
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 182-201.

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A method for constructing a geometric solution of the Riemann problem is described for a scalar conservation law perturbed by a rarefaction wave. The phase flow of the associated autonomous system is described topologically, and an explicit formula for the Hausdorff limit defining a geometric solution is presented. 
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     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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V. V. Palin. Limit Passage in the Construction of a Geometric Solution: The Case of a Rarefaction Wave. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 182-201. http://geodesic.mathdoc.fr/item/TM_2021_315_a12/

[1] Burago D., Burago Yu., Ivanov S., A course in metric geometry, Grad. Stud. Math., 33, Amer. Math. Soc., Providence, RI, 2001 | DOI

[2] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer, Dordrecht, 1988

[3] P. D. Lax, Hyperbolic Partial Differential Equations, Courant Lect. Notes Math., 14, Am. Math. Soc., Providence, RI, 2006 | DOI

[4] Palin V.V., “Konstruktsiya geometricheskogo resheniya v sluchae volny razrezheniya”, Zap. nauchn. sem. POMI, 489, 2020, 55–66 | MR

[5] V. V. Palin, “On the passage to the limit in the construction of geometric solutions of the Riemann problem”, Math. Notes, 108:3–4 (2020), 356–369 | DOI | MR