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When a plane shock hits a two-dimensional wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The experimental, computational, and asymptotic analysis has indicated that various patterns occur, including regular reflection and Mach reflection. The von Neumann's conjectures on the transition from regular to Mach reflection involve the existence, uniqueness, and stability of regular shock reflection-diffraction configurations, generated by concave cornered wedges for compressible flow. In this paper, we discuss some recent developments in the study of the von Neumann's conjectures. More specifically, we discuss the uniqueness and stability of regular shock reflection-diffraction configurations governed by the potential flow equation in an appropriate class of solutions. We first show that the transonic shocks in the global solutions obtained in Chen-Feldman [19] are convex. Then we establish the uniqueness of global shock reflection-diffraction configurations with convex transonic shocks for any wedge angle larger than the detachment angle or the critical angle. Moreover, the solution under consideration is stable with respect to the wedge angle. Our approach also provides an alternative way of proving the existence of the admissible solutions established first in [19].
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Shock interactions for the Burgers-Hilbert equation
This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with H2 regularity away from the shocks plus a corrector term having an asymptotic behavior like x ln x close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.
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- Award ID(s):
- 2006884
- PAR ID:
- 10455939
- Date Published:
- Journal Name:
- Communications in partial differential equations
- Volume:
- 47
- Issue:
- 9
- ISSN:
- 0360-5302
- Page Range / eLocation ID:
- 1795–1844
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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