Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Piecewise smooth solutions to the Burgers–Hilbert equation

Pages: 165 – 184

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n1.a7

Authors

Alberto Bressan (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Tianyou Zhang (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Abstract

The paper is concerned with the Burgers–Hilbert equation $u_t + {(u^2 / 2)}_x = \mathbf{H}[u]$, where the right-hand side is a Hilbert transform. Unique entropy admissible solutions are constructed locally in time, having a single shock. In a neighborhood of the shock curve, a detailed description of the solution is provided.

Keywords

piecewise smooth solutions, Burgers–Hilbert equation, shock, uniqueness

2010 Mathematics Subject Classification

35B65, 76B15

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Published 10 January 2017