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Generic regularity of conservative solutions to a nonlinear wave equation

Abstract: The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt}−c(u)\big(c(u)u_x\big)_x=0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient ux can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom’s transversality theorem.

Reference
[7] J. Damon, Generic properties of solutions to partial differential equations. Arch. Rational Mech. Anal. 140 (1997) 353–403 [MR1489320].
Paper:
Available as PDF (335 Kbytes).
Author(s):
Alberto Bressan,
Geng Chen,
Submitted by:
; 2015-02-09.