Shock formation for the Burgers-Hilbert equation

Abstract

We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^{\infty }$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar shock at one single point with an explicitly computable blowup profile. The blowup profile is a cusp with Hölder $1/3$ continuity. The blowup time and location are described in terms of explicit ODEs. Our proof uses a transformation to modulated self-similar variables, the quantitative properties of the stable self-similar solution to the inviscid Burgers equation, an $L^2$-estimate in self-similar variables, and pointwise estimates for Hilbert transform and for transport equations.

References in corpus (3)

• The wave breaking for Whitham-type equations revisited
• Formation of shocks for 2D isentropic compressible Euler
• On the approximation of vorticity fronts by the Burgers-Hilbert equation

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