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Annals of PDE

, 4:10 | Cite as

On the Formation of Shock for Quasilinear Wave Equations with Weak Intensity Pulse

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Abstract

In this paper we continue to study the shock formation for the 3-dimensional quasilinear wave equation
with \(G''(0)\) being a non-zero constant. Since (\(\star \)) admits global-in-time solution with small initial data, to present shock formation, we consider a class of large data. Moreover, no symmetric assumption is imposed on the data. Compared to our previous work (Miao and Yu in Invent Math 207(2):697–831, 2017), here we pose data on the hypersurface \(\{(t,x)|t=-\,r_{0}\}\) instead of \(\{(t,x)|t=-\,2\}\), with \(r_{0}\) being arbitrarily large. We prove an a priori energy estimate independent of \(r_{0}\). Therefore a complete description of the solution behavior as \(r_{0}\rightarrow \infty \) is obtained. This allows us to relax the restriction on the profile of initial data which still guarantees shock formation. Since (\(\star \)) can be viewed as a model equation for describing the propagation of electromagnetic waves in nonlinear dielectric, the result in this paper reveals the possibility to use wave pulse with weak intensity to form electromagnetic shocks in laboratory. A main new feature in the proof is that all estimates in the present paper do not depend on the parameter \(r_{0}\), which requires different methods to obtain energy estimates. As a byproduct, we prove the existence of semi-global-in-time solutions which lead to shock formation by showing that the limits of the initial energies exist as \(r_{0}\rightarrow \infty \). The proof combines the ideas in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2007) where the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established, and the short pulse method introduced in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012), where the formation of black holes in general relativity is proved.

Notes

Acknowledgements

The author would like to thank the anonymous referees, who carefully read a previous version of this paper and suggested many valuable improvements and corrections. This work was supported by NSF grant DMS-1253149 to The University of Michigan and in its initial phase by ERC Advanced Grant 246574 “Partial Differential Equations of Classical Physics”.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bâtiment des MathématiquesEPFLLausanneSwitzerland

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