Abstract
The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation
where \(G^{\prime \prime }(0)\) is a non-zero constant. We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation. Since \((\star )\) satisfies the classical null condition, it admits global smooth solutions for small data. Therefore, we will work with large data (in energy norm). Moreover, no symmetry condition is imposed on the initial datum. We emphasize the geometric perspectives of shock formation in the proof. More specifically, the key idea is to study the interplay between the following two objects: (1) the energy estimates of the linearized equations of \((\star )\); (2) the differential geometry of the Lorentzian metric \(g=-\frac{1}{(1+3G^{\prime \prime }(0) (\partial _t\phi )^2)} d t^2+dx_1^2+dx_2^2+dx_3^2\). Indeed, the study of the characteristic hypersurfaces (implies shock formation) is the study of the null hypersurfaces of g. The techniques in the proof are inspired by the work (Christodoulou in The Formation of Shocks in 3-Dimensional Fluids. Monographs in Mathematics, European Mathematical Society, 2007) in which the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established. We also use the short pulse method which is introduced in the study of formation of black holes in general relativity in Christodoulou (The Formation of Black Holes in General Relativity. Monographs in Mathematics, European Mathematical Society, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012).
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Notes
In a future work, we will consider a more general case for which the data is prescribed at past infinity. Therefore, we have to let \(r_0\) go to \(\infty \) and the dependence of the estimates (of the current work) on \(r_0\) will be crucial.
For a multi-index \(\alpha \), the symbol \(\alpha -1\) means another multi-index \(\beta \) with degree \(|\beta | = |\alpha |-1\).
This sentence always means that, there exists \(\varepsilon = \varepsilon (M)\) so that for all \(\delta \le \varepsilon \), we have ...
We emphasize that the trace \({\text {tr}}\) is defined with respect to g.
We emphasize that the traceless part of is defined with respect to .
The inequality is up to a constant depending only on the bootstrap constant M.
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Acknowledgments
The authors are grateful to three anonymous referees, who carefully read a previous version of this paper and suggested many valuable improvements and corrections. S. Miao is supported by NSF Grant DMS-1253149 to The University of Michigan. P. Yu is supported by NSFC 11522111 and NSFC 11271219. The research of S. Miao was in its initial phase supported by ERC Advanced Grant 246574 “Partial Differential Equations of Classical Physics”.
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Miao, S., Yu, P. On the formation of shocks for quasilinear wave equations. Invent. math. 207, 697–831 (2017). https://doi.org/10.1007/s00222-016-0676-2
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DOI: https://doi.org/10.1007/s00222-016-0676-2