On the formation of shocks for quasilinear wave equations
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- 作者：Shuang Miao ; Pin Yu
- 刊名：Inventiones mathematicae
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：207
- 期：2
- 页码：697-831
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics, general;
- 出版者：Springer Berlin Heidelberg
- ISSN：1432-1297
- 卷排序：207
文摘
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).References1.Alinhac, S.: Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux. Invent. Math. 111(3), 627–670 (1993)MathSciNetCrossRefGoogle Scholar2.Alinhac, S.: Blowup of small data solutions for a quasilinear wave equation in two space dimensions. Ann. Math. (2) 149(1), 97–127 (1999)3.Alinhac, S.: Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. II. Acta Math. 182(1), 1–23 (1999)MathSciNetCrossRefMATHGoogle Scholar4.Christodoulou, D.: The Action Principle and Partial Differential Equations. Annals of Mathematics Studies, vol. 146. Princeton University Press, Princeton, NJ (2000)5.Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. Monographs in Mathematics. European Mathematical Society, Zürich (2007)6.Christodoulou, D.: The Formation of Black Holes in General Relativity. Monographs in Mathematics. European Mathematical Society, Zürich (2009)7.Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of Minkowski Space. Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton, NJ (1993)8.Christodoulou, D., Miao, S.: Compressible Flow and Euler’s Equations. (Monograph, 602 pp.). Surveys in Modern Mathematics, vol. 9. International Press, Somerville, MA; Higher Education Press, Beijing (2014)9.Holzegel, G., Klainerman, S., Speck, J., Wong, W.: Shock Formation in Small-Data Solutions to \(3D\) Quasilinear Wave Equations: An Overview. J. Hyperbolic Differ. Equ. 13(1), 1–105 (2016)10.John, F.: Blow-up of radial solutions of \(u_{tt}=c^2(u_t)\Delta u\) in three space dimensions. Mat. Apl. Comput. 4(1), 3–18 (1985)MathSciNetGoogle Scholar11.John, F.: Nonlinear Wave Equations, Formation of Singularities. University Lecture Series, vol. 2. AMS, Providence (1990)12.Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33(1), 43–101 (1980)MathSciNetCrossRefMATHGoogle Scholar13.Klainerman, S., Rodnianski, I.: On the formation of trapped surfaces. Acta Math. 208(2), 211–333 (2012)MathSciNetCrossRefMATHGoogle Scholar14.Speck, J.: Shock formation in small-data solutions to \(3D\) quasilinear wave equations. arXiv:1407.6320v1 (preprint)Copyright information© Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsShuang Miao1Email authorPin Yu21.Department of MathematicsUniversity of MichiganAnn ArborUSA2.Department of Mathematics, Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina About this article CrossMark Publisher Name Springer Berlin Heidelberg Print ISSN 0020-9910 Online ISSN 1432-1297 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips