Pure and Applied Analysis

Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

Jared Speck

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/paa.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In n1 spatial dimensions, we study the Cauchy problem for a genuinely nonlinear quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology) of regular initial data that are close to the data of a simple plane wave, we give a sharp, constructive proof of shock formation in which the transport variable remains bounded but its first-order Cartesian coordinate partial derivatives blow up in finite time. Moreover, we prove that, at least at the low derivative levels, the singularity does not propagate into the symmetric hyperbolic variables: they and their first-order Cartesian coordinate partial derivatives remain bounded, even though they interact with the transport variable all the way up to its singularity. The formation of the singularity is tied to the finite-time degeneration, relative to the Cartesian coordinates, of a system of geometric coordinates adapted to the characteristics of the transport operator. Two crucial features of the proof are that relative to the geometric coordinates, all solution variables remain smooth, and that the finite-time degeneration coincides with the intersection of the transport characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to quasilinear wave equations, the main new features of the present work are: (i) we develop a theory of nonlinear geometric optics for transport operators, which is compatible with the coupling and which allows us to implement a quasilinear geometric vector field method, even though the regularity properties of the corresponding eikonal function are less favorable compared to the wave equation case and (ii) we allow for a full quasilinear coupling; i.e., the principal coefficients in all equations are allowed to depend on all solution variables.

Article information

Source
Pure Appl. Anal., Volume 1, Number 3 (2019), 447-514.

Dates
Received: 17 February 2019
Revised: 14 April 2019
Accepted: 22 April 2019
First available in Project Euclid: 31 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.paa/1564538459

Digital Object Identifier
doi:10.2140/paa.2019.1.447

Mathematical Reviews number (MathSciNet)
MR3985091

Zentralblatt MATH identifier
07114664

Subjects
Primary: 35L67: Shocks and singularities [See also 58Kxx, 76L05]
Secondary: 35L45: Initial value problems for first-order hyperbolic systems

Keywords
blowup characteristics eikonal equation eikonal function simple wave singularity stable blowup vector field method wave breaking

Citation

Speck, Jared. Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation. Pure Appl. Anal. 1 (2019), no. 3, 447--514. doi:10.2140/paa.2019.1.447. https://projecteuclid.org/euclid.paa/1564538459


Export citation

References

  • S. Alinhac, “Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, II”, Acta Math. 182:1 (1999), 1–23.
  • S. Alinhac, “Blowup of small data solutions for a quasilinear wave equation in two space dimensions”, Ann. of Math. $(2)$ 149:1 (1999), 97–127.
  • S. Alinhac, “The null condition for quasilinear wave equations in two space dimensions, II”, Amer. J. Math. 123:6 (2001), 1071–1101.
  • D. Christodoulou, The action principle and partial differential equations, Annals of Math. Studies 146, Princeton Univ. Press, 2000.
  • D. Christodoulou, The formation of shocks in 3-dimensional fluids, Eur. Math. Soc., Zürich, 2007.
  • D. Christodoulou, The formation of black holes in general relativity, Eur. Math. Soc., Zürich, 2009.
  • D. Christodoulou, The shock development problem, Eur. Math. Soc., Zürich, 2019.
  • D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton Math. Series 41, Princeton Univ. Press, 1993.
  • D. Christodoulou and A. Lisibach, “Shock development in spherical symmetry”, Ann. PDE 2:1 (2016), art. id. 3.
  • D. Christodoulou and S. Miao, Compressible flow and Euler's equations, Surveys of Modern Math. 9, Int. Press, Somerville, MA, 2014.
  • D. Christodoulou and D. R. Perez, “On the formation of shocks of electromagnetic plane waves in non-linear crystals”, J. Math. Phys. 57:8 (2016), art. id. 081506.
  • C. M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Math. Wissenschaften 325, Springer, 2010.
  • G. Holzegel, S. Klainerman, J. Speck, and W. W.-Y. Wong, “Small-data shock formation in solutions to 3D quasilinear wave equations: an overview”, J. Hyperbolic Differ. Equ. 13:1 (2016), 1–105.
  • F. John, “Formation of singularities in one-dimensional nonlinear wave propagation”, Comm. Pure Appl. Math. 27 (1974), 377–405.
  • F. John, “Blow-up for quasilinear wave equations in three space dimensions”, Comm. Pure Appl. Math. 34:1 (1981), 29–51.
  • S. Klainerman, “Uniform decay estimates and the Lorentz invariance of the classical wave equation”, Comm. Pure Appl. Math. 38:3 (1985), 321–332.
  • S. Klainerman, “The null condition and global existence to nonlinear wave equations”, pp. 293–326 in Nonlinear systems of partial differential equations in applied mathematics, I (Santa Fe, NM, 1984), edited by B. Nicolaenko et al., Lectures in Appl. Math. 23, Amer. Math. Soc., Providence, RI, 1986.
  • S. Klainerman and I. Rodnianski, “Improved local well-posedness for quasilinear wave equations in dimension three”, Duke Math. J. 117:1 (2003), 1–124.
  • S. Klainerman and I. Rodnianski, “Rough solutions of the Einstein-vacuum equations”, Ann. of Math. $(2)$ 161:3 (2005), 1143–1193.
  • S. Klainerman, I. Rodnianski, and J. Szeftel, “The bounded $L^2$ curvature conjecture”, Invent. Math. 202:1 (2015), 91–216.
  • P. D. Lax, “Development of singularities of solutions of nonlinear hyperbolic partial differential equations”, J. Math. Phys. 5 (1964), 611–613.
  • P. D. Lax, “The formation and decay of shock waves”, Amer. Math. Monthly 79:3 (1972), 227–241.
  • P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS Regional Conf. Series Appl. Math. 11, Soc. Indust. Appl. Math., Philadelphia, 1973.
  • J. M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Math. 218, Springer, 2013.
  • H. Lindblad and I. Rodnianski, “The global stability of Minkowski space-time in harmonic gauge”, Ann. of Math. $(2)$ 171:3 (2010), 1401–1477.
  • J. Luk and J. Speck, “The hidden null structure of the compressible Euler equations and a prelude to applications”, 2016. To appear in J. Hyperbolic Differ. Equ.
  • J. Luk and J. Speck, “Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity”, Invent. Math. 214:1 (2018), 1–169.
  • S. Miao, “On the formation of shock for quasilinear wave equations with weak intensity pulse”, Ann. PDE 4:1 (2018), art. id. 10.
  • S. Miao and P. Yu, “On the formation of shocks for quasilinear wave equations”, Invent. Math. 207:2 (2017), 697–831.
  • J. Rauch, “BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one”, Comm. Math. Phys. 106:3 (1986), 481–484.
  • B. Riemann, “Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”, Abh. Königlichen Ges. Wiss. Göttingen 8 (1860), 43–66.
  • H. Ringström, The Cauchy problem in general relativity, Eur. Math. Soc., Zürich, 2009.
  • J. Sbierski, “On the existence of a maximal Cauchy development for the Einstein equations: a dezornification”, Ann. Henri Poincaré 17:2 (2016), 301–329.
  • T. C. Sideris, “Formation of singularities in solutions to nonlinear hyperbolic equations”, Arch. Ration. Mech. Anal. 86:4 (1984), 369–381.
  • T. C. Sideris, “Formation of singularities in three-dimensional compressible fluids”, Comm. Math. Phys. 101:4 (1985), 475–485.
  • H. F. Smith and D. Tataru, “Sharp local well-posedness results for the nonlinear wave equation”, Ann. of Math. $(2)$ 162:1 (2005), 291–366.
  • J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, Math. Surveys and Monographs 214, Amer. Math. Soc., Providence, RI, 2016.
  • J. Speck, “A new formulation of the $3D$ compressible Euler equations with dynamic entropy: remarkable null structures and regularity properties”, preprint, 2017. To appear in Arch. Ration. Mech. Anal.
  • J. Speck, “Shock formation for $2D$ quasilinear wave systems featuring multiple speeds: blowup for the fastest wave, with non-trivial interactions up to the singularity”, Ann. PDE 4:1 (2018), art. id. 6.
  • J. Speck, G. Holzegel, J. Luk, and W. Wong, “Stable shock formation for nearly simple outgoing plane symmetric waves”, Ann. PDE 2:2 (2016), art. id. 10.
  • W. W.-Y. Wong, “A comment on the construction of the maximal globally hyperbolic Cauchy development”, J. Math. Phys. 54:11 (2013), art. id. 113511.