Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

Holzegel, Gustav; Klainerman, Sergiù; Speck, Jared; Wong, Willie Wai Yeung

doi:10.1142/s0219891616500016

Cited by 62 publications

(93 citation statements)

References 60 publications

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“…Christodoulou's results were extended [60] to a larger class of equations and data by Speck (see also the survey article [23], joint with Holzegel, Klainerman, and Wong). In particular, for data given on R 3 , he proved a sharp small-data shock formation result for equations (1.1a) or (1.3a) whenever the null condition fails.…”

Section: Detailed Blowup-results In More Than One Spatial Dimensionmentioning

confidence: 95%

“…1.3 was essentially a simple version of this method. Readers may consult [23,60] for detailed examples derived with the help of sharp techniques paralleling the ones employed in the present article.…”

Section: Blowup-results In One Spatial Dimensionmentioning

confidence: 99%

“…In particular, the solution's first derivatives relative to the geometric coordinates do not blow up! The blowup of the solution's first rectangular partial derivatives is a "low-level" effect that could be obtained 23 by transforming back to the rectangular coordinates and showing that μ = 0 causes a degeneracy in the change of variables (see Lemma 2.7).…”

Section: Overview Of the Analysismentioning

confidence: 99%

“…Previous work [1,2,15,60] in more than one spatial dimension, which is summarized in the survey article [23], has shown shock formation in solutions to various quasilinear wave equations in a different regime: that of solutions generated by small data supported in a compact subset of R 2 or R 3 . Recently, Miao and Yu proved a related Footnote 14 continued spatial dimensions, the wave equation corresponding to the Lagrangian L = 1 − 1 + (m −1 ) αβ ∂ α ∂ β admits global solutions whenever the data are small, smooth, and compactly supported.…”

Section: Remark 19 (Possibly Allowing Itself To Be Larger)mentioning

confidence: 99%

“…22 The possibility that the high derivatives might behave worse is a fundamental difficulty that permeates our analysis. 23 We use a slightly different, more direct argument to prove the blowup; see Subsubsect. 1.4.1 for an overview.…”

Section: Overview Of the Analysismentioning

confidence: 99%

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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

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Section: Detailed Blowup-results In More Than One Spatial Dimensionmentioning

confidence: 95%

Section: Blowup-results In One Spatial Dimensionmentioning

confidence: 99%

Section: Overview Of the Analysismentioning

confidence: 99%

Section: Remark 19 (Possibly Allowing Itself To Be Larger)mentioning

confidence: 99%

Section: Overview Of the Analysismentioning

confidence: 99%

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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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On the Global Behavior of Weak Null Quasilinear Wave Equations

Deng

Pusateri

2019

Comm Pure Appl Math

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We consider a subclass of those quasilinear wave equations in 3 + 1 space‐time dimensions that satisfy the “weak null condition” as defined by Lindblad and Rodnianski , and study the large‐time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is −∂t2u+()1+uitalicΔu=0. Global solutions for such equations have been constructed by Lindblad and Alinhac. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solutions close to the light cone, including the blowup at infinity. © 2019 Wiley Periodicals, Inc.

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Formation of Shocks for2DIsentropic Compressible Euler

Buckmaster

Shkoller

Vicol

2020

Comm Pure Appl Math

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We consider the 2D isentropic compressible Euler equations, with pressure law p./ h . 1 =/ , with > 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope 1 =", for " > 0 taken sufficiently small relative to the O.1/ amplitude, there exist smooth solutions to the Euler equations which form a shock in time O."/. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp-type, with Hölder C 1 =3 regularity.Our objective is the construction of solutions with inherent O.1/ vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann-type variables to obtain a system of forced transport equations. Using a transformation to modulated self-similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self-similar time, of a smooth blowup profile.

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Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

Cited by 62 publications

References 60 publications

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

On the Global Behavior of Weak Null Quasilinear Wave Equations

Formation of Shocks for2DIsentropic Compressible Euler

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