Shock Formation in Small-Data Solutions to 3D
                    Quasilinear Wave Equations

Speck, Jared

doi:10.1090/surv/214

Cited by 75 publications

(173 citation statements)

References 14 publications

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“…This remarkable structure was first 35 observed 36 by Klainerman and Rodnianski in their proof of low regularity well-posedness for quasilinear wave equations [36] and was also used in [15,52,60]. Combining, we find that L μtr g / χ − G L LX + μP = · · · .…”

Section: Energy Estimates At the Highest Ordersupporting

confidence: 66%

“…1.4.2, the energies (1.16a) and null fluxes (1.16b) control geometric torus derivatives with μ weights, which makes them too weak to control certain error integrals involving torus derivatives that lack μ weights, at least in regions where μ is small. The saving grace is that as in [15,52,60], our energy estimates generate a spacetime integral with a good sign. Under appropriate bootstrap assumptions, the integral is strong in regions where μ is small and controls geometric torus derivatives without μ weights.…”

Section: The Coercive Spacetime Integralmentioning

confidence: 89%

“…That is, we could allow for quasilinear terms such as · ∂ 2 . Moreover, we could also allow for the presence of semilinear terms verifying the strong null condition (see [60] for the definition) on RHS (1.1a) or (1.3a). In the regime close to a plane symmetric simple wave, these terms would make only a negligible contribution to the dynamics and in particular, they would not interfere with the shock formation processes.…”

Section: Remark 18 (Additional Nonlinearities That We Could Allow)mentioning

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%

“…To bound the top-order factor L Pχ in L 2 , one can derive elliptic estimates on the n − 1 dimensional surfaces analogous to the t,u in the present article; see, for example, [11,15,36,52,60] for more details.…”

Section: Energy Estimates At the Highest Ordermentioning

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: Energy Estimates At the Highest Ordersupporting

confidence: 66%

Section: The Coercive Spacetime Integralmentioning

confidence: 89%

Section: Remark 18 (Additional Nonlinearities That We Could Allow)mentioning

confidence: 99%

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

confidence: 99%

Section: Energy Estimates At the Highest Ordermentioning

confidence: 99%

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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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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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Shock Formation and Vorticity Creation for 3d Euler

Buckmaster

Shkoller

Vicol

2022

Comm Pure Appl Math

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We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3,4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces.

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Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

Cited by 75 publications

References 14 publications

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Formation of Shocks for2DIsentropic Compressible Euler

Shock Formation and Vorticity Creation for 3d Euler

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