The null condition for quasilinear wave equations in two space dimensions I

Alinhac, Serge

doi:10.1007/s002220100165

Cited by 236 publications

(389 citation statements)

References 6 publications

Supporting

Mentioning

380

Contrasting

Unclassified

Order By: Relevance

“…Alinhac was the first [1,2] to give a sharp description of singularity formation in solutions to quasilinear wave equations in more than one spatial dimension without symmetry assumptions. He addressed a compactly supported small-data regime in which dispersive effects are eventually overcome by sufficiently strong quadratic nonlinearities.…”

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

confidence: 99%

“…Recently, Miao and Yu proved [52] a related shock formation result for the wave equation 2 ] φ = 0 in three spatial dimensions with data that are compactly supported in an annular region of radius ≈ 1 and thin width δ, where δ is a small positive parameter. The data's amplitude and their functional dependence on a radial coordinate are rescaled by powers of δ. Consequently, the data and their derivatives verify a hierarchy of estimates featuring various powers of δ.…”

Section: Blowup In a Large-data Regime Featuring A One-parameter Scalmentioning

confidence: 96%

“…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%

“…Another way to think about (1.10) is: by bringing μ under the outer L differentiation, we have generated a product term of the form −(Lμ) · · · . This leads to the cancellation of the worst term on the RHS, which was proportional to μ −1 (X ) 2 . Put differently, the term 1 2 G L LX from the RHS of equation (1.12) below generates complete, nonlinear cancellation of a term proportional to μ −1 (X ) 2 .…”

Section: Overview Of the Analysismentioning

confidence: 99%

“…On the other hand, we have the matrix identity M = N · g · N (where g is viewed as a 2 × 2 matrix expressed relative to the spatial rectangular coordinates), which implies that detM = detg(detN ) 2 . Combining these identities, we conclude (2.48).…”

Section: Lemma 27 (Basic Properties Of the Change Of Variables Map)mentioning

confidence: 99%

See 4 more Smart Citations

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 99%

Section: Blowup In a Large-data Regime Featuring A One-parameter Scalmentioning

confidence: 96%

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

confidence: 99%

Section: Overview Of the Analysismentioning

confidence: 99%

Section: Lemma 27 (Basic Properties Of the Change Of Variables Map)mentioning

confidence: 99%

See 3 more Smart Citations

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

View full text Add to dashboard Cite

show abstract

Global Well‐Posedness of Incompressible Elastodynamics in Two Dimensions

Lei

2016

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

Cai

Lei

Lin

et al. 2019

Comm Pure Appl Math

View full text Add to dashboard Cite

This paper studies the inviscid limit of the two‐dimensional incompressible viscoelasticity, which is a system coupling a Navier‐Stokes equation with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known since the work of [35]. The inviscid case was solved recently by the second author [28]. While the latter was solely based on the techniques from the studies of hyperbolic equations, and hence the two‐dimensional problem is in general more challenging than that in higher dimensions, the former was relied crucially upon a dissipative mechanism. Indeed, after a symmetrization and a linearization around the equilibrium, the system of the incompressible viscoelasticity reduces to an incompressible system of damped wave equations for both the fluid velocity and the deformation tensor. These two approaches are not compatible. In this paper, we prove global existence of solutions, uniformly in both time t ∈ [0, +∞) and viscosity μ ≥ 0. This allows us to justify in particular the vanishing viscosity limit for all time. In order to overcome difficulties coming from the incompatibility between the purely hyperbolic limiting system and the systems with additional parabolic viscous perturbations, we introduce in this paper a rather robust method that may apply to a wide class of physical systems of similar nature. Roughly speaking, the method works in the two‐dimensional case whenever the hyperbolic system satisfies intrinsically a “strong null condition.” For dimensions not less than three, the usual null condition is sufficient for this method to work. © 2019 Wiley Periodicals, Inc.

show abstract

The null condition for quasilinear wave equations in two space dimensions I

Cited by 236 publications

References 6 publications

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Global Well‐Posedness of Incompressible Elastodynamics in Two Dimensions

Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

Contact Info

Product

Resources

About