Degeneracy in Finite Time of 1D Quasilinear Wave Equations

Sugiyama, Yuusuke

doi:10.1137/15m1016369

Cited by 10 publications

(18 citation statements)

References 24 publications

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“…Specifically, he showed that all axially symmetric solutions (without any smallness assumption) lead to a finite-time degeneracy caused by the vanishing of a principal coefficient in the evolution equations. We also note that in the case of one spatial dimension, results similar to ours are obtained in [53,[95][96][97] using proofs by contradiction that rely on the method of Riemann invariants. However, since the method of Riemann invariants is not applicable in more than one spatial dimension and since we are interested in direct proofs, our approach here is quite different.…”

Section: Introductionsupporting

confidence: 74%

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Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Speck

2017

Analysis & PDE

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Abstract. In three spatial dimension, we study the Cauchy problem for the wave equation −∂ 2 t Ψ + (1 + Ψ) P ∆Ψ = 0 for P ∈ {1, 2}. We exhibit a form of stable Tricomi-type degeneracy formation that has not previously been studied in more than one spatial dimension. Specifically, using only energy methods and ODE techniques, we exhibit an open set of data such that Ψ is initially near 0 while 1 + Ψ vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this phenomenon. The solution remains regular in the following sense: there is a high-order L 2 -type energy, featuring degenerate weights only at the top-order, that remains bounded. When P = 1, we show that any C 1 extension of Ψ to the future of a point where 1 + Ψ = 0 must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the Lorentzian metric corresponding to the wave equation blows up at those points. Thus, our results show that curvature blowup does not always coincide with singularity formation in the solution variable. Similar phenomena occur when P = 2, where the vanishing of 1 + Ψ corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all values of Ψ.The data are compactly supported and are allowed to be large or small as measured by an unweighted Sobolev norm. However, we assume that initially, the spatial derivatives of Ψ are nonlinearly small relative to |∂ t Ψ|, which allows us to treat the equation as a perturbation of the ODE d 2 dt 2 Ψ = 0. We show that for appropriate data, ∂ t Ψ remains quantitatively negative, which simultaneously drives the degeneracy formation and yields a favorable spacetime integral in the energy estimates that is crucial for controlling some top-order error terms. Our result complements those of Alinhac and Lindblad, who showed that if the data are small as measured by a Sobolev norm with radial weights, then the solution is global.

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Section: Introductionsupporting

confidence: 74%

“…Speck would like to thank Yuusuke Sugiyama for bringing the references [53,[95][96][97] to his attention, to Michael Dreher for pointing out the references [70,71], to Willie Wong for pointing out the relevance of his work [99], and to an anonymous referee for pointing out the work [65].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Speck

2017

Analysis & PDE

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“…However, when p ′ goes to zero, the equations loss the strictly hyperbolicity. The author's papers [25,26] give a sufficient condition that the lack of the strictly hyperbolicity occurs in finite time. The p-system with γ < 1 would be meaningful in the study of elastic-plastic materials (e.g.…”

Section: Plan Of This Paper and Notationsmentioning

confidence: 99%

Singularity formation for the 1D compressible Euler equations with variable damping coefficient

Sugiyama

2018

Nonlinear Analysis

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“…Johnson [7] and Yamaguchi and Nishida [23]). On the other hand, in [18,19] (see Remark 5 in [18] and Theorem 4.1 in [19]), the author has shown that the degeneracy (8) occurs in finite time, if…”

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confidence: 99%

“…The main theorem of this paper removes the compactness condition on initial data and extends the result in [8,17] to (1) with more general c(θ) and 0 ≤ λ < 2. In [17,18], the generalization on λ has already been pointed out without a proof. In fact, applying the method in [8,17] to the equation in (1), we can generalize the result in [17,18] to (1) with 0 ≤ λ < 2 and c(θ) = 1 + θ.…”

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confidence: 99%