On the Global Behavior of Weak Null Quasilinear Wave Equations

Deng, Yu; Pusateri, Fabio

doi:10.1002/cpa.21881

Cited by 21 publications

(20 citation statements)

References 47 publications

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“…Remark 1.3. We compare the results in this paper with those in Deng-Pusateri [5]. First, the approximation result (i.e.…”

Section: B)mentioning

confidence: 93%

“…In [25], Lindblad and Schlue proved the existence of the wave operators for the semilinear models of Einstein's equations. In [5], Deng and Pusateri used the original Hörmander's asymptotic system (1.9) to prove a partial scattering result for (1.1). In their proof, they applied the spacetime resonance method; we refer to [26,27] for some earlier applications of this method to the first order systems of wave equation.…”

Section: Introductionmentioning

confidence: 99%

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Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

2021

Commun. Math. Phys.

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In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in [33]. Starting from a global solution u to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system. 2020 Mathematics Subject Classification. 35L70.

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“…Remark 1.3. We compare the results in this paper with those in Deng-Pusateri [5]. First, the approximation result (i.e.…”

Section: B)mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

2021

Commun. Math. Phys.

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“…The main difficulty is then in understanding what happens in regions where the oscillations match with the oscillations of the linear flow (time resonances) and in regions where wave packets interact for a long time (space resonances). Among the vast literature concerning the spacetime resonances method, we mention in particular the hyperbolic applications found in the works [35] and [13]. The first concerns the long time dynamics of solutions to equations satisfying the null condition, while the second concerns the long time dynamics of solutions to equations satisfying the weak null condition.…”

Section: Nonlinear Wave Equations and Related Resultsmentioning

confidence: 99%

Global stability for a nonlinear system of anisotropic wave equations

Anderson¹

2021

Preprint

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In this paper, we initiate the study of global stability for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of crystal optics, and they exhibit birefringence. We introduce a physical space strategy based on bilinear energy estimates that allows us to prove decay for the nonlinear problem. This uses decay for the homogeneous wave equation as a black box. The proof also requires us to interface this strategy with the vector field method and take advantage of the scaling vector field. A careful analysis of the spacetime geometry of the interaction between waves is necessary in the proof.

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“…REMARK 31. One may ask whether the higher energy growth is associated to the blowup at infinity described by Alinhac [Ali03], and which seems generic for wave equations with weak null quasilinearities [Lin08,LR05,DP18].…”

Section: Closing the Bootstrapmentioning

confidence: 99%

Global Nearly-Plane-Symmetric Solutions to the Membrane Equation

Abbrescia

Wong

2020

Forum of Mathematics, Pi

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We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.

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On the Global Behavior of Weak Null Quasilinear Wave Equations

Cited by 21 publications

References 47 publications

Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

Modified Wave Operators for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

Global stability for a nonlinear system of anisotropic wave equations

Global Nearly-Plane-Symmetric Solutions to the Membrane Equation

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