Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

Germain, Pierre; Harrop-Griffiths, Benjamin; Marzuola, Jeremy L.

doi:10.1002/cpa.21828

Cited by 12 publications

(14 citation statements)

References 61 publications

(86 reference statements)

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“…Note that in this definition, we completely sidestep the important issue for local/global wellposedness of the corresponding Cauchy problems. The local aspect of the theory is discussed in the recent paper [2], but the global well-posedness theory (which is more relevant as far as stability is concerned), seems lacking at the moment.…”

Section: Proposition 1 (Existence and Uniqueness Of Bell-shaped Compa...mentioning

confidence: 99%

“…Recall that f n ⊥ Φ ′ , whence their weak limit f n U also satisfies U ⊥ Φ ′ . According to (3.22) however, this implies that L + ( ΦU ) = 0, at least in a distributional sense 2 , against the compactly supported test functions. Standard elliptic theory, together with the decay properties of ΦU proves that ΦU is indeed an L 2 eigenfunction for L + .…”

Section: Remarkmentioning

confidence: 99%

“…2 ] = 0. Thus, the Schrödinger operator L − has an eigenvalue at zero, with corresponding positive eigenfunction Φ 3 2 . By Sturm-Liouville's theorem, this means that zero is at the bottom of the spectrum for L − .…”

Section: Remarkmentioning

confidence: 99%

“…The well-posedness of the Cauchy problem has been studied quite recently in [2]. In particular, these models conserve the mass M[u] = R |u| 2 and the Hamiltonian, which in this case takes the form…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the stability of the compacton waves for the degenerate KdV and NLS models

Hakkaev¹,

Ramadan²,

Stefanov³

2021

Preprint

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In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p > 8. This extends previous work of Germain, Harrop-Griffits and Marzuola, [3], who have previously established orbital stability for some specific waves, in the range p < 8.

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Section: Proposition 1 (Existence and Uniqueness Of Bell-shaped Compa...mentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the stability of the compacton waves for the degenerate KdV and NLS models

Hakkaev¹,

Ramadan²,

Stefanov³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Moreover, the study of Lie symmetries and conservation laws of equation 1is also motivated by the research in [18]. It is presented that equation 1admits a Hamiltonian structure, it has translation, reflection and scaling symmetries, and it conserves besides the Hamiltonian, the mass and the momentum.…”

mentioning

confidence: 99%

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Bruzón¹,

Recio²,

Garrido³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - S

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On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

Jeong

2022

Ann. PDE

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We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig-Ponce-Rolvung-Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola-Metcalfe-Tataru in the case of elliptic principal term.A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig-Ponce-Rolvung-Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola-Metcalfe-Tataru to the present case of nonelliptic principal term.4 Although, pedantically, Section 7.5 considers the paralinearized system, which is more difficult than the simplified problem discussed here.5 This was done without having uniqueness corresponding to the initial data, which is still an open problem.

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Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

Cited by 12 publications

References 61 publications

On the stability of the compacton waves for the degenerate KdV and NLS models

On the stability of the compacton waves for the degenerate KdV and NLS models

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

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