Dispersive Limit of the Euler--Poisson System in Higher                     Dimensions

Pu, Xueke

doi:10.1137/120875648

Cited by 31 publications

(31 citation statements)

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“…The long wave scaling. In some recent independent works, Lannes, Linares and Saut [29] in the first hand, and Guo and Pu [22,38] in the other hand, have rigorously studied the long wave limit of the pressureless Euler-Poisson system. Precisely, one looks for solutions to (1.4) under the form:…”

Section: The Long Wave Scaling Of the Vlasov-poisson Equationmentioning

confidence: 99%

“…In 2D and 3D, which corresponds to the setting of (1.8), [29] and [38] give the derivation of a higher dimensional generalization of the KdV equation, which is referred to as the Zakharov-Kuznetsov (in short ZK) equation:…”

Section: The Long Wave Scaling Of the Vlasov-poisson Equationmentioning

confidence: 99%

“…With another kind of anisotropic (in space) scaling, Pu has also derived in 2D the Kadomstev-Petviashvili II (in short KP-II) equation [38]:…”

Section: The Long Wave Scaling Of the Vlasov-poisson Equationmentioning

confidence: 99%

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From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

Han-Kwan

2013

Commun. Math. Phys.

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Section: The Long Wave Scaling Of the Vlasov-poisson Equationmentioning

confidence: 99%

Section: The Long Wave Scaling Of the Vlasov-poisson Equationmentioning

confidence: 99%

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From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

Han-Kwan

2013

Commun. Math. Phys.

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“…Even at the linear level, there are new ion-acoustic waves, Langmuir waves, as well as light waves etc. At the nonlinear level, the Euler-Maxwell system is the origin of many well-known dispersive PDE, such as KdV [24], KP [36,39], Zakharov [45], Zakharov-Kuznetsov [36,39] and NLS, which can be derived from (1.1) and (1.2) via different scaling and asymptotic expansions. We also refer to [3,11,12] for derivation of the cold-ion and quasi-neutral equations.…”

Section: Introductionmentioning

confidence: 99%

Global solutions of the Euler--Maxwell two-fluid system in 3D

2016

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The fundamental "two-fluid" model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler-Maxwell system. Our construction applies equally well to other plasma models such as the Euler-Poisson system for two-fluids and a relativistic Euler-Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.Contents 1 4 Indeed, the ratio me/M i is no bigger than the ratio of the electron mass to the proton mass which equals 1/1836. 5 This is called the electrostatic approximation. 6 YAN GUO, ALEXANDRU D. IONESCU, AND BENOIT PAUSADER Klein-Gordon equations (faster time decay of linear waves like t −3/2 , absence of quadratic resonances), global smooth irrotational flows were constructed in [22] via the normal form method of Shatah [41]: Theorem 1.3 (Stability of a neutral equilibrium [22]). Solutions of (1.15) with initial data (n 0 , v 0 ) small, smooth, neutral and irrotational in the sense thatremain globally smooth and decay to 0 in L ∞ as t → +∞.The neutral assumption was later removed in [15] and this result was extended to two spatial dimensions independently in [28,37] (see also [30,31]). Theorem 1.3 was the first positive result indicating that the dispersive effect alone in the two-fluid theory may prevent shock formation 6 and it started an investigation to understand to which extent the introduction of electromagnetic forces could stabilize the full Euler-Maxwell system.Recently, further progress was made in this direction in the study of another simplified model: the Euler-Poisson equation for the ions 7 :(1.17)Here the electron dynamics with constant temperature is decoupled from the ion dynamics via the Boltzmann relation. The model equation then becomesThis system has intermediate behavior between (1.14) and (1.16). The linearized solutions decay slowly (like t −4/3 ) and create many strong degeneracies near the zero frequency, where the dispersion relation is similar to the wave dispersion up to third order (see λ i in Lemma A.4). Nevertheless, the first and third authors were able to obtain an analogue of Theorem 1.3 for perturbations of a neutral equilibrium by using a variation on the normal form method, controlling bilinear multipliers with rough coefficients using arguments inspired by [26]. Here, a crucial property is the fact that the nonlinearity is an exact derivative, which helps compensate for the degeneracy at the 0 frequency.1.2.2. Two-fluid models with different speeds. Both systems (1.15) and (1.17) can be reduced (under the irrotational assumption) to a (complex) scalar quasilinear equation with one speed. This is no longer the case for more complicated two-fluid mo...

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“…For the Euler-Poisson equation (1.5), Guo and Pausader [8] constructed global smooth irrotational solutions with small amplitude for this equation with fixed ε > 0 and T i > 0. Very recently, Guo and Pu [9] derived the KdV equation from (1.5) for the full range of T i ≥ 0, and Pu [16] derived the Kadomtsev-Petviashvili II equation and the Zakharov-Kuznetsov equation via the Gardner-Morikawa type transformations. Guo et al [7] made a breakthrough for the Euler-maxwell two-fluid system in 3D and proved that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions.…”

Section: Introductionmentioning

confidence: 99%

Quasineutral limit of the pressureless Euler-Poisson equation for ions

Guo

2016

Quart. Appl. Math.

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Abstract. In this paper, we consider the quasineutral limit of the Euler-Poisson equation for a clod, ion-acoustic plasma when the Debye length tends to zero. When the ionacoustic plasma is cold, the Euler-Poisson equation is pressureless and hence fails to be Friedrich symmetrisable, which excludes the application of the classical energy estimates method. This brings new difficulties in proving uniform estimates independent of ε. The main novelty in this article is to introduce new ε-weighted norms of the unknowns and to combine energy estimates in different levels with weights depending on ε. Finally, that the quasineutral regimes are the incompressible Euler equations is proven for well prepared initial data.

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Dispersive Limit of the Euler--Poisson System in Higher Dimensions

Cited by 31 publications

References 23 publications

From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

Global solutions of the Euler--Maxwell two-fluid system in 3D

Quasineutral limit of the pressureless Euler-Poisson equation for ions

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