Effective dynamics for boson stars

Fröhlich, Jürg; Jonsson, B. L. G.; Lenzmann, Enno

doi:10.1088/0951-7715/20/5/001

Cited by 51 publications

(50 citation statements)

References 40 publications

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“…The author plans to pursue this question in future work. Our next result proves a so-called nondegeneracy condition, which was introduced in [Fröhlich et al 2007b] as a spectral assumption supported by numerical evidence. There, the effective motion of solitary waves for (1-7) with an slowly varying external potential was studied.…”

Section: Resultssupporting

confidence: 69%

“…There, the effective motion of solitary waves for (1-7) with an slowly varying external potential was studied. Furthermore, the following nondegeneracy result allows us to give an unconditional proof for the cylindrical symmetry of traveling solitary waves for the time-dependent pseudorelativistic Hartree equation (1-7); see [Fröhlich et al 2007b] for more details. The precise nondegeneracy statement reads as follows.…”

Section: Resultsmentioning

confidence: 99%

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Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann

2009

APDE

195

235

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We prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations. IntroductionThe pseudorelativistic Hartree energy functional, given (in appropriate units) byarises in the mean-field limit of a quantum system describing many self-gravitating, relativistic bosons with rest mass m > 0. Such a physical system is often referred to as a boson star, and various models for these -at least theoretical -objects have attracted a great deal of attention in theoretical and numerical astrophysics over the past years. In order to gain some rigorous insight into the theory of boson stars, it is of particular interest to study ground states (that is, minimizers) for the variational problemwhere the parameter N > 0 plays the role of the stellar mass. Provided that problem (1-2) has indeed a ground state Q ∈ H 1/2 ‫ޒ(‬ 3 ), one readily finds that it satisfies the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, (1-3) with µ = µ(Q) ∈ ‫ޒ‬ being some Lagrange multiplier.MSC2000: 35Q55.

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Section: Resultssupporting

confidence: 69%

Section: Resultsmentioning

confidence: 99%

Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann

2009

APDE

195

235

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“…Let us also mention that the dispersive nonlinear PDE (1-7) exhibits a rich variety of phenomena, such as stable and unstable traveling solitary waves, as well as finite-time blowup solutions indicating the "gravitational collapse" of a boson star; see [Fröhlich et al 2007a;2007b;. For well-posedness results concerning (1-7) and its rigorous derivation from many-body quantum mechanics, we refer to [Cho and Ozawa 2006; and [Elgart and Schlein 2007], respectively.…”

Section: Uniqueness Of Ground States For Pseudorelativistic Hartree Ementioning

confidence: 99%

“…This coercivity estimate plays a central role in the stability analysis of solitary waves for NLStype equations and their effective motion in an external potential; see, for example, [Weinstein 1985;Bronski and Jerrard 2000;Fröhlich et al 2004;2007b;Jonsson et al 2006;Abou Salem 2007;Holmer and Zworski 2008].…”

Section: It Is Convenient To View the Operator L (Which Is Not ‫-ރ‬Limentioning

confidence: 99%

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Untitled

2009

APDE

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UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS ENNO LENZMANNWe prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation,in the regime of Q with sufficiently small L 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = |Q| 2 1 except for at most countably many N . Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrödinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the socalled nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

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A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

Wang

2021

Math Methods in App Sciences

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In this paper, we consider the pseudo‐relativistic Hartree equation i∂tψ=−△+m2ψ−1|x|∗|ψ|2ψonℝ3 and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where v∈ℝ3 denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider v=false(β,0,0false) with 0 < β < 1, and let Ncfalse(βfalse)=Ncfalse(vfalse)false|v=false(β,0,0false). We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with false‖φβfalse‖L22=false(1−βfalse)Ncfalse(βfalse) satisfy limβ→0+false‖φβfalse‖H1false/2→+∞. The explicit blow‐up profile and rate will be computed.

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Effective dynamics for boson stars

Cited by 51 publications

References 40 publications

Uniqueness of ground states for pseudorelativistic Hartree equations

Uniqueness of ground states for pseudorelativistic Hartree equations

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A blow‐up result for the travelling waves of the pseudo‐relativistic Hartree equation with small velocity

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