Scattering Theory Below Energy for a Class of Hartree Type Equations

Chae, Myeongju; Hong, Sungeon; Kim, Joonil; Yang, Chan Woo

doi:10.1080/03605300701629427

Cited by 9 publications

(14 citation statements)

References 24 publications

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“…Proof. This lemma is proved in the case n = 3 and V ∈ L q 1 in [9].It is not hard to see that their argument is dimensional independent and can be extended to higher dimensions directly. Moreover, in that proof, Young's inequality was used to obtain the following estimate ∞ j=1 |V * (g j h j )| 2…”

Section: From This Corollary and The Embedding Relationmentioning

confidence: 85%

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Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

Miao

Yang

2019

Commun. Contemp. Math.

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By I-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equationwith radial data in H s R 4 for s > sc := γ/2 − 1. It is a sharp global result except of the critical case s = sc, which is a very difficult open problem. 2000 Mathematics Subject Classification. Primary: 35Q40; Secondary: 35Q55. 1 is corresponding to the H 1 2 -regularity of the solution in the interaction Morawetz estimate (see Proposition 2.12).Remark 1.3. Analogous unconditional global existence and scattering for the critical case s = s c for 2 < γ < 4 is much more difficult. On one hand, there is no conserved quantity to be used. On the other hand, the I-method would also break down as can be seen in our proof. In this case, it is well-known in the litterature that for the semilinear Schrödinger equations with radial data, the uniform boundedness of the critical normḢ sc implies scattering [29]. An interesting problem is to relax this assumption by considering a discrete time sequence tending to the maximal time of existence, along which the solution is bounded in certain critical Sobolev norms, and showing that this weaker assumption also implies scattering, as investigated by Duyckaerts and the third author in [15] for wave equations.Remark 1.4. The result is restricted to the radial setting because we need the radial Sobolev inequality in the frequency localized version, see Proposition 2.7. The argument here can also be extended to all higher dimensions n ≥ 5, where by using the double Duhamel formula, Miao Xu and Zhao [44] established the scattering theory for energy critical case γ = 4. Thus, it is interesting to remove the radial assumption in this low regularity problems for γ smaller than but close to 4.

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Section: From This Corollary and The Embedding Relationmentioning

confidence: 85%

“…For other works on the global well-posedness and scattering for the Hartree equation, see [9,16,20,21,22,24,43,53].…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

Miao

Yang

2019

Commun. Contemp. Math.

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“…Existence of global solutions in R 3 to (1.1) corresponding to initial data below the energy threshold was recently obtained in [5] by using the method of "almost conservation laws" or "I-method" (for a detailed description of this method, see [25] or Section 3 below) and the interaction Morawetz estimate for the solution u, where global wellposedness was obtained in H s (R 3 ) with s > max(1/2, 4(γ − 2)/(3γ − 4)) (see Fig. 1).…”

Section: Introductionmentioning

confidence: 98%

“…In order to obtain the low regularity global solution of the IVP (1.1), we combine I-method with an interaction Morawetz estimate for the smoothed out version I u of the solution. By comparison with the interaction Morawetz estimate for the solution in [5], such a Morawetz estimate for an almost solution is the main novelty of this paper, which helps us to lower the need on the regularity of the initial data. In addition, we do not use the monotonicity property of the multiplier m(ξ ) · ξ p in the proof of the almost conservation law.…”

Section: Introductionmentioning

confidence: 98%

“…1). Since authors in [5] used the interaction Morawetz estimate, which involvesḢ 1/2 norm of the solution, the restriction condition s 1 2 is prerequisite. In order to resolve IVP (1.1) in H s , s < 1 2 by still using the interaction Morawetz estimate, we need return to the interaction Morawetz estimate for the smoothed out version I u of the solution, which is used simultaneously in [2] and [6].…”

Section: Introductionmentioning

confidence: 99%

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Global well-posedness and scattering for the defocusing \( H^{\frac{1}{2}} \)-subcritical Hartree equation in \( R^{d} \)

Miao

Zhao

2009

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We prove the global well-posedness and scattering for the defocusing H 1 2 -subcritical (that is, 2 < γ < 3) Hartree equation with low regularity data in R d , d 3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev spacewhich also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321-348], where the global well-posedness was established for any s > max(1/2, 4(γ − 2)/(3γ − 4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution I u, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ ) · ξ p . As a byproduct of our proof, we obtain that the H s norm of the solution obeys the uniform-in-time bounds.

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Quantified hydrodynamic limits for Schrödinger-type equations without the nonlinear potential

Kim

Moon

2023

J. Evol. Equ.

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Scattering Theory Below Energy for a Class of Hartree Type Equations

Cited by 9 publications

References 24 publications

Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

Global well-posedness for the defocusing Hartree equation with radial data in ℝ4

Global well-posedness and scattering for the defocusing \( H^{\frac{1}{2}} \)-subcritical Hartree equation in \( R^{d} \)

Quantified hydrodynamic limits for Schrödinger-type equations without the nonlinear potential

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