The Hartree–Fock equations in modulation spaces

Bhimani, Divyang G.; Grillakis, Manoussos G.; Okoudjou, Kasso A.

doi:10.1080/03605302.2020.1758721

Cited by 10 publications

(8 citation statements)

References 41 publications

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“…Let us prove (11). As a consequence of the characterization (7) and the embeddings in (8), for s large enough we have…”

Section: Proof Of the Main Resultsmentioning

confidence: 93%

“…1.1] finite time blow-up has been established in some modulation spaces. On the other hand, Bhimani et al proved in [7] global well-posedness for the Hartree-Fock equations associated to harmonic oscillator in some modulation spaces; see also [25]. There is a large literature dealing with the analysis of PDEs on modulation spaces; we refer to the surveys [2,30] and the monograph [41], and the references therein (see also [3,10]); we also mention the article [23] for results on the Hermite operator obtained using phase-space methods.…”

Section: Introduction and Discussion Of The Resultsmentioning

confidence: 99%

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Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Bhimani¹,

Manna²,

Nicola³

et al. 2020

Preprint

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We study the Hermite operator H = −∆ + |x| 2 in R d and its fractional powers H β , β > 0 in phase space. Namely, we represent functions f via the socalled short-time Fourier, alias Fourier-Wigner or Bargmann transform V g f (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of V g f , that is in terms of membership to modulation spaces M p,q , 0 < p, q ≤ ∞. We prove the complete range of fixed-time estimates for the semigroup e −tH β when acting on M p,q , for every 0 < p, q ≤ ∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing.As an application, we establish global well-posedness for the nonlinear heat equation for H β with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e −ct as the solution of the corresponding linear equation, where c = d β is the bottom of the spectrum of H β . This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data -hence in M ∞,1 .

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“…Let us prove (11). As a consequence of the characterization (7) and the embeddings in (8), for s large enough we have…”

Section: Proof Of the Main Resultsmentioning

confidence: 93%

Section: Introduction and Discussion Of The Resultsmentioning

confidence: 99%

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Bhimani¹,

Manna²,

Nicola³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We note that Bhimani et al in [7,Theorem 1.1] established local well-posedness for (1.3) in M p,q pR d q Ă L 2 pR d q for 1 ď p ď 2, 1 ď q ď 2d d`γ . Their approach was based on trilinear estimates and boundedness of Fourier multiplier in M p,q pR d q.…”

Section: Introductionmentioning

confidence: 77%

“…In recent years Cauchy problem for nonlinear dispersive equations with low regularity initial data have been studied by many authors, see [2,7,9,10,24,25,30,39,40]. In this paper, we establish a local and global well-posedness for (1.1) with Cauchy data in Fourier amalgam and modulation spaces.…”

Section: Introductionmentioning

confidence: 88%

“…where ϕpξq " ξ 1`ξ 2 , D x " 1 i B x and ϕpD x q is the Fourier multiplier operator defined by p w p,q s pMq " $ ' ' ' ' & ' ' ' ' % F L q s pMq (Fourier-Lebesgue spaces) if p " q M 2,q s pMq (modulation spaces) if p " 2 M 2,2 s pMq " W 2,2 s pMq " H s pMq (Sobolev spaces) if p " q " 2 F L q s pMq " M p,q s pMq " W p,q s pMq if M " T d . These time-frequency spaces are proven to be very fruitful in handling various problems in analysis and have gained prominence in nonlinear dispersive PDEs, e.g., [7][8][9][10][11][12][13][14][15]. We now state our main theorem.…”

Section: Introductionmentioning

confidence: 99%

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Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Bhimani

Haque

2021

Mathematics

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We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

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The Hartree and Hartree–Fock Equations in Lebesgue $$L^p$$ and Fourier–Lebesgue $$\widehat{L}^p$$ Spaces

Bhimani

Haque

2022

Ann. Henri Poincaré

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The Hartree–Fock equations in modulation spaces

Cited by 10 publications

References 41 publications

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

The Hartree and Hartree–Fock Equations in Lebesgue $$L^p$$ and Fourier–Lebesgue $$\widehat{L}^p$$ Spaces

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