Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

Antonelli, Paolo; Michelangeli, Alessandro; Scandone, Raffaele

doi:10.1007/s00033-018-0938-5

Cited by 5 publications

(3 citation statements)

References 43 publications

(75 reference statements)

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“…3 is a weak solution to (1.1). By means of Proposition 3.4, we have also that Ã ∈ W 1,6 T L 3 . Therefore, after choosing T possibly smaller (depending on R 1 , R 2 ), we have (ũ, Ã) ∈ Z, whence (ũ, Ã) = (u, A).…”

Section: As a Consequence Of Lemma 43 We Can Define Formentioning

confidence: 80%

“…Altough magnetic Strichartz estimates are well understood for time independent potentials [13,14,15,16,17] (see also [37] and references therein), in the time dependent case much less is known, and the only results available require the smallness of suitable scale invariant space-time norms [19,48]. In particular, even for a non-linear Schrödinger equation with a given external time dependent magnetic potential, the well-posedness in the energy space is in general an open question (global existence of weak solutions can be proved using the method of parabolic regularization [6]). Concerning the Maxwell-Schrödinger system with focusing non-linearities, one can also study the existence and stability of standing waves, see for instance [12] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Global well-posedness for the non-linear Maxwell-Schrödinger system

Antonelli,

Marcati,

Scandone

2019

Preprint

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In this paper we study the Cauchy problem associated to the Maxwell-Schrödinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic quantum plasma, interacting with its self-generated electro-magnetic potential.We show the global well-posedness at high regularity for the sub-cubic case, and we provide polynomial bounds for the growth of the Sobolev norm of the solutions, for a certain range of non-linearities. The main tools are suitable a priori estimates, obtained by means of Koch-Tzvetkov type bounds for the non-homogeneous Schrödinger equation, which overcome the lack of Strichartz estimates for the magnetic-Schrödinger flow. Then we use a classical argument from the Kato school involving modified energies, which combined with the a priori estimates allows us to control the non-linearity globally in time.As a byproduct of our analysis, we show that the Lorentz force associated to the electro-magnetic field is well defined for solutions slightly more regular than being finite energy. This aspect is of fundamental importance since all the related physical models require the observability of electro-magnetic effects. The well posedness of the Lorentz force still appears to be an open problem in the case of solutions of finite energy only.

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Section: As a Consequence Of Lemma 43 We Can Define Formentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Global well-posedness for the non-linear Maxwell-Schrödinger system

Antonelli,

Marcati,

Scandone

2019

Preprint

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“…Our second main focus concerns the local and global well-posedness of (1.12) in the energy space. This field is under a comprehensive and well-established control in the classical, unperturbed case α = ∞, including also when (1.12) contains much more singular and non-symmetric convolution potentials, together with electric and magnetic potentials, possibly depending on time (see, e.g., [21,45,12] and the references therein).…”

Section: Strictly Positive and Strictly Radially Decreasing;mentioning

confidence: 99%

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

Georgiev¹,

Michelangeli²,

Scandone³

2022

Preprint

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We study a class of two-dimensional non-linear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a non-trivial adaptation of the standard properties of Schwartz symmetrisation for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

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Global Solutions to the Nonlinear Maxwell-Schrödinger System

Scandone

2022

Trends in Mathematics

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Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

Cited by 5 publications

References 43 publications

Global well-posedness for the non-linear Maxwell-Schrödinger system

Global well-posedness for the non-linear Maxwell-Schrödinger system

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

Global Solutions to the Nonlinear Maxwell-Schrödinger System

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