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Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

Michael Ruzhansky (UGent) and Niyaz Tokmagambetov
(2016) LETTERS IN MATHEMATICAL PHYSICS. 107(4). p.591-618
Author
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Abstract
In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.
Keywords
Wave equation, Well-posedness, Electromagnetic field, Cauchy problem, Landau Hamiltonian, MAGNETIC SCHRODINGER-OPERATORS, SPECTRAL PROPERTIES, CAUCHY-PROBLEM, TRACE FORMULA, COEFFICIENTS, EIGENFUNCTIONS, PERTURBATIONS, ASYMPTOTICS, REGULARITY

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MLA
Ruzhansky, Michael, and Niyaz Tokmagambetov. “Very Weak Solutions of Wave Equation for Landau Hamiltonian with Irregular Electromagnetic Field.” LETTERS IN MATHEMATICAL PHYSICS, vol. 107, no. 4, 2016, pp. 591–618, doi:10.1007/s11005-016-0919-6.
APA
Ruzhansky, M., & Tokmagambetov, N. (2016). Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. LETTERS IN MATHEMATICAL PHYSICS, 107(4), 591–618. https://doi.org/10.1007/s11005-016-0919-6
Chicago author-date
Ruzhansky, Michael, and Niyaz Tokmagambetov. 2016. “Very Weak Solutions of Wave Equation for Landau Hamiltonian with Irregular Electromagnetic Field.” LETTERS IN MATHEMATICAL PHYSICS 107 (4): 591–618. https://doi.org/10.1007/s11005-016-0919-6.
Chicago author-date (all authors)
Ruzhansky, Michael, and Niyaz Tokmagambetov. 2016. “Very Weak Solutions of Wave Equation for Landau Hamiltonian with Irregular Electromagnetic Field.” LETTERS IN MATHEMATICAL PHYSICS 107 (4): 591–618. doi:10.1007/s11005-016-0919-6.
Vancouver
1.
Ruzhansky M, Tokmagambetov N. Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. LETTERS IN MATHEMATICAL PHYSICS. 2016;107(4):591–618.
IEEE
[1]
M. Ruzhansky and N. Tokmagambetov, “Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field,” LETTERS IN MATHEMATICAL PHYSICS, vol. 107, no. 4, pp. 591–618, 2016.
@article{8585404,
  abstract     = {{In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.}},
  author       = {{Ruzhansky, Michael and Tokmagambetov, Niyaz}},
  issn         = {{0377-9017}},
  journal      = {{LETTERS IN MATHEMATICAL PHYSICS}},
  keywords     = {{Wave equation,Well-posedness,Electromagnetic field,Cauchy problem,Landau Hamiltonian,MAGNETIC SCHRODINGER-OPERATORS,SPECTRAL PROPERTIES,CAUCHY-PROBLEM,TRACE FORMULA,COEFFICIENTS,EIGENFUNCTIONS,PERTURBATIONS,ASYMPTOTICS,REGULARITY}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{591--618}},
  title        = {{Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field}},
  url          = {{http://doi.org/10.1007/s11005-016-0919-6}},
  volume       = {{107}},
  year         = {{2016}},
}
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