Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field |
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Authors: | Michael Ruzhansky Niyaz Tokmagambetov |
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Institution: | 1.Department of Mathematics,Imperial College London,London,UK;2.Al-Farabi Kazakh National University,Almaty,Kazakhstan |
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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. |
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