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

TL;DR

This paper introduces a framework for defining and analyzing very weak solutions to the wave equation associated with the Landau Hamiltonian under irregular electromagnetic fields, extending solution concepts to distributional coefficients.

Contribution

It develops a notion of very weak solutions for wave equations with distributional coefficients and demonstrates existence and consistency with classical solutions.

Findings

Very weak solutions exist for the Landau Hamiltonian wave equation with irregular coefficients.

The approach generalizes classical solutions to distributional electromagnetic fields.

The method ensures solutions are consistent with classical solutions when regularity conditions are met.

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.