Global Lp solutions of the Boltzmann equation with an angle-potential concentrated collision kernel and convergence to a Landau solution
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We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases, however, the motivation of such a collision kernel is to treat the (extreme) case of Coulomb potentials, on which this particular form of collision operator is well defined. We show that the scaled angle-potential singular concentration in a grazing collisions limit makes the Boltzmann operator converge in the sense of distributions to the Landau operator acting on the Boltzmann solutions, and also that solutions of this type of Boltzmann equation converge to solutions of the Landau equation that conserve mass, momentum and energy.