Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

Abstract

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the L ∞ (R 3 )-norm at a finite time T can occur only if the L 3/2 (R 3 )-norm of the solution concentrates for times close to T . The bounds are obtained using the comparison principle for the Landau equation and for the associated mass function. This method provides long-time existence results for the isotropic version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.

Global Existence of Weak Even Solutions for an Isotropic Landau Equation with Coulomb Potential

“…The fact that neither (7) not (8) can be proven at the moment is rather unsatisfactory. And consequently the results stated in Theorem 2 should be viewed as conditional.…”

“…Recently the first author and collaborator showed in [8] global in time existence of smooth and bounded radial and monotone decreasing solutions to (1) for initial data that have finite mass, energy and entropy. This last result puts in evidence how solutions to a non-linear equation with a non-local diffusivity such as a[u] behave drastically different from (and better than) Keller-Segel or semilinear heat equation.…”

“…The condition of radial symmetry and monotone decreasing simplify the analysis in [8] considerably: bounds on the solution are obtained using comparison principle for (1) and for the associated mass function. Pointwise upper bounds are obtained using barriers: these arguments are based on the observation that certain functions are supersolutions for the elliptic operator under certain assumptions on the solution.…”

Spectral gap and exponential convergence to equilibrium for a multi-species Landau system

“…Recently the set of initial data for which this theory is valid has been improved by Carrapatoso and Mischler [9] via a linearization method.Recently the first author and Guillen have shown, for the Coulomb case, global in time existence of classical solution for a modified isotropic homogeneous Landau equationin the case of radially symmetric (but no smallness assumptions!) initial data 55 [22]. Moreover, using the theory of A p weights, they showed that solutions to the original Landau equations with general initial data for γ > −2 have an instantaneous regularization which does not deteriorate as time increases, with bounds that only depend on the physical quantities, mass, momentum and energy [21].…”

“…in the case of radially symmetric (but no smallness assumptions!) initial data 55 [22]. Moreover, using the theory of A p weights, they showed that solutions to the original Landau equations with general initial data for γ > −2 have an instantaneous regularization which does not deteriorate as time increases, with bounds that only depend on the physical quantities, mass, momentum and energy [21].…”