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

Abstract

In this manuscript we consider an isotropic modification for the Landau equation with Coulomb potential in three space dimensions. Global in time existence of weak solutions for even initial data is shown by employing a time semi-discretization of the equation, an entropy inequality and a uniform estimate for the second moment of the solution to the discretized problem. Moreover, under an additional condition that has to be satisfied uniformly over time, uniform boundedness of the solution is proved, with bounds depending solely on the mass, second moment and entropy of the solution. A byproduct of our analysis is a proof of improved regularity for weak solutions to the Landau equation with Coulomb potential.and u satisfies the following weak formulation of (1)

Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions

“…Recently the authors in [11] improved the regularity of the weak solutions: let f be a weak solution to the Landau equation as in (6); then…”

“…are uniformly bounded and one can take f η R (v) as test function in (7). Since ∇η R → 0 as R → +∞ and both A[f ] and a[f ] are uniformly bounded one can pass to the limit R → +∞ and obtain (11).…”

“…First we do not require the weak solutions to be bounded, but only belonging to some L p space with p > 3/2. Secondly our method uses the weak representation of (1)-(2) provided in [11] and a new weighted Poincaré inequality (10) recently introduced in [10]. This inequality is shown to be valid for any solution f to the Landau equation that is uniformly in time L p (R 3 )-integrable, for some p > 3/2 (and has high enough bounded moments).…”

A Review for an Isotropic Landau Model

“…All the computations here are formal, meaning we assume that u and all related quantities have enough regularity for the mathematical manipulations to make sense. We refer to [11] for the detailed calculations. Let…”

The Landau equation as a Gradient Flow

“…To tackle this and other problems an array of novel methods have been employed. Here is an incomplete sample of the contributions made in this direction which highlight the difficulties of the soft potential case [22,21,1,12,11,47,33,31,34,32,43,30,42,29]. A brief glance at some of these references illustrates the breadth of techniques that have found partial success at answering the open questions; probability-based arguments, kinetic and parabolic theory, and many more.…”