Global Smooth Solutions to the Landau–Coulomb Equation in \(L^{3/2}\)

We consider the homogeneous Landau equation in \({\mathbb {R}}^3\) with Coulomb potential and initial data in polynomially weighted \(L^{3/2}\). We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to \(L^p\) with \(p>3/2\), there is a unique solution. At the crux of the result is a new \(\varepsilon \)-regularity criterion in the spirit of the Caffarelli–Kohn–Nirenberg theorem: a solution which is small in weighted \(L^{3/2}\) is regular. Although the \(L^{3/2}\) norm is a critical quantity for the Landau–Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the \(L^{3/2}\) norm alone is not enough to control the \(L^\infty \) norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau–Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau–Coulomb equation with rough initial data.