A non-local inequality and global existence

Abstract

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty )$ we obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. $$ We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \cite{ksLM} to include all $\alpha \in (0,74/75)$.

6 pages, to appear in Advances in Mathematics

Cited by in corpus (6)

• Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential
• Entropy dissipation estimates for the relativistic Landau equation, and applications
• De Giorgi-Nash-Moser and H{ö}rmander theories: new interplay
• Hardy's inequality and (almost) the Landau equation
• On the gradient flow structure of the isotropic Landau equation
• Global well-posedness for the homogeneous Landau equation