Anisotropic Caffarelli-Kohn-Nirenberg type inequalities

Abstract

Caffarelli, Kohn and Nirenberg considered in 1984 the interpolation inequalities \[\||x|^{γ_1}u\|_{L^s(\mathbb{R}^n)}\le C\||x|^{γ_2}\nabla u\|_{L^p(\mathbb{R}^n)}^a\||x|^{γ_3}u\|_{L^q(\mathbb{R}^n)}^{1-a} \] in dimension $n\ge 1$, and established necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Motivated by our study of the asymptotic stability of solutions to the Navier-Stokes equations, we consider a more general and improved anisotropic version of the interpolation inequalities \[ \||x|^{γ_1}|x'|^αu\|_{L^s(\mathbb{R}^n)}\le C\||x|^{γ_2}|x'|^μ\nabla u\|_{L^p(\mathbb{R}^n)}^{a}\||x|^{γ_3}|x'|^βu\|_{L^q(\mathbb{R}^n)}^{1-a} \] in dimensions $n\ge 2$, where $x=(x^{\prime },x_n)$ and $x^{\prime }=(x_1,...,x_{n-1})$, and give necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Moreover we extend the Caffarelli-Kohn-Nirenberg inequalities from $q\ge 1$ to $q>0$. This extension, together with a nonlinear Poincaré inequality which we obtain in this paper, has played an important role in our proof of the above mentioned anisotropic interpolation inequalities.

Improvement on exposition

References in corpus (1)

• Stability of singular solutions to the Navier-Stokes system