Ladyzhenskaya's inequality
In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions. There is an analogous inequality for functions of three real variables, but the exponents are slightly different; much of the difficulty in establishing existence and uniqueness of solutions to the three-dimensional Navier–Stokes equations stems from these different exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities.
Let Ω be a Lipschitz domain in Rn for n = 2 or 3, and let u: Ω → R be a weakly differentiable function that vanishes on the boundary of Ω in the sense of trace. Then there exists a constant C depending only on Ω such that, in the case n = 2,
and, in the case n = 3,Generalizations
- Both the two- and three-dimensional versions of Ladyzhenskaya's inequality are special cases of the Gagliardo–Nirenberg interpolation inequality
- A simple modification of the argument used by Ladyzhenskaya in her 1958 paper yields the following inequality for u: R2 → R, valid for all r ≥ 2:
- The usual Ladyzhenskaya inequality on Rn, n = 2 or 3, can be generalized to use the weak L2 “norm” of u in place of the usual L2 norm:
