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 (for smooth enough initial data). 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 formula_1 be a Lipschitz domain in formula_2 for formula_3 and let formula_4 be a weakly differentiable function that vanishes on the boundary of formula_1 in the sense of trace (that is, formula_6 is a limit in the Sobolev space formula_7 of a sequence of smooth functions that are compactly supported in formula_1). Then there exists a constant formula_9 depending only on formula_1 such that, in the case formula_11:
formula_12
and in the case formula_13:
formula_14
formula_15
which holds whenever
formula_16
Ladyzhenskaya's inequalities are the special cases formula_17 formula_18 when formula_11 and formula_20 when formula_13. 
formula_24
formula_29