Ladyzhenskaya's inequality

rdf:langString Ladyzhenskaya's inequality

rdf:langString 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 be a Lipschitz domain in for and let be a weakly differentiable function that vanishes on the boundary of in the sense of trace (that is, is a limit in the Sobolev space of a sequence of smooth functions that are compactly supported in ). Then there exists a constant depending only on such that, in the case : and in the case :