Ladyzhenskaya Inequality
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, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the
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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 \Omega be a
Lipschitz domain In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The ...
in \mathbb R^ for n = 2 \text 3 and let u: \Omega \rightarrow \mathbb R be a weakly differentiable function that vanishes on the boundary of \Omega in the sense of
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(that is, u is a limit in the
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
H^1(\Omega) of a sequence of
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s that are compactly supported in \Omega). Then there exists a constant C depending only on \Omega such that, in the case n = 2: : \, u \, _ \leq C \, u \, _^ \, \nabla u \, _^ and in the case n = 3: : \, u \, _ \leq C \, u \, _^ \, \nabla u \, _^


Generalizations

* Both the two- and three-dimensional versions of Ladyzhenskaya's inequality are special cases of the
Gagliardo–Nirenberg interpolation inequality In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the L^p-norms of different weak derivatives of a function through an interpolat ...
:: \, u \, _ \leq C \, u \, _^\alpha \, u \, _^, :which holds whenever :: p > q \geq 1, s > n ( \tfrac - \tfrac ), \text \tfrac = \tfrac + (1 - \alpha) ( \tfrac - \tfrac ). :Ladyzhenskaya's inequalities are the special cases p = 4, q = 2, s = 1 \alpha = \tfrac when n = 2 and \alpha = \tfrac when n = 3. * A simple modification of the argument used by Ladyzhenskaya in her 1958 paper (see e.g. Constantin & Seregin 2010) yields the following inequality for u: \mathbb R^ \rightarrow \mathbb R, valid for all r \ge 2: :: \, u \, _ \leq C r \, u \, _^ \, \nabla u \, _^. * The usual Ladyzhenskaya inequality on \mathbb R^, n = 2 \text 3, can be generalized (see McCormick & al. 2013) to use the weak L^ "norm" of u in place of the usual L^ norm: :: \, u \, _ \leq \begin C \, u \, _^ \, \nabla u \, _^, & n = 2, \\ C \, u \, _^ \, \nabla u \, _^, & n = 3. \end


See also

*
Agmon's inequality In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,Lemma 13.2, in: Agmon, Shmuel, ''Lectures on Elliptic Boundary Value Problems'', AMS Chelsea Publishing, Providence, RI, 2010. . consist of two closely related interpolati ...


References

* * [] * {{cite journal , last1 = McCormick , first1 = D. S. , last2 = Robinson , first2 = J. C. , last3 = Rodrigo , first3 = J. L. , title = Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO , journal = Milan J. Math. , volume = 81 , issue = 2 , pages = 265–289 , year = 2013 , doi = 10.1007/s00032-013-0202-6 , arxiv = 1303.6351 , citeseerx = 10.1.1.758.7957 , s2cid = 44022084 Inequalities Fluid dynamics Sobolev spaces