In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the
Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
Russian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
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 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
for
and let
be a
weakly differentiable function that vanishes on the boundary of
in the sense of
trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
(that is,
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 ...
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
). Then there exists a constant
depending only on
such that, in the case
:
:
and in the case
:
:
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 ...
::
:which holds whenever
::
:Ladyzhenskaya's inequalities are the special cases
when
and
when
.
* A simple modification of the argument used by Ladyzhenskaya in her 1958 paper (see e.g. Constantin & Seregin 2010) yields the following inequality for
, valid for all
:
::
* The usual Ladyzhenskaya inequality on
, can be generalized (see McCormick & al. 2013) to use the
weak "norm" of
in place of the usual
norm:
::
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