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 ...
, the Aubin–Lions lemma (or theorem) is the result in the theory of
Sobolev spaces of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
-valued functions, which provides a
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
criterion that is useful in the study of nonlinear evolutionary
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a
Galerkin method or by
mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution.
The result is named after the
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
mathematicians
Jean-Pierre Aubin and
Jacques-Louis Lions
Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture pr ...
. In the original proof by Aubin, the spaces ''X''
0 and ''X''
1 in the statement of the lemma were assumed to be
reflexive, but this assumption was removed by Simon, so the result is also referred to as the Aubin–Lions–Simon lemma.
Statement of the lemma
Let ''X''
0, ''X'' and ''X''
1 be three Banach spaces with ''X''
0 ⊆ ''X'' ⊆ ''X''
1. Suppose that ''X''
0 is
compactly embedded in ''X'' and that ''X'' is
continuously embedded in ''X''
1. For
, let
:
(i) If
then the embedding of into
is compact.
(ii) If
and
then the embedding of into
is compact.
See also
*
Lions–Magenes lemma
Notes
References
*
*
* (Theorem II.5.16)
*
* (Sect.7.3)
* (Proposition III.1.3)
*
*
{{DEFAULTSORT:Aubin-Lions lemma
Banach spaces
Theorems in functional analysis
Lemmas in analysis
Measure theory