In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Aubin–Lions lemma is a result in the theory of Sobolev space

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 itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

s of Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

-valued functions. More precisely, it is a compactness

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 criterion that is very useful in the study of nonlinear evolutionary partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s. The result is named after the French

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s Thierry Aubin

Thierry Aubin

Thierry Aubin was a French mathematician at Centre de Mathématiques de Jussieu who worked on Riemannian geometryand non-linear partial differential equations...

 and Jacques-Louis Lions

Jacques-Louis Lions

Jacques-Louis Lions ForMemRS 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 prize in 1986. Lions is listed as an ISI highly cited researcher.-Biography:After...

.

Statement of the lemma

Let X₀, X and X₁ be three Banach spaces with X₀ ⊆ X ⊆ X₁. Suppose that X₀ is compactly embedded

Compactly embedded

In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.-Definition :...

 in X and that X is continuously embedded

Continuously embedded

In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space...

 in X₁; suppose also that X₀ and X₁ are reflexive space

Reflexive space

In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...

s. For 1 < p, q < +∞, let


Then the embedding of W into L^p([0, T]; X) is also compact