Lions–Magenes lemma explained

In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself.

Statement of the lemma

Let X0, X and X1 be three Hilbert spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is continuously embedded in X and that X is continuously embedded in X1, and that X1 is the dual space of X0. Denote the norm on X by || ⋅ ||X, and denote the action of X1 on X0 by

\langle ⋅ , ⋅ \rangle

. Suppose for some

T>0

that

u\inL2([0,T];X0)

is such that its time derivative
•
u

\inL2([0,T];X1)

. Then

u

is almost everywhere equal to a function continuous from

[0,T]

into

X

, and moreover the following equality holds in the sense of scalar distributions on

(0,T)

:
1
2
d
dt
2
\|u\|
X

=\langle

•
u

,u\rangle

The above equality is meaningful, since the functions

t →

2,
\|u\|
X

  t → \langle

•
u

(t),u(t)\rangle

are both integrable on

[0,T]

.

See also

Notes

It is important to note that this lemma does not extend to the case where

u\inLp([0,T];X0)

is such that its time derivative
•
u

\inLq([0,T];X1)

for

1/p+1/q>1

. For example, the energy equality for the 3-dimensional Navier–Stokes equations is not known to hold for weak solutions, since a weak solution

u

is only known to satisfy

u\inL2([0,T];H1)

and
•
u

\inL4/3([0,T];H-1)

(where

H1

is a Sobolev space, and

H-1

is its dual space, which is not enough to apply the Lions–Magnes lemma (one would need
•
u

\inL2([0,T];H-1)

, but this is not known to be true for weak solutions).

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Lions–Magenes lemma".

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