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