Lions–Magenes lemma

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.

## Contents

* 1 Statement of the lemma
* 2 See also
* 3 Notes
* 4 References

## 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 ⟨⋅,⋅⟩. Suppose for some T>0 that u∈L2([0,T];X0) is such that its time derivative u˙∈L2([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):

12ddt‖u‖X2=⟨u˙,u⟩

The above inequality is meaningful, since the functions

t->‖u‖X2,t->⟨u˙(t),u(t)⟩

are both integrable on [0,T].

## See also

* Aubin–Lions lemma

## Notes

It is important to note that this lemma does not extend to the case where u∈Lp([0,T];X0) is such that its time derivative u˙∈Lq([0,T];X1) for p≠2, q≠2. 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∈L2([0,T];H1) and u˙∈L4/3([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˙∈L2([0,T];H−1), but this is not known to be true for weak solutions). [1]

## References

1. ↑ Constantin, Peter; Foias, Ciprian I. (1988), Navier–Stokes Equations, Chicago Lectures in Mathematics, Chicago, IL: University of Chicago Press

* Temam, Roger (2001). Navier-Stokes Equations: Theory and Numerical Analysis. Providence, RI: AMS Chelsea Publishing. pp. 176–177. (Lemma 1.2)

* Lions, Jacques L.; Magenes, Enrico (1972). Nonhomogeneous boundary values problems and applications. Berlin, New York: Springer-Verlag.

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