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
[edit]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 that is such that its time derivative . Then is almost everywhere equal to a function continuous from into , and moreover the following equality holds in the sense of scalar distributions on :
The above equality is meaningful, since the functions
are both integrable on .
See also
[edit]Notes
[edit]It is important to note that this lemma does not extend to the case where is such that its time derivative for . For example, the energy equality for the 3-dimensional Navier–Stokes equations is not known to hold for weak solutions, since a weak solution is only known to satisfy and (where is a Sobolev space, and is its dual space, which is not enough to apply the Lions–Magnes lemma (one would need , but this is not known to be true for weak solutions). [1]
References
[edit]- ^ 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.