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.

Statement of the lemma

Let ''X''₀, ''X'' and ''X''₁ be three Hilbert spaces with ''X''₀ ⊆ ''X'' ⊆ ''X''₁. Suppose that ''X''₀ is continuously embedded in ''X'' and that ''X'' is continuously embedded in ''X''₁, and that ''X''₁ is the dual space of ''X''₀. Denote the norm on ''X'' by , ,  ⋅ , , _''X'', and denote the action of ''X''₁ on ''X''₀ by $\langle\cdot,\cdot\rangle$. Suppose for some $T>0$ that $u \in L^2 ($, T X_0) is such that its time derivative $\dot \in L^2 ($, T X_1). Then $u$ is almost everywhere equal to a function continuous from ,T/math> into $X$, and moreover the following equality holds in the sense of scalar distributions on $(0,T)$:

:$\frac\frac \, u\, _X^2 = \langle\dot,u\rangle$

The above equality is meaningful, since the functions

:$t\rightarrow \, u\, _X^2, \quad t\rightarrow \langle \dot(t),u(t)\rangle$

are both integrable on ,T/math>.

See also

* Aubin–Lions lemma

Notes

It is important to note that this lemma does not extend to the case where $u \in L^p ($, T X_0) is such that its time derivative $\dot \in L^q ($, T X_1) 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 $u \in L^2 ($, T H^1) and $\dot \in L^($, T H^) (where $H^1$ is a Sobolev space, and $H^$ is its dual space, which is not enough to apply the Lions–Magnes lemma. For this case, one would need $\dot \in L^2($, T H^), but this is not known to be true for weak solutions.

References

* (Lemma 1.2)

*

{{DEFAULTSORT:Lions-Magenes lemma
Lemmas in mathematical analysis