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 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 [math]\displaystyle{ \langle\cdot,\cdot\rangle }[/math]. Suppose for some [math]\displaystyle{ T\gt 0 }[/math] that [math]\displaystyle{ u \in L^2 ([0, T]; X_0) }[/math] is such that its time derivative [math]\displaystyle{ \dot{u} \in L^2 ([0, T]; X_1) }[/math]. Then [math]\displaystyle{ u }[/math] is almost everywhere equal to a function continuous from [math]\displaystyle{ [0,T] }[/math] into [math]\displaystyle{ X }[/math], and moreover the following equality holds in the sense of scalar distributions on [math]\displaystyle{ (0,T) }[/math]:

[math]\displaystyle{ \frac{1}{2}\frac{d}{dt} \|u\|_X^2 = \langle\dot{u},u\rangle }[/math]

The above inequality is meaningful, since the functions

[math]\displaystyle{ t\rightarrow \|u\|_X^2, \quad t\rightarrow \langle \dot{u}(t),u(t)\rangle }[/math]

are both integrable on [math]\displaystyle{ [0,T] }[/math].

See also

• Aubin–Lions lemma

Notes

It is important to note that this lemma does not extend to the case where [math]\displaystyle{ u \in L^p ([0, T]; X_0) }[/math] is such that its time derivative [math]\displaystyle{ \dot{u} \in L^q ([0, T]; X_1) }[/math] for [math]\displaystyle{ p\neq 2 }[/math], [math]\displaystyle{ q\neq 2 }[/math]. For example, the energy equality for the 3-dimensional Navier–Stokes equations is not known to hold for weak solutions, since a weak solution [math]\displaystyle{ u }[/math] is only known to satisfy [math]\displaystyle{ u \in L^2 ([0, T]; H^1) }[/math] and [math]\displaystyle{ \dot{u} \in L^{4/3}([0, T]; H^{-1}) }[/math] (where [math]\displaystyle{ H^1 }[/math] is a Sobolev space, and [math]\displaystyle{ H^{-1} }[/math] is its dual space, which is not enough to apply the Lions–Magnes lemma (one would need [math]\displaystyle{ \dot{u} \in L^2([0, T]; H^{-1}) }[/math], but this is not known to be true for weak solutions).

References

• 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.