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 $⟨\cdot ,\cdot ⟩$. Suppose for some $T>0$ that $u\in L^2([0,T];X_0)$ is such that its time derivative $\dot{u}\in L^2([0,T];X_1)$. 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 $(0,T)$:

$\frac{1}{2}\frac{d}{dt}\Vert u{\Vert }_X^2=⟨\dot{u},u⟩$

The above equality is meaningful, since the functions

$t\to \Vert u{\Vert }_X^2,t\to ⟨\dot{u}(t),u(t)⟩$

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 $u\in L^p([0,T];X_0)$ is such that its time derivative $\dot{u}\in L^q([0,T];X_1)$ 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 $u\in L^2([0,T];H^1)$ and $\dot{u}\in L^{4/3}([0,T];H^{-1})$ (where $H^1$ is a Sobolev space, and $H^{-1}$ is its dual space, which is not enough to apply the Lions–Magnes lemma (one would need $\dot{u}\in L^2([0,T];H^{-1})$, but this is not known to be true for weak solutions).

References

• Template:Cite book (Lemma 1.2)

• Template:Cite book