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 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 $\langle \cdot ,\cdot \rangle$ . 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}}\|u\|_{X}^{2}=\langle {\dot {u}},u\rangle

The above equality is meaningful, since the functions

t\rightarrow \|u\|_{X}^{2},\quad t\rightarrow \langle {\dot {u}}(t),u(t)\rangle

are both integrable on $[0,T]$ .

Notes edit

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). ^[1]