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.

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

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

The above inequality is meaningful, since the functions

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

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

• Aubin–Lions lemma

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

• 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. 

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