Lions–Magenes lemma

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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 ${\displaystyle \langle \cdot ,\cdot \rangle }$. 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 ${\displaystyle (0,T)}$:

${\displaystyle {\frac {1}{2}}{\frac {d}{dt}}\|u\|_{X}^{2}=\langle {\dot {u}},u\rangle }$

The above equality is meaningful, since the functions

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

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

• Aubin–Lions lemma

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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 ${\displaystyle 1/p+1/q>1}$. For example, the energy equality for the 3-dimensional Navier–Stokes equations is not known to hold for weak solutions, since a weak solution ${\displaystyle u}$ 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. For this case, 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]