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 ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } . Suppose for some T > 0 {\displaystyle T>0} that u ∈ L 2 ( [ 0 , T ] ; X 0 ) {\displaystyle u\in L^{2}([0,T];X_{0})} is such that its time derivative u ˙ ∈ L 2 ( [ 0 , T ] ; X 1 ) {\displaystyle {\dot {u}}\in L^{2}([0,T];X_{1})} . Then u {\displaystyle u} is almost everywhere equal to a function continuous from [ 0 , T ] {\displaystyle [0,T]} into X {\displaystyle X} , and moreover the following equality holds in the sense of scalar distributions on ( 0 , T ) {\displaystyle (0,T)} :

1 2 d d t ‖ u ‖ X 2 = ⟨ u ˙ , u ⟩ {\displaystyle {\frac {1}{2}}{\frac {d}{dt}}\|u\|_{X}^{2}=\langle {\dot {u}},u\rangle }

The above equality is meaningful, since the functions

t → ‖ u ‖ X 2 , t → ⟨ u ˙ ( t ) , u ( t ) ⟩ {\displaystyle t\rightarrow \|u\|_{X}^{2},\quad t\rightarrow \langle {\dot {u}}(t),u(t)\rangle }

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

See also Aubin–Lions lemma

Notes This lemma does not extend to the case where u ∈ L p ( [ 0 , T ] ; X 0 ) {\displaystyle u\in L^{p}([0,T];X_{0})} is such that its time derivative u ˙ ∈ L q ( [ 0 , T ] ; X 1 ) {\displaystyle {\dot {u}}\in L^{q}([0,T];X_{1})} for 1 / p + 1 / q > 1 {\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 u {\displaystyle u} is only known to satisfy u ∈ L 2 ( [ 0 , T ] ; H 1 ) {\displaystyle u\in L^{2}([0,T];H^{1})} and u ˙ ∈ L 4 / 3 ( [ 0 , T ] ; H − 1 ) {\displaystyle {\dot {u}}\in L^{4/3}([0,T];H^{-1})} (where H 1 {\displaystyle H^{1}} is a Sobolev space, and H − 1 {\displaystyle H^{-1}} is its dual space, which is not enough to apply the Lions–Magnes lemma. For this case, one would need u ˙ ∈ L 2 ( [ 0 , T ] ; H − 1 ) {\displaystyle {\dot {u}}\in L^{2}([0,T];H^{-1})} , 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.