Energy estimates for Galerkin approximate solution.

roposition

(Energy
estimates for the Galerkin approximate solution). The solution
of the equations (
Galerkin problem
) satisfies
the
estimates
for all
and constants
depending only on
and the functions
.

We drop the term
in the formula (**), recall the starting condition for
at
:
and apply the proposition (
Differential
inequality
2
):
Thus, we proved the first desired inequality.

Next, we drop the term
from the formula (**) and integrate in
:
hence, we proved the second inequality.

To prove the final inequality, we seek to estimate the
quantity

We introduce
: the projection on the linear span of
in
.
We deduce from the formula (
Galerkin problem
)
that
and apply the formula (
Holder inequality
)
and the proposition
(
Energy estimates for the
bilinear form B
) to estimate the
RHS:
Note that
,
hence
We integrate the last inequality in
and use the first inequality of this
proposition