Then
with the following
bounds
where the constant
depends only on
and
the coefficients of
.

Proof

We multiply the equations (
Galerkin problem
)
with
and sum for
.
We
obtain
where
We note
that
According to the formula (
Cauchy
inequality
), smoothness of
and boundedness of
Therefore,
for any
.
We integrate the above for
then and obtain (for some
):
According to the proposition
(
Energy
estimates for the Galerkin approximate
solution
),
and by uniform positive definiteness of the matrix
there is a constant
such
that
Therefore,

We pass to the limit
and find the first and third desired inequalities.

To prove the remaining inequality we note that the
identity
may be rewritten as
where the
is already shown to be in
.
Hence, the proposition (
Boundary
elliptic regularity
) applies and we
derive
we integrate for
and the obtained in this proof estimates to arrive to the second desired
inequality.

Proposition

(Parabolic regularity 2). Let
be a bounded subset of
.
Assume that the functions
of the operator
(see the definition (
Parabolic
differential operator
)) are smooth on
and independent of
.
Let
Suppose that the following compatibility conditions
hold:
Then the weak solution
of the problem (
Parabolic Dirichlet
problem
) satisfies the
estimates
where the constant
depends only on
and
the coefficients of
.

The compatibility conditions may be understood as follows. Since
then
we should have for smoothness
.
Since
then we should have
and so fourth. The assumption of time independency of the coefficients of
keeps it from getting very verbose.