I've been studying linear parabolic equations from Evan's PDE text (chapter 7) and I seem to have shown something that's too strong. I will list the setup here:
Let $U \subset \mathbb R^n$ be a bounded domain, $T>0$ and set $U_T = U \times (0,T).$ Consider the divergence form parabolic PDE of the form,
$$ u_t + Lu = u_t - \sum_{i,j=1}^n \left(a^{ij}u_{x_i}\right)_{x_j} + \sum_{i=1}^n b^iu_{x_i} + cu = f $$
on $U_T,$ where $a^{ij},b^i,c \in C^1(\overline U_T)$ with $a^{ij}$ uniformly elliptic ($\sum_{i,j}a^{ij}\xi_i\xi_j \geq \theta|\xi|^2$) and $f \in L^2(U_T),$ subject to boundary conditions $u = \psi$ on $U \times \{0\}$ for some $\psi \in L^2(U)$ and $u=0$ on $\partial U \times (0,T).$
When showing existence via the Galerkin method, we fix a sequence $\{\phi_j\}_{j=1}^n$ in $H^1_0(U)$ which forms an orthonormal basis of $L^2(U)$ and is orthogonal in $H^1(U).$ We can then (assuming $f, \psi$ are sufficiently regular) construct approximate solutions,
$$ u^N(x,t) = \sum_{i=1}^N d_i^N(t) \phi(x), $$
such that $u^N_t + Lu^N = f_N$ weakly on $U_T$ and $u^N = \psi_N$ on $U \times \{0\},$ where $f_N$ and $\psi_N$ are orthogonal projections onto the relevant subspaces spanned by $\langle\phi_1,\dots,\phi_N\rangle.$ That is,
$$ f_N(x,t) = \sum_{i=1}^N \langle f(\cdot,t), \phi_N \rangle_{L^2(U)} \phi_N(x), \quad \psi_N(x) = \sum_{i=1}^N \langle \psi,\phi_N\rangle_{L^2(U)} \phi_N. $$
The idea is to show that as $N \rightarrow \infty,$ the sequence $u^N$ converges to a weak solution.
When doing this, we prove an energy estimate of the form,
$$ \max_{0\leq t\leq T} \lVert u^N(t) \rVert_{L^2(U)} + \lVert u^N \rVert_{L^2(0,T,H^1(U))} \leq C \left(\lVert f_N\rVert_{L^2(U_T)} + \lVert \psi_N \rVert_{L^2(U)} \right).$$
This is theorem 2 in 7.1 of Evans (p354). Although Evans has $f$ and $\psi$ on the RHS, the analogous argument also proves this estimate. If we instead take $\tilde f = f_N - f_M$ and $\tilde \psi = \psi_N - \psi_M$ however, by uniqueness of weak solutions (theorem 4) we get the unique weak solution must be $u_N - u_M.$ Hence we get the estimate,
$$ \max_{0\leq t\leq T} \lVert u^N(t)-u^M(t) \rVert_{L^2(U)} + \lVert u^N-u^M \rVert_{L^2(0,T,H^1(U))} \leq C \left(\lVert f_N-f_M\rVert_{L^2(U_T)} + \lVert \psi_N - \psi_M \rVert_{L^2(U)} \right).$$
Now if we let $N,M \rightarrow \infty,$ the right hand side vanishes. So our sequence $u^N$ is actually Cauchy in $C([0,T],L^2(U))$ and $L^2(0,T;H^1_0(U)),$ so it converges strongly to some $u$ in both spaces.
My question: What I have done wrong in the above?
When proving existence theorems, Evans instead observes the sequence is uniformly bounded in the spaces and hence extracts a weak limit. This however is not only a bit more delicate to work with, but it only guarantees the limit $u$ is in $L^{\infty}(0,T;L^2(U)).$
Precisely because my above argument isn't mentioned in the text, I have a strong suspicion that it is incorrect. I cannot find any fault in my answer however, despite having looked over it several times.
Note: I assumed earlier that $f,\psi$ are actually regular when proving the existence, but that was only to apply Picard-Lindelöf. For the actual energy estimates, I don't need to assume this. Hence taking $\tilde f, \tilde \psi$ as I did isn't an issue. Also once we have proved existence, we can remove this hypothesis by a density argument. I can elaborate on this if necessary, but I don't believe there is any issue here.
As it turns out this is correct, and there's no issues with this kind of reasoning. Since we have an a-priori estimate and the equation is linear, nothing goes wrong.
There are two points worth noting here however:
This kind of reasoning completely breaks down when dealing with nonlinear equations. Often in that setting we find suitable weak solutions by taking a sequence of approximate solutions, establishing a uniform estimate and taking limits. In this setting the best we can hope for is weak(*) convergence in the suitable spaces. I presume Evans opted for this argument to illustrate this technique of passing to weak limits.
Even in the nonlinear setting, one usually can still establish $L^2$-continuity in time. It is proven in section 5.9, theorem 3 that if $u \in L^2(0,T;H^1_0(U))$ with $\partial_t u \in L^2(0,T;H^{-1}(U)),$ then $u \in C([0,T];L^2(U)).$ Since these spaces are reflexive, if we can bound our approximating sequences with respect to these norms (note: this applies to the setup of my question also) then we obtain $L^2$-continuity in time.