Imagine we have a nonlinear PDE, e.g. some velocity model

$$\begin{aligned} -\Delta v + (v \cdot \nabla) v + \nabla p = f \\ \text{div} \, v = 0 \end{aligned}$$

which has a weak solution for some initial & boundary data if we are in the stationary case. In the existence proof via Galerkin we can use compact embeddings of the kind $H^1 \hookrightarrow \hookrightarrow L^2$ to get strong convergence and we can do the limit process in the nonlinear term.

Now, let us couple that PDE to some time-dependent PDE so that the right-hand side $f$ now depends on time, e.g. we couple it to a convective reaction-diffusion equation as follows $$\begin{aligned}-\Delta v + (v \cdot \nabla) v + \nabla p &= \nabla u \\ \text{div} \, v &= 0 \\ \partial_t u - \Delta u+ \nabla u \cdot v &= 0 \end{aligned}$$ Hence our velocity model got instationary and $v$ will depend on time.

In my opinion it is now impossible to show existence of a weak solution via Galerkin's method. We'd need strong convergence, but for this we'd need a bound on the (fractional) time derivative of $v$. Note that the embedding $L^2(0,T;H^1) \hookrightarrow L^2(0,T;L^2)$ is not compact. Instead we want to use something like $L^2(0,T;H^1) \cap H^1(0,T;(H^1)') \hookrightarrow \hookrightarrow L^2(0,T;L^2).$ This is not directly available. Of course, adding a term $\partial_t v$ to the velocity model would save the day but we don't want to do that. Maybe adding some $\varepsilon \partial_t$ and later letting $\varepsilon \to 0$ would help.

I can hardly imagine that there is theorem that if a weak solution exist then one can prove it with Galerkin's method. So maybe some fixed point theorem could help. But here in that case I can't believe it that there exists a weak solution. It seems so far away.

So, can we prove the non-existence of weak solution in such a case?