# On the non-existence of weak solutions to nonlinear evolutionary PDEs without time derivative

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?

• Is there some kind of continuous or bounded (mapping bounded sets into bounded sets) solution mapping for the stationary equation? – Hannes Nov 8 '18 at 15:36
• What PDE do you want to couple with? Everything depends on that. The equation you wrote down doesn't touch $t$ at all, so it's not parabolic in $(x,t)$, and I see no reason to expect Galerkin to work. – user126920 Nov 8 '18 at 15:54
• @StanleySnelson Let us say that $f(x,t)=\nabla u(x,t)$ where $u$ is the weak solution to the coupled PDE $\partial_t u - \Delta u + \nabla u \cdot v = 0$ or something. So the equation will get the $t$ through $f$, that's the point. – Fritz Nov 8 '18 at 16:25