In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the usual definition is given bellow. More information could be found, for example, in [Dafermos's book](https://www.springer.com/la/book/9783662494493).


Let's consider a problem:


$$ \hspace{1.1cm} u(x,t)\_t + f(u(x,t))\_x = g(u(x,t)), \; x \in \mathbb{R}, t \in [0,T],\label{1}\tag{1} $$
$$\hspace{1cm} u(x,0)=u\_{0} (x), \; x \in \mathbb{R}. \label{2}\tag{2}$$


*The weak solution* $u$ of problem \eqref{1}-\eqref{2} is given with: 


$$ 
\int\_{0}^T \int\_{\mathbb{R}} [u \psi\_{t} + f(u) \psi\_{x}] \; dx dt + \int\_{\mathbb{R}} u\_{0}(x) \psi (x,0) \; dx = - \int\_{0}^T \int\_{\mathbb{R}} g(u) \psi \; dx dt,\label{3}\tag{3}$$


for every test function $\psi(x,t) \in C\_{c}^{\infty}(\mathbb{R} \times [0,T))$. **So here the test functions are smooth real valued functions with compact support.** 


A few days ago I was reading a paper ([Sueur](https://arxiv.org/pdf/1203.5187.pdf)) and I stumbled upon some interesting technique. That paper talks about strong solutions of Euler system and weak solution of Navier-Stokes system and their connection. 


The sentence that made me think was:**"One may wish to apply the weak formulation of the Navier-Stokes equations with the solution of the Euler equations as a test function."** 


**Using a (relatively smooth) solution of one system as a test function for the other system** sounds unusual and amazing at the same time. I then remembered the problem that I was working on a few years ago. So I have two questions.


1. Instead of the usual real valued test functions $\psi(x,t) \in C\_{c}^{\infty}(\mathbb{R} \times [0,T))$ could we use some Banach space-valued test functions such as $\psi(t)(x) \in C^{\infty}([0,T);H^m(\mathbb{R}))$ or $\psi(t)(x) \in C^{2}([0,T);H^m(\mathbb{R}))$ or $\psi(t)(x) \in C([0,T);H^m(\mathbb{R}))$? The last one is the one I am the most interested in and those would be "smooth solutions". Here $H^m(\mathbb{R})$ represents Sobolev space $W^{m,2}(\mathbb{R})$ - those were the one I was working on a few years ago.
2. What would be the rules based on which we could decide whether some space of test functions is acceptable or not? For example could we use $\psi(t)(x) \in L^{\infty}([0,T);H^m(\mathbb{R}))$ or $\psi(t)(x) \in L^{1}([0,T);H^m(\mathbb{R}))$ or whatever else? I am sure that the answer would be NO because of that the t derivative maybe doesn't even exist (just see \eqref{3}). But I am maybe wrong too.


Any help with this would be great (whether it is some reference in the literature or good old fashioned way by writting the answer).



Banach space-valued test functions in the definition of a weak solution of a PDE problem
Writing your equation (3) as $a(u,\psi)=0$, it is indeed common to call $u\in L^1\_\mathrm{loc}$ a `weak solution' to your problem if and only it satisfies


$$
a(u,\psi) = 0 \mbox{ for all } \psi \in C\_c^\infty(\mathbb{R}\times [0,T))
$$


However, this is a *definition* which implicitly encodes $C^\infty\_c(\mathbb{R}\times [0,T))$ as the base set of 'test functions' for your problem. I mean this in the sense that only once you have verified that $a(u,\psi)=0$ for all $\psi\in C\_c^\infty(\mathbb{R}\times [0,T))$ can you declare $u$ to be a (weak) 'solution' your problem. 


In the other direction, this notion of 'solution' gives us a very powerful mechanism for learning things about anything we know to be a solution. Namely, we know that any weak solution satisfies $a(u,\psi)= 0$, no matter what smooth compactly supported function we choose as $\psi$. By making clever *particular* choices of $\psi$ (which I think is what Sueur is proposing), one can learn a great deal about functions $u$ which are 'solutions' in the above sense. 


People do sometimes use 'test function' in slightly looser senses as well (see [Why is this test function admissible? [Paper explanation]](https://mathoverflow.net/questions/331039/why-is-this-test-function-admissible-paper-explanation/331074?noredirect=1#comment825572_331074)), but that's a separate issue. 


**Some additional points:**


* Even if your putative solutions only *a priori* belong a 'big' space which includes lots of coarse functions, it is often the case that being a member of this big space *and* being a weak or distributional solution to your PDE actually implies much greater smoothness. As you may be aware, this idea is 'regularity theory'. I don't have a reference to hand, but I'd imagine there are existing regularity theorems which are applicable to your PDE. Once you know your solutions are actually smooth, you can multiply them by smooth cutoff functions to get genuine test functions, at which point you can test them against whatever you like.
* It's often possible to make some headway by approximating a problem of interest by a sequence of approximate problems (see the proof of theorem 4 in the paper discussed in the post associated with the answer I linked to above).
* I know you're quite keen to see things in contexts other than that of the Navier Stokes equations, but the notes <http://homepages.warwick.ac.uk/~masdh/Campinas.pdf> discuss what looks at first glance like a slightly different notion of 'weak solution' from the one discussed above (here I'm thinking particularly of definition 1.2 and the discussion leading up to it). The two notions end up being the same, but this takes a little bit of deduction (the argument is sketched in the notes).


