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.

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) 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.

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.

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).