I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn much about PDEs in my master study, only for instance the solution of the four fundamental types of PDEs and some Sobolev space theory. It was a challenge for me to learn the PDE theory at the beginning, since one doesn't only need to know a lot of things involving mathematics, but should also learn something about the physical modelling. This was a little bit hard for me since my minor subject was economics. But after the one year learning I think studying physical models is also very attractive and interesting. But as more as I learned, I know one can generally not conclude a very central theory for all the models, which causes a problem to my phd topic. My topic is regularity theory for a general setting of physical models. As one knows, the general trick is to consider the Nirenberg-difference quotient in such regularity theory, supposing the difference quotient is still in the space of test function. But now, if we give some first order differential conditions to the test functions, for instance $\nabla\cdot\phi=0$ or $\nabla\times\phi=0$ for a test function $\phi$, then one needs not to have that the difference quotient is still divergence free. My task is to find a trick to solve these problems. But as more as I learned, I have more and more confusion:
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I can not find any models which use restricted test functions. The restrictions here are mostly given by divergence or curl free conditions. But these conditions are usually to solutions but not to test functions. Examples are the Maxwell's equations and piezoelectric model.
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My advisor asked me to give a generalization which concludes a class of such problems, but as you know, PDE questions are usually formulated very differently. Indeed, I can give some sort of generalization to the conditions, but it seems that these generalization does not make any sense in physics. I can not see that a generalization makes a contribution to physical models, since in general only the gradient, divergence or curl appear in the most models and have a clear physical meaning.
I will have a talk to my advisor in next week, eventually I will ask her to give me some real world models that relate to my topic, which seems not actually possible since I have asked her once earlier and got no answer… If this is the situation, I would ask her to give me a new topic which actually comes from physics and for which one can actually work on any unknown issues.
But before I meet her, I have a question to my confusion for which I might find a solution here: Does anyone know some real world models which indeed use restricted test functions like first order linear differential restrictions as described above?
Any kind of help is welcome!
EDIT: Thanks for the comments! Ok, let me explain more what I have researched. Let us consider for instance the following equations: Find a solution $u\in M$
\begin{equation}
\int_\Omega u:v=\int_\Omega f:v
\end{equation}
for all $v\in M$ and a given $f\in L_2$, where $M$ is defined by $M:=\{v\in L_2:\nabla\cdot v=0\}$, and the divergence is understood as distribution. The existence and uniqueness of solution is guaranteed by Lax-Milgram. Now the question is, can $u$ be as regular as $f$ if $f$ is for instance in $H^1_{loc}$? Since the test functions do not include all smooth functions with compact support, we can not say that $u=f$ almost everywhere. But in fact, one can still prove $u$ is in $H^1_{loc}$ if $f$ is in $H^1_{loc}$. The suggestion from my adviser was to use Piola transformation, but I found another approach which is easier to handle this problem and can be used for distribution like $\nabla\times v=0$ or $\nabla v=0$. Using my approach one can even derive a criterium for the general distributional restrction
\begin{equation}
(\sum_{ij}a_{ijk}\partial_i v_j)_k=0,
\end{equation}
where $a_{ijk}$ are constants and fullfill certain properties. Eventually one can do the same trick to an elliptic equation or a Maxwell's system (but I haven't seen such models, namely with restricted test functions, in any references.) Now to be honest, I dont have any physical models related to this. But in general, this could relate to constrained optimal PDE problems (they have a similar formulation, but still, I can not find any concrete models so far). My advisor didn't tell me anything about models, she just said one may find some… At that time I wasn't really familiar with the models, so I accepted this topic, but for this moment, after having some basic modeling knowledge, I still can't find any appropriate models.
Best Answer
Well, I'm not sure if I got your question correctly. If I understood apropriately, you are asking for real life models where the test functions have restrictions as being divergence/curl free. Fluid dynamics is one of the fields where this restriction appear, not only on the solution, but also on the test function. Check the book "Navier-Stokes Equations and Turbulence" by Foias, Manly, Rosa and Temam for instance. Let me also copy the link to the appropriate page (the page is in google books)
https://books.google.com/books?id=f4zaccq9KREC&pg=PA39&lpg=PA39&dq=divergence+free+test+function&source=bl&ots=ZboqaQ8IOo&sig=wzvufxa7QsBZHK14JUUuNnaa8HE&hl=es&sa=X&ved=0CDYQ6AEwA2oVChMI5-STvsXuyAIVQcFjCh03yghh#v=onepage&q=divergence%20free%20test%20function&f=false