Funnily enough (given that it's quite a distinct expression that has a meaning of its own) I don't think there's a commonly accepted word in English to describe it.
I would go with @Eran's 'Sturgeon Face'.
(To clarify, a sturgeon is a fish that looks like it has that facial expression.)
Even though the definition that @Eran
quoted calls it an 'excessive frown', a frown is generally thought to involve the eyebrows more than the mouth.
This just demonstrates the paucity of vocabulary in English when it comes to talking about such expressions.
For the second photo, I would call that a furrowed brow .
That doesn't necessarily imply that it's compassionate, but it does imply worry.
You could say a brow furrowed with compassion .
ell.stackexchange.com/users/59888/Igid
The first is what I would call a grimace to make an expression of pain, strong dislike , etc.
in which the face twists in an ugly way: Either the verb or noun, scowl , would be appropriate too to look at someone or something with a very annoyed expression
The second picture showing Bernie Sanders is simply a frown , or you could call the furrows in his forehead, worry lines to bring your eyebrows together so that there are lines on your face above your eyes to show that you are annoyed or worried
ell.stackexchange.com/users/1694/Mari-Lou A
Although this isn't a word for it per se , most books I read express this emotion by saying "X looked impressed".
Although the "looked impressed" may be vague, this sort of expression is usually associated with looking impressed.
Here are some google image search results when searching for "impressive meme" and "impressed expression"
This expression is very similar to a shrug (minus the hand gesture), but a shrug expresses an entirely opposite reaction, usually indifference.
The second one is best described as a frown , as people have already pointed out.
ell.stackexchange.com/users/60103/ColonD
Google search yielded the term "Sturgeon Face".
Sturgeon Face is a participatory photo meme that involves turning down one's corners of mouth so that it forms an excessive frown (From
Know Your Meme )
ell.stackexchange.com/users/47040/Eran
I would say that first reaction falls under Impressed .
Given the severity of the expression, maybe very impressed .
Impressed - The feeling of amazement which arises when it is difficult for a person to imagine something that he or she has encountered being any better than it is.
ell.stackexchange.com/users/66784/Sid
How to desribe a person reacting with this face?   Alternatively, I'd appreciate if you suggested me a word to describe this kind of compassionate frowning:
What do you call Obama's face expression "not bad"?
ell.stackexchange.com
<word-request><phrase-request><vocabulary><image-identification>
ell.stackexchange.com/questions/150522
ell.stackexchange.com/users/26941/Probably
This expresison might be described as "impressed". Impressed - The feeling of amazement
Impressed - The feeling of amazement which arises when it is  difficult for a person to imagine something that he or she has  encountered being any better than it is.  