The spelling for the adjective derived from the name Shaw is Shavian and not Shawian. Similarly you can find Arrow → Arrovian and Harrow → Harrovian. This strikes me as odd.
First of all, I accept that the adjective for /ʃɔː/ is pronounced /ʃɔːvɪən/, so this is not about the word, just the spelling. Now, I see two possible rationales:
Shavian is phonemically closer since v usually represents /v/ in English orthography and w represents /w/.
Shawian is etymologically closer, without being too distant phonemically since /v/ and /w/ are pretty close.
What strikes me about this is that English orthography usually champions etymology over phonemics, in particular when it comes to proper names. Thus, my question is: Has phonemics just won for once, or is there any other argument for spelling these adjectives with a v?
It is said that Shaw disliked the adjective “Shawian” (which does look and sound awkward) and proposed to invent a new one.
He Latinized (sort of) his surname, from Shaw to Shavius (there is no “w” in Latin, while “u” is spelled as “v”).
Shavius naturally lends itself to Shavian.
At least that is the explanation given in the Dictionary of Eponyms
by Martin Manser, which is available online in fragments (and also in its complete form for a fee).