Is there a term for negligent spelling of words such as 'you' as 'u'?

This is known as txtspk, from the shortened forms of text and speak.

A more scientific name is SMS language.

Wikipedia suggests the name "SMS Language", but also offers "textese", "texting language", and "chatspeak", among others.

I (and others) have been known to refer to it as Princespeak, since The Artist Formerly Known As The Artist Formerly Known As Prince was a pioneer in the area, inflicting it on the mass consciousness as early as 1985 with the initial release of "Nothing Compares 2 U", though Sinead O'Connor's 1990 cover was when it really got out there.

As Guffa and Kosmonaut point out wikipedia has entry for SMS language and lists following terms:

Textese (also known as txtese, chatspeak, txt, txtspk, txtk, txto, txtk, texting language, txt lingo, or txt talk

Urban dictionary has interesting entries for text speak, textese, etc..

Let me present another perspective, which is not necessarily better, but might shed another light on the whole subject.

I would say that this practice is an example of shorthand. Though the term is usually used for

a quick way of writing that uses symbols to represent letters, words, or phrases, used especially when you write what someone is saying as they are talking

where symbols are usually meant to be graphemes different from regular alphabet, I find the definition very relevant for 'SMS language'.

Especially "you write what someone is saying as they are talking", which I think is the reason why such abbreviations are attractive

• any increase in speed will make the chat (instant messaging) more interactive and more similar to conversation
• in text messaging use of such abbreviations becomes more frequent due to inferior input methods on most mobile phones
• possibly there is a desire to be very informal in attempt for the communication to resemble conversation (justification for this idea is: "u" and "you" are indistinguishable in speaking, by using "u" in written communication it can be said that one actually pretends it is indistinguishable from "you" i.e. pretends that one is in verbal conversation.)
• then there are all emoticons and related words used to convey nonverbal signs and signals, which introduce certain degree of flexibility towards attempts to be 'creative' with spelling and grammar

Certainly you can also find influences such as need to be as short as possible (text messages, tweets) or attempts to be recognized as part of some group by using ingroup memes (leet, lolspeak) as well as you will find bad spellings due to international character of the internet and varying level of proficiency in English. However, I think that the desire for the communication to resemble conversation is often neglected when looking at the reasons for shortening of words.

All in all I would consider using the term shorthand, since shorthand can also be:

• aesthetically substandard
• cryptic
• can use different methods to shorten same message
• is appropriate to write spoken language

The difference is that shorthand has a definitive use in capturing spoken words, where in written communication shorthand represents a problem if both parties are not familiar with the codes used and does not improve it.

If I was to coin a new term maybe something like stenotyping might give some idea about nature of the process (even though it mixes Gk. stenos, narrow and English sense of the word typing in construction).