A verb phrase that describes someone that doesn't pay their debt

ie. Knowing that they have an unpaid debt, yet doesn't care.

I suppose "he is renouncing his debt" would be fine, but it doesn't seem to flow well.

"he is abandoning his debt" doesn't sound too good either, so are there any better alternatives?

In BrE, deliberately running away from a debt is "welshing". You welsh on a debt.

A fairly proper and all-encompassing verb would be "renege":

He is reneging on his debts.

It does somewhat depends on why he is not paying his debts. Does he no longer believe debt is a valid measure in society? Is he now jobless and unable to pay debts he would like to otherwise pay? Different verbs will convey those different intents, but "reneging" should satisfy any of them.

Deadbeat -- a very common american usage.

He's being delinquent on his debts/in paying his debts

delinquent

: failing to pay an amount of money that is owed

M-W

: a status that suggests to a lender that you are late in paying your debts. Being delinquent on your debts often means being more than 30 days late twice or 60 days late once. Being delinquent often means your account is handed over to a collection agency for collecting payment. This step not only results in you being called at home for payment, but also undermines your credit history.

Provident Credit Union

: North American In arrears: delinquent accounts

: formal Failing in one’s duty.

Oxford Dictionaries

... you will learn that once a person in our society has developed a history of being delinquent on his debts he will, most likely, continue those payment habits for the rest of his adult life.

Cash Or Credit?: A Nuts and Bolts Guide to Effective Credit Management

Also, He's being derelict in paying his debts

derelict

: neglectful of duty; delinquent; negligent.

Random House

: chiefly North American (Of a person) shamefully negligent in not having done what one should have done: he was derelict in his duty to his country

Oxford Dictionaries