Good question.
The usual answer is transaction fees but I don't think it'll be that simple.
One thing I consider certain. Fees high enough to satisfy the current amount of miners will be unacceptable for the users.

Fortunately, 2140 is inconceivably far away and I don't worry about it one bit.

They will have had four years of mining .00000001 BTC block rewards before the reward becomes zero; the reward amount halves every year four years so there will not be a drastic and dire change at a particular date.

Heh, even if a bitcoin becomes 1M$ by then I doubt people will get used to mining 1 cent per block.
Anyways it's nothing to worry about. There's enough time to think of a proper solution.

*sigh*
Ok. I'll take it slower for you.

The current block reward is 25k$ @ 1k$/BTC. Currently, there are 100-600 transactions in a block.
That means, if we were to sustain the current network with transaction fees alone, each user would have to pay a transaction fee of 25k$/0.6k = 41.67$ per transaction. Would you pay that?
10$ sounds to me like a barely acceptable transaction fee. At 10$ per transaction, 10$*600 = 6k$ block reward. Would you mine for that at the current difficulty? Not everybody has free electricity.