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So right now, as of 3rd January 2019, the US government is shut down, due to disagreements between Trump and the Democrats on border security funding.

Both parties are in an apparent stalemate. Trump can't back down because if he doesn't get his wall, it'll be a massive failure on his part. Democrats can't back down because they've managed to make their opposition to a border wall a symbol of their so-called "humane" policies, and so allowing funding for that will be an own goal of epic proportions.

So my question is simply, what happens if neither party backs down? Does the shut-down just go on indefinitely? How does this end?

If necessary, Congress can pass a spending bill without the president's support.

Currently Trump claims he will veto any bill which doesn't include funding for the wall. However if two-thirds of each chamber of Congress agree, they can override a presidential veto and end the shutdown.

The current Senate has already passed a spending bill without funding for Trump's wall once with a veto-proof majority of 100-0. Instead of voting on that bill though, the Republican-lead House of Representatives voted on a different bill with wall funding that was sure to fail in the Senate.

The Senate won't change much in the next Congress, but the House of Representatives will have new leadership and can try to pass a similar bill to what has previously passed the Senate. If the Senate votes similarly and the House gets enough votes they can end the shutdown with or without Trump.

The government shutdown is caused by a lapse in funding for government operations and agencies.

The Antideficiency Act mandates that the government cannot incur "obligations or the making of expenditures (outlays) in excess of amounts available in appropriations or funds". As explained by The Hill, this act "provides the framework for which government functions temporarily cease and how employees are impacted".

Thus, the only way to end a government shutdown is to pass appropriations legislation. This can be in the form of a regular appropriations bill (a yearly budget) or a continuing resolution (CR; a short-term funding bill). Without any appropriations legislation passed, the shutdown will continue indefinitely.

Currently, both parties are looking to pass a continuing resolution. However, since there was no agreement on the amount of funding to provide for border security, the Senate passed a CR which does not include funding for border security while the House passed a CR which includes $5.7 billion for border security. Since there was no agreement between the Senate and the House, the government shutdown continues to date.